Variance-Based Uncertainty Relations: A Concise Review of Inequalities Discovered Since 1927

Abstract

A brief review of various existing mathematical formulations of the uncertainty relations in quantum mechanics, containing variances of two or more non-commuting operators, is given. In particular, inequalities for the products of higher-order moments of a coordinate and a momentum are considered, as well as inequalities making the uncertainty relations more accurate when additional information about a quantum system is available (for example, the correlation coefficient or the degree of mixing of a quantum state characterized by the trace of the squared statistical operator). The special cases of two, three, and four operators are discussed in detail.

1. Introduction

The famous “uncertainty relation” (UR)

$$\Delta x\Delta p\ge \hslash /2$$

(1)

was introduced by Heisenberg [1] in 1927 as an approximate (qualitative) inequality. Later in the same year, one of its quantitative formulations (actually, the simplest possible one) was proven rigorously in the frameworks of the wave function description of quantum systems by Kennard [2]. Since then, it is frequently considered as one of the cornerstones of quantum mechanics. (For example, Feynman told in his lectures [3]: “The uncertainty principle “protects” quantum mechanics. Heisenberg recognized that if it were possible to measure the momentum and the position simultaneously with greater accuracy, the quantum mechanics would collapse.”).

The physical and philosophical meanings of the uncertainty relations (or the uncertainty principle) were discussed intensively by Bohr and Heisenberg [4,5,6,7], as well as by numerous other outstanding authors [8,9,10,11,12]. The initial history is described in detail in the book [13]. Since this subject is discussed (at different levels of rigor and completeness) in every textbook on quantum mechanics, one could imagine that it was closed many decades ago. Nonetheless, it is remarkable (and, perhaps, surprising for somebody) that new papers, devoted to generalizations of the UR and their consequences, still appear in the physical and mathematical literature. Moreover, some “bursts” of publications on this subject are observed during the past few years, related mainly to problems of quantum information theory.

By now, it has become especially clear that Heisenberg and Kennard considered, as a matter of fact, two different types of uncertainty relations. Using the modern terminology, Kennard considered the preparation uncertainties, i.e., possible inequalities for some quantities characterizing admissible quantum states (let us call them PURs; see, e.g., [14,15,16,17,18,19,20,21,22]). On the other hand, Heisenberg (as well as Bohr, Einstein, and many other founders of the quantum theory) considered the measurement uncertainties. For several decades, only vague, approximate (qualitative) relations were considered in this connection, while rigorous quantitative relations (let us call them MURs) were found much later (see, e.g., Refs. [23,24,25,26,27] for reviews).

The initial aim of this review was to present all known forms of UR, generalizing inequality (1). A previous attempt to collect available results was made almost 40 years ago in review [28]. However, a lot of new relations have been obtained since that time. Moreover, new possibilities for the literature search have appeared due to the power of the internet. The main difficulty in preparing a comprehensive review consists in the great number of generalizations or modifications introduced by many authors over recent decades. Therefore, here I consider PURs only. Moreover, even in this class of URs, I concentrate on the relations containing variances of quantum operators and their simplest extensions or modifications as measures of “uncertainties”. Other existing families of URs are only mentioned in the last section of this short review. The results are presented at the “physical level” of rigor. Mathematically inclined readers can find rigorous definitions and theorems, e.g., in books [29,30,31,32,33] or review articles [34,35].

The paper is divided into the following sections and subsections.

Section 2. Preparation uncertainty relations based on variances: the beginning of the story.

Section 3. Robertson’s inequalities for N arbitrary operators and their generalizations. Section 3.1: $2n$ coordinate and momentum operators and some generalizations. Section 3.1.1—Inequalities for the partial and full traces of covariance matrices. Section 3.1.2—Inequalities for sums of uncertainties. Section 3.1.3—Symplectic invariants and resulting uncertainty relations. Section 3.2: Inequalities without covariances.

Section 4. Specific inequalities for two operators. Section 4.1: Three canonical observables with a linear dependence.

Section 5. Inequalities for three independent operators: a general case. Section 5.1: Two observables coupled with a third one. Section 5.2: Inequalities without covariances for three operators.

Section 6. Concrete families of three operators. Section 6.1: Angular momentum operators. Section 6.2: A charged particle in a magnetic field. Section 6.3: “Degrees of uncertainty” of the angular momentum. Section 6.4: Spin-1/2. Section 6.5: Angular momentum, sine and cosine operators. Section 6.6: Bi-products of the coordinate and momentum in one dimension. Section 6.7: Fourth-order moments in the Gaussian states.

Section 7. Four and more operators. Section 7.1: Four quadratures. Section 7.2: Inequalities without covariances for four operators. Section 7.3: Five operators. Section 7.4: Compact sum and product inequalities for N operators.

Section 8. Inequalities based on the coordinate probability density alone. Section 8.1: Connections with the “Fisher information” and generalized mean values.

Section 9. State-extended and state-independent UR. Section 9.1: Trifonov’s inequalities. Section 9.2: Maccone–Pati inequalities and their generalizations. Section 9.3: “Weighted-like”, “tighten”, “reverse” and “improved” uncertainty relations.

Section 10. Bargmann–Faris inequalities and their generalizations.

Section 11. Uncertainty relations for mixed and non-Gaussian states. Section 11.1: Inequalities containing the “second-order purity”. Section 11.2: Inequalities for modified “uncertainties”. Section 11.3: Generalizations to several dimensions. Section 11.4: Inequalities containing the “skew information”. Section 11.5: Simple examples. Section 11.6: Purity- and Gaussianity-constrained uncertainty relations.

Section 12. Inequalities for higher moments.

Section 13. Concluding remarks about other families of “uncertainty relations”.

2. Preparation Uncertainty Relations Based on Variances: The Beginning of the Story

Following Heisenberg [1], the “uncertainty” of a quantity A is frequently defined as the square root of its variance (or mean squared deviation), i.e., $\Delta A\equiv \sqrt{{\sigma }_A}$, ${\sigma }_A\equiv \langle {\widehat{A}}^2\rangle -{\langle \widehat{A}\rangle }^2$. Here, $\widehat{A}$ is a Hermitian operator corresponding to the observable A, and the angular brackets mean averaging over the state of the quantum system. For a pure state, described by means of the wave function $\psi$ with the normalization $\int {\psi }^{*}\psi dV=1$, the standard definition is $\langle \widehat{A}\rangle =\int {\psi }^{*}\widehat{A}\psi dV$. For a mixed state, described by means of the Hermitian positive definite statistical operator (density matrix) $\widehat{\rho }$ with the normalization $\mathrm{T}\mathrm{r}\left(\widehat{\rho }\right)=1$, we have the definition

$$\langle \widehat{A}\rangle =\mathrm{T}\mathrm{r}\left(\widehat{\rho }\widehat{A}\right).$$

(2)

Note that Heisenberg considered only Gaussian wave packets, for which relation (1) is an equality, whereas in more general situations, he confined himself to the analysis of “thought experiments,” like “Heisenberg’s microscope” and some others (considered in a more detailed form by Bohr [4,6,7]). For arbitrary pure states, the inequality

$${\sigma }_p{\sigma }_x\ge {\hslash }^2/4$$

(3)

was obtained for the first time apparently by Kennard [2] and Weyl [36]. I shall use the name Heisenberg–Kennard–Weyl (HKW) inequality for inequality (3). (It has been pointed out in [13] that the proof of inequality (3) was suggested at the same time also by Pauli. In particular, one can read on page 77 of Weyl’s book, after the strict inequality (1), the following words: “I am indebted to W. Pauli for this remark” (concerning the proof). Probably, Pauli did not publish his proof in any paper during that period, although inequality (1) was sometimes called “the Pauli inequality,” according to [13]. In some papers, inequality (1) is called “the Heisenberg–Pauli–Weyl inequality”).

A generalization of relation (3) to the case of “classical” observables A and B, i.e., some functions of the canonically conjugate coordinates and momenta of the form $f_i\left(\mathbf{x}\right)p_i$ (provided $\mathrm{d}\mathrm{i}\mathrm{v}\mathbf{f}\left(\mathbf{x}\right)=0$) was made by Robertson [37] (for pure quantum states) (Note that Robertson translated Weyl’s book [36], when he was acting as assistant to Weyl in 1928–1929):

$${\sigma }_A{\sigma }_B\ge {\displaystyle \frac{1}{4}}{\left|\langle [\widehat{A},\widehat{B}]\rangle \right|}^2.$$

(4)

In particular, he proved the known inequality for the angular momentum projection operators ${\widehat{L}}_x$ and ${\widehat{L}}_y$ [see Equation (12) below]. A few months later, Robertson noticed in [38] that (4) holds for arbitrary Hermitian operators (in particular, for the spin operators). (Actually, ref. [38] is the 8-line abstract of the presentation made during the meeting of the American Physical Society on 21–22 February 1930).

However, simple examples show that in many cases the left-hand side of (4) turns out bigger than the right-hand one. This means that, probably, one should add some extra terms to the right-hand side, taking into account some additional parameters or specific properties of concrete quantum systems under consideration. The first step in this direction was made by Schrödinger [39], (The English translation of the original Schrödinger’s paper can be found, e.g., in [40]), who obtained a more precise version of (4), taking into account the anticommutator of operators $\widehat{A}$ and $\widehat{B}$:

$${\sigma }_A{\sigma }_B\ge {\sigma }_{AB}^2+{\displaystyle \frac{1}{4}}{\left|〈[\widehat{A},\widehat{B}]〉\right|}^2\equiv G_{AB}^2,$$

(5)

$${\sigma }_{AB}\equiv {\displaystyle \frac{1}{2}}\langle \widehat{A}\widehat{B}+\widehat{B}\widehat{A}\rangle -\langle \widehat{A}\rangle \langle \widehat{B}\rangle \equiv {\displaystyle \frac{1}{2}}〈\left\{\delta \widehat{A},\delta \widehat{B}\right\}〉,\delta \widehat{A}\equiv \widehat{A}-\langle \widehat{A}\rangle .$$

(6)

The same inequality (5) was given by Robertson [38], but he wrote the right-hand side in a different form, which is equivalent to $G_{AB}^2\equiv {\left|〈\left(\delta \widehat{A}\right)\left(\delta \widehat{B}\right)〉\right|}^2$. It is worth mentioning that inequalities (4) and (5) are valid under the important restriction, that vector $|\psi \rangle$ belongs to the domains of operators $\widehat{A}\widehat{B}$ and $\widehat{B}\widehat{A}$, in addition to the domains of definition of operators $\widehat{A}$ and $\widehat{B}$. Otherwise, inequality (5) must be replaced by the modified relation [41]

$${\sigma }_A{\sigma }_B\ge {\sigma }_{AB}^2+{\left(Im〈\widehat{A}\psi ,\widehat{B}\psi 〉\right)}^2.$$

(7)

Applying (5) to the coordinate and momentum operators, one arrives at the inequality (Note, however, that inequality (8) was not written explicitly in [38,39]. It seems that (8) remained not well known for a long time, since it was frequently rediscovered without citing [38,39]: see, e.g., Refs. [42,43,44])

$${\sigma }_p{\sigma }_x-{\sigma }_{xp}^2\ge {\hslash }^2/4,$$

(8)

Inequality (8) was written [45]

$${\sigma }_p{\sigma }_x\ge {\displaystyle \frac{{\hslash }^2}{4\left(1-r^2\right)}},r={\displaystyle \frac{{\sigma }_{xp}}{\sqrt{{\sigma }_p{\sigma }_x}}},$$

(9)

emphasizing the role of the “correlation coefficient” r as an additional parameter, responsible for the increase of product ${\sigma }_p{\sigma }_x$. One could treat the relation (9) as though an “effective Planck constant” $\hslash {\left(1-r^2\right)}^{-1/2}$ is occurring instead of the usual constant ℏ.

The derivation of inequality (8) without using commutators was given in paper [46]. Modifications of the uncertainty relation (8) in the case of “discrete quantum mechanics”, based on finite differences, were studied in paper [47].

To simplify formulas for operators labeled with indexes, we shall often use the notation

$$z_j\equiv \langle {\widehat{z}}_j\rangle ,z_{jk}=z_{kj}={\displaystyle \frac{1}{2}}\langle {\widehat{z}}_j{\widehat{z}}_k+{\widehat{z}}_k{\widehat{z}}_j\rangle -\langle {\widehat{z}}_j\rangle \langle {\widehat{z}}_k\rangle .$$

(10)

We shall also use the notation

$$\overline{A}\equiv \langle \widehat{A}\rangle ,\overline{BC}\equiv {\displaystyle \frac{1}{2}}\langle \widehat{B}\widehat{C}+\widehat{C}\widehat{B}\rangle -\overline{B}·\overline{C}.$$

(11)

A disadvantage of inequality (4) is that it often becomes trivial for operators different from the coordinate and momentum ones, due to the very simple reason: for many states and operators, the right-hand side of (4) equals zero, while the left-hand side is obviously positive. For example, this is the case for the angular momentum operators, when relation (4) assumes the form

$$L_{xx}{L}_{yy}\ge ({\hslash }^2/4)L_z^2.$$

(12)

If the average value of the operator ${\widehat{L}}_z$ equals zero (and this can happen for many quantum states), then relation (12) gives no information about the variances $L_{xx}$ and $L_{yy}$. (Nonetheless, inequality (12) is important, because it tells us that for any eigenstate of operator ${\widehat{L}}_j$ (with $j=x,y,z$), at least one average value $\langle {\widehat{L}}_k\rangle$ with $k\ne j$ must be zero.).

An insufficient efficiency of relation (12) may be explained, partially, by the fact that three equivalent noncommuting operators ${\widehat{L}}_x$, ${\widehat{L}}_y$, and ${\widehat{L}}_z$ enter this relation on an unequal footing. Therefore, there exists a need to generalize inequalities (4) or (5) to systems of many (more than two) operators. This problem was considered for the first time by Robertson [48]. His results and their generalizations [28,49] are given in the next section.

3. Robertson’s Inequalities for N Arbitrary Operators and Their Generalizations

Let us consider N arbitrary (not necessarily Hermitian) operators ${\widehat{z}}_1$, ${\widehat{z}}_2$, …, ${\widehat{z}}_N$. Following Robertson, we construct the operator $\widehat{f}={\sum }_{j=1}^N{\alpha }_j({\widehat{z}}_j-\langle {\widehat{z}}_j\rangle )$, where ${\alpha }_j$ are arbitrary complex numbers. All the following results are based on the fundamental fact that the inequality $\langle {\widehat{f}}^{†}\widehat{f}\rangle \ge 0$ must be satisfied for any pure or mixed quantum state (the symbol ${\widehat{f}}^{†}$ means the Hermitian conjugated operator). In the explicit form, this inequality is the condition of positive semi-definiteness of the quadratic form ${\alpha }_j^{*}{F}_{jm}{\alpha }_m$, whose coefficients $F_{jm}=〈\left({\widehat{z}}_j^{†}-{\langle {\widehat{z}}_j\rangle }^{*}\right)\left({\widehat{z}}_m-\langle {\widehat{z}}_m\rangle \right)〉$ form the Hermitian matrix $F = \parallel F_{jm}\parallel$. One has only to use the known conditions of the positive semi-definiteness of Hermitian quadratic forms (see, for example, [50,51,52,53]) to write the explicit inequalities for the elements of matrix F. All such inequalities can be considered as generalizations of inequality (4) to the case of more than two operators.

If all operators ${\widehat{z}}_j$ are Hermitian, then it is convenient to split matrix F as $F=X+iY$, where X and Y are real symmetric and antisymmetric matrices, respectively, consisting of the elements

$$X_{mn}={\displaystyle \frac{1}{2}}〈\left\{\left({\widehat{z}}_m-\langle {\widehat{z}}_m\rangle \right),\left({\widehat{z}}_n-\langle {\widehat{z}}_n\rangle \right)\right\}〉,Y_{mn}={\displaystyle \frac{1}{2i}}〈\left[{\widehat{z}}_m,{\widehat{z}}_n\right]〉.$$

(13)

The symbols $\{,\}$ and $[,]$ mean, as usual, the anticommutator and the commutator.

The fundamental inequality ensuring the positive semi-definiteness of matrix F is

$$detF=det\parallel X+iY\parallel \ge 0.$$

(14)

Another consequence of the positive semi-definiteness of matrix F is Hadamard’s inequality [50,51,52,53,54,55]

$$X_{11}{X}_{22}\dots X_{NN}\ge detF.$$

(15)

The following inequality is also valid [51,53]:

$$detX\ge detF.$$

(16)

However, it is easy to see that inequalities (15) and (16) are trivial in the case of $N=2$.

The following inequalities were derived by Robertson in [48]:

$$X_{11}{X}_{22}\dots X_{NN}\ge detX\ge detY.$$

(17)

It is worth noting that $detY\ge 0$ for any antisymmetric matrix Y. Inequality (17) becomes useless if N is an odd number, since $detY=0$ in this case. This drawback can be fixed by redefining the meaning of mean values and covariances: see Section 8.1.

Generalizations of (15) and (17) are Mirsky’s inequalities [53,55]

$$detF\le detF(1,2,\dots ,p)detF(p+1,\dots ,N),$$

(18)

$${\displaystyle \frac{detF(1,\dots ,N)}{detF(2,\dots ,N)}}\le {\displaystyle \frac{detF(1,3,\dots ,N)}{detF(3,\dots ,N)}}\le \cdots \le {\displaystyle \frac{detF(1,N)}{detF\left(N\right)}}\le detF\left(1\right).$$

(19)

Here, the symbol $F(p,q,\dots ,m)$ means the matrix consisting of those elements of the matrix F that stand at the intersections of the rows labeled by the numbers $p,q,\dots ,m$ and the columns with the same numbers. It is clear that the order of the rows or columns in (19) can be arbitrary. Actually, inequalities (18) can also be considered as some kinds of generalizations of the fundamental Hadamard’s inequality: see, e.g., [50] (Chap. 9, Sec. 5).

Explicit forms of the inequalities given above can be found, e.g., in the paper by Wünsche [56]. Geometrical interpretations of some inequalities for N continuous variables were considered in [18]. Some other inequalities for N observables (rather involved) can be found in [57].

Several families of inequalities can be obtained as consequences of the condition that all principal minors of the positive semi-definite matrix F must be non-negative. However, many such relations lack a symmetry between all N operators: see inequality (174) as an example for $N=4$. One of the possibilities to restore the symmetry was studied by Trifonov and Donev [58]. They considered characteristic coefficients $C_r^{\left(N\right)}\left(X\right)$ and $C_r^{\left(N\right)}\left(Y\right)$ of matrices X and Y, defined in a standard way by the series expansions [50]

$$det\parallel X-\lambda I\parallel =\sum _{r=0}^N{C}_r^{\left(N\right)}\left(X\right){(-\lambda )}^{N-r},$$

(20)

$$det\parallel Y-\lambda I\parallel =\sum _{r=0}^N{C}_r^{\left(N\right)}\left(Y\right){(-\lambda )}^{N-r},$$

where I is the $N\times N$ unit matrix. Their first observation was the equality

$$C_r^{\left(N\right)}\left(X\right)=\sum _{1\le i_1<i_2<\dots <i_r\le N}detX\left(i_1,i_2,\dots ,i_r\right),$$

where the structure of $r\times r$ matrix $X\left(i_1,i_2,\dots ,i_r\right)$ was described above. The second observation was the inequality $detX\left(i_1,i_2,\dots ,i_r\right)\ge detY\left(i_1,i_2,\dots ,i_r\right)$, which is the Robertson inequality (17) for the subset of operators ${\widehat{Z}}_{i_1},{\widehat{Z}}_{i_2},\dots ,{\widehat{Z}}_{i_r}$. The consequence is the set of inequalities

$$C_r^{\left(N\right)}\left(X\right)\ge C_r^{\left(N\right)}\left(Y\right),$$

(21)

called “characteristic uncertainty relations” in [58]. Note that the right-hand sides of (21) are non-negative for arbitrary sets of operators.

3.1. 2n Coordinate and Momentum Operators and Some Generalizations

For a system of n coordinate operators and n momentum operators, matrix F can be represented in the form $F=Q-i\hslash \mathrm{\Sigma }/2$, where the $2n\times 2n$ symmetric matrix Q consists of all possible second-order centered moments of the variables $p_1,p_2,\dots ,p_n$, and $x_1,x_2,\dots ,x_n$, while the only nonzero elements of the antisymmetric matrix $\mathrm{\Sigma }$ are ${\mathrm{\Sigma }}_{m,n+m}=-{\mathrm{\Sigma }}_{n+m,m}=1$, $m=1,2,\dots ,n$. The positive semi-definiteness of matrix F, both in this case and in the case of arbitrary Hermitian operators, was first established by Robertson [48]. The condition $F\ge 0$ was also used in later studies [59,60,61] for arbitrary (Hermitian) operators and in [30,62] for the coordinate and momentum operators. The simplest inequality in the case under study is that resulting from relation (17):

$$\left(\overline{x_1^2}\right)\left(\overline{x_2^2}\right)\cdots \left(\overline{x_n^2}\right)\left(\overline{p_1^2}\right)\left(\overline{p_1^2}\right)\cdots \left(\overline{p_n^2}\right)\ge det\left(Q\right)\ge {\left({\hslash }^2/4\right)}^n,$$

(22)

since $detY={(\hslash /2)}^{2n}det\mathrm{\Sigma }={({\hslash }^2/4)}^n$.

It is natural to represent matrix Q in the block form:

$$Q=∥\begin{array}{cc}{Q}_p& Q_{px}\\ Q_{xp}& Q_x\end{array}∥,Q_p={\tilde{Q}}_p,Q_x={\tilde{Q}}_x,Q_{px}={\tilde{Q}}_{xp}$$

(23)

(the tilde represents a transposed matrix). Note that the consequence of the condition $F\ge 0$ is the condition $F^{\prime }\ge 0$, where matrix $F^{\prime }$ is obtained from F by means of deleting the blocks $Q_{px}$ and $Q_{xp}$ (or similar blocks in the case of arbitrary $2n$ Hermitian operators). To prove this, one should write the condition of the positive semi-definiteness of the form $w_j^{*}{F}_{jm}{w}_m$ for $2n$-dimensional vectors $\mathbf{w}$ of the form $\mathbf{w}=(\mathbf{u},i\mathbf{v})$, where $\mathbf{u}$ and $\mathbf{v}$ are arbitrary real n-dimensional vectors. Therefore, we obtain the condition (see also [60,62])

$$det\left(Q_p\right)det\left(Q_x\right)\ge {\left({\hslash }^2/4\right)}^n.$$

(24)

The positive semi-definiteness of matrix F results in the positive semi-definiteness of all its submatrices of the form $F(p,q,\dots ,m)$ [see the notation after Equation (19)]. For the proof, it is sufficient to consider the form ${\mathbf{w}}^{*}F\mathbf{w}$ in the subspace of the vectors $\mathbf{w}$ with the only nonzero elements labeled with the indexes $p,q,\dots ,m$. In particular, $Q_p\ge 0$ and $Q_x\ge 0$, although these relations are trivial.

One more set of relations can be obtained if one notices that for any positive definite non-singular matrix F, the inverse matrix $F^{-1}$ is positive definite, as well (it is sufficient to consider vectors $\mathbf{w}$ of the form $\mathbf{w}=F^{-1}\mathbf{y}$ to prove this statement). Then, using the Frobenius formula for the inverse block matrix (see, for example, [50]),

$${∥\begin{array}{cc}A& B\\ C& D\end{array}∥}^{-1}=∥\begin{array}{cc}{A}^{-1}+A^{-1}B{H}^{-1}C{A}^{-1}& -A^{-1}B{H}^{-1}\\ -H^{-1}C{A}^{-1}& H^{-1}\end{array}∥,$$

(25)

$$H=D-C{A}^{-1}B,$$

one can conclude that submatrix H must also be positive definite. Therefore, the following matrix must be positive definite in the case of the coordinate and momentum operators:

$$Q_x-\left(Q_{xp}-ihI/2\right)Q_p^{-1}\left(Q_{px}+ihI/2\right)>0.$$

(26)

Making similar manipulations with matrix $F^{\prime }$, one can obtain the condition (see also [60,62])

$$Q_x-{\displaystyle \frac{{\hslash }^2}{4}}{Q}_p^{-1}>0.$$

(27)

We assumed above that $F>0$, i.e., that matrix F is nonsingular. It is clear, however, that this restriction can be relaxed, so that the sign of the strict inequality in relations (26) and (27) can be replaced with the sign of the unstrict inequality $\ge 0$, although the proof would be more complicated (for example, some limiting procedure could be used). We do not dwell upon this subject. More rigorous discussions, including such mathematical subtleties as domains of definition of operators, self-adjointness, and so on, can be found, for example, in [29,30,31,32,34,35,60,63,64]. We simply suppose hereafter that all quantities appearing in formulas do exist. Several relations generalizing those discussed in this subsection (or illustrating them) can be found, e.g., in Ref. [65].

If one applies inequalities (19) to the matrix on the left-hand side of relation (27), the following inequality can be obtained [60]:

$$\left(\overline{x_j^2}\right)\left(\overline{p_j^2}\right)\ge {\displaystyle \frac{{\hslash }^2\left(\overline{x_j^2}\right)det{Q}_x\left(x_1,\dots ,x_{j-1},x_{j+1},\dots ,x_n\right)}{4det\left(Q_x\right)}}.$$

(28)

It is stronger than inequality (3), because the right-hand side of relation (28) is usually greater than ${\hslash }^2/4$: it coincides with ${\hslash }^2/4$ if and only if the variable $x_j$ is uncorrelated with all other coordinates. The significance of inequality (28) is the same as that of inequality (9): if certain additional information concerning the quantum system is available (for example, correlation coefficients between the coordinate and the momentum or between different coordinates), then the product of uncertainties of the coordinate and the momentum can be greater than ${\hslash }^2/4$.

Inequalities (24), (26) and (27) can be generalized in an obvious way for arbitrary systems of Hermitian operators divided into two groups [(1) and (2)], if one splits total $N\times N$ matrices X and Y (forming the matrix $F=X+iY$) into blocks as follows:

$$X=∥\begin{array}{cc}{X}_1& X_{12}\\ X_{21}& X_2\end{array}∥,Y=∥\begin{array}{cc}{Y}_1& Y_{12}\\ Y_{21}& Y_2\end{array}∥,$$

(29)

$$X_1={\tilde{X}}_1,X_2={\tilde{X}}_2,X_{12}={\tilde{X}}_{21},Y_1=-{\tilde{Y}}_1,Y_2=-{\tilde{Y}}_2,Y_{12}=-{\tilde{Y}}_{21}.$$

Then, instead of (24) we obtain the inequality [28]

$$det{X}_{1}det{X}_2\ge detY.$$

(30)

Generalizations of relations (26) and (27) are the conditions of non-negativity of the following matrices:

$$X_2-\left(X_{21}+i{Y}_{21}\right)X_1^{-1}\left(X_{12}+i{Y}_{12}\right)\ge 0,$$

(31)

$$X_2-{\tilde{Y}}_{12}{X}_1^{-1}{Y}_{12}\ge 0.$$

(32)

The dimensions of the blocks $X_1$ and $X_2$ (the same as the dimensions of the $Y_1$ and $Y_2$ submatrices) need not coincide.

3.1.1. Inequalities for the Partial and Full Traces of Covariance Matrices

A weakness of the product inequality (4) is that its left-hand side turns into zero for eigenstates of operators $\widehat{A}$ ot $\widehat{B}$. Therefore, several authors looked for inequalities whose left-hand sides contain, instead of products, sums of variances (or their square roots—“uncertainties”) of the observables. Perhaps, one of the first papers in this direction was published by Turner and Snider [66]. They considered the special case of three space dimensions. Here, we give generalizations of their results for n spatial dimensions. The starting points are the identity

$$\langle \delta {\widehat{x}}_k\delta {\widehat{p}}_l\rangle =(i\hslash /2){\delta }_{kl}+\overline{x_k{p}_l}$$

(33)

and the Schwarz inequality

$$\left|\langle \delta {\widehat{x}}_k\delta {\widehat{p}}_l\rangle \right|\le {\left[\langle {\left(\delta {\widehat{x}}_k\right)}^2\rangle \langle {\left(\delta {\widehat{p}}_l\right)}^2\rangle \right]}^{1/2}.$$

(34)

The notation used here is the same as in (6) and (11). Squaring both sides of (34) and performing the summation over k and l we get the relation

$${\sigma }_{\mathbf{x}}{\sigma }_{\mathbf{p}}\equiv \mathrm{T}\mathrm{r}\left(Q_x\right)\mathrm{T}\mathrm{r}\left(Q_p\right)\ge n{\hslash }^2/4+\mathrm{T}\mathrm{r}\left(Q_{xp}{Q}_{px}\right),$$

(35)

where n is the number of degrees of freedom of the system, i.e., the dimension of vectors $\mathbf{x}$ or $\mathbf{p}$. A slightly different inequality can be obtained if one calculates the sum ${\sum }_{k,l}\langle \delta {\widehat{x}}_k\delta {\widehat{p}}_l\rangle {\langle \delta {\widehat{x}}_l\delta {\widehat{p}}_k\rangle }^{*}$ with account of (33) and (34):

$${\sigma }_{\mathbf{x}}{\sigma }_{\mathbf{p}}\equiv \mathrm{T}\mathrm{r}\left(Q_x\right)\mathrm{T}\mathrm{r}\left(Q_p\right)\ge n{\hslash }^2/4+\mathrm{T}\mathrm{r}\left(Q_{xp}^2\right).$$

(36)

However, both inequalities, (35) and (36) seem to be not strong enough. For example, they only tell us that ${\sigma }_{\mathbf{x}}{\sigma }_{\mathbf{p}}$ must be greater than $n{\hslash }^2/4$ in the absence of correlations, whereas we know that ${\sigma }_{\mathbf{x}}{\sigma }_{\mathbf{p}}=n^2{\hslash }^2/4$ for the ground state of the n-dimensional harmonic isotropic oscillator. A stronger inequality with factor $n^2$ instead of n does exist, indeed. To prove it, one should put $k=l$ in (34) and calculate the sum over k. The following chain of relations arise:

$$\begin{array}{ccc}& & \left|\sum _{k=1}^n\langle \delta {\widehat{x}}_k\delta {\widehat{p}}_k\rangle \right|=\left|n(i\hslash /2)+\mathrm{T}\mathrm{r}\left(Q_{xp}\right)\right|\equiv \sqrt{n^2{\hslash }^2/4+{\left[\mathrm{T}\mathrm{r}\left(Q_{xp}\right)\right]}^2}\hfill \\ & & \le \sum _{k=1}^n\left|\langle \delta {\widehat{x}}_k\delta {\widehat{p}}_k\rangle \right|\le \sum _{k=1}^n\sqrt{\langle {\left(\delta {\widehat{x}}_k\right)}^2\rangle \langle {\left(\delta {\widehat{p}}_k\right)}^2\rangle }\le \sqrt{\sum _{k=1}^n\langle {\left(\delta {\widehat{x}}_k\right)}^2\rangle \sum _{l=1}^n\langle {\left(\delta {\widehat{p}}_l\right)}^2\rangle }.\hfill \end{array}$$

The last inequality in this chain is again a special case of the Schwarz inequality. Thus we arrive at the inequality

$$\mathrm{T}\mathrm{r}\left(Q_x\right)\mathrm{T}\mathrm{r}\left(Q_p\right)\ge n^2{\displaystyle \frac{{\hslash }^2}{4}}+{\left[\mathrm{T}\mathrm{r}\left(Q_{xp}\right)\right]}^2.$$

(37)

Extensions of inequality ${\sigma }_{\mathbf{x}}{\sigma }_{\mathbf{p}}\ge n^2{\hslash }^2/4$ in n spatial dimensions, taking into account the difference between the quantum state and its closest Gaussian partner, were obtained in papers [67,68]. The most clear inequality given in [68] has the following form (using $\hslash =1$):

$${\sigma }_{\mathbf{x}}{\sigma }_{\mathbf{p}}\ge n^2/4+G^2/4+G^4/16,$$

(38)

$$G^2=\mathrm{i}\mathrm{n}\mathrm{f}\left\{\int {\left[\psi \left(\mathbf{x}\right)-cexp\left(-\lambda {\mathbf{x}}^2\right)\right]}^{2}d\mathbf{x};c\in R,\lambda >0\right\}.$$

Generalizations of inequalities (35)–(37) to sets of arbitrary operators can be obtained if the dimensions of all blocks in matrices (29) are the same, i.e., when the number of operators is even. The scheme of deriving these generalizations is the same as above. The resultant inequalities are [28]:

$$\begin{array}{ccc}& & \mathrm{T}\mathrm{r}\left(X_1\right)\mathrm{T}\mathrm{r}\left(X_2\right)-\mathrm{T}\mathrm{r}\left(X_{12}{X}_{21}\right)\ge \mathrm{T}\mathrm{r}\left(Y_{12}{\tilde{Y}}_{12}\right),\hfill \end{array}$$

(39)

$$\begin{array}{ccc}& & \mathrm{T}\mathrm{r}\left(X_1\right)\mathrm{T}\mathrm{r}\left(X_2\right)-\mathrm{T}\mathrm{r}\left(X_{12}^2\right)\ge \mathrm{T}\mathrm{r}\left(Y_{12}^2\right),\hfill \end{array}$$

(40)

$$\begin{array}{ccc}& & \mathrm{T}\mathrm{r}\left(X_1\right)\mathrm{T}\mathrm{r}\left(X_2\right)-{\left[\mathrm{T}\mathrm{r}\left(X_{12}\right)\right]}^2\ge {\left[\mathrm{T}\mathrm{r}\left(Y_{12}\right)\right]}^2.\hfill \end{array}$$

(41)

If two observables A and B have the same physical dimensions (or they are rescaled somehow), then an immediate consequence of the Robertson—Schrödinger inequality (5) is

$${\sigma }_A+{\sigma }_B\ge 2\left|G_{AB}\right|.$$

(42)

Taking the sum of such inequalities with respect to all pairs of observables $z_1,z_2,\dots ,z_N$, one obtains the inequality [69,70]

$$\mathrm{T}\mathrm{r}\left(X\right)\ge {\displaystyle \frac{2}{N-1}}\sum _{1\le j<k\le N}\left|G_{jk}\right|\ge {\displaystyle \frac{2}{N-1}}\sum _{1\le j<k\le N}\left|Y_{jk}\right|,$$

(43)

containing $N(N-1)/2$ terms in the right-hand side. For the even number of operators $N=2m$, the same scheme yields several families of inequalities, containing m terms in the right-hand side [69,70]:

$$\mathrm{T}\mathrm{r}\left(X\right)\ge 2\sum _{j=1}^m\left|G_{j,j+m}\right|\ge 2\sum _{j=1}^m\left|Y_{j,j+m}\right|.$$

(44)

The strongest inequality is obtained if m pairs $(z_j,z_{j+m})$ (with $j=1,2,\dots ,m$) are chosen in such a way that $\left|Y_{1,m+1}\right|\ge \left|Y_{2,m+2}\right|\ge \dots \ge \left|Y_{m,2m}\right|$. For this choice, inequality (44) is stronger than (43) [a simple example is the set of four canonical coordinates and momenta $x,y,p_x,p_y$]. However, it seems difficult to make this choice explicit if the mean values of commutators depend on quantum states.

Another generalization of inequality (43) was found in Ref. [71]. Its authors considered two families of Hermitian operators: ${\widehat{A}}_1,\dots ,{\widehat{A}}_N$ and ${\widehat{B}}_1,\dots ,{\widehat{B}}_M$. They proved the inequality

$$\sqrt{\mathrm{T}\mathrm{r}\left(A\right)\mathrm{T}\mathrm{r}\left(B\right)}\ge {\lambda }^{-1}\sum _{jk}\sqrt{Y_{jk}^2+{\left(\overline{A_j{B}_k}\right)}^2},$$

(45)

where $\lambda$ is the maximal singular value of the $N\times M$ matrix $\mathrm{\Lambda }=\parallel \langle \left(\delta {\widehat{A}}_j\right)\left(\delta {\widehat{B}}_k\right)\rangle /(\Delta A_j\Delta B_k\parallel$ (if some $\Delta z_i=0$, then one should put ${\mathrm{\Lambda }}_{ij}=0$). (Remember that singular values of matrix G are non-negative square roots of eigenvalues of matrix $G^{†}G$). The elements $Y_{jk}$ are given by formula (13) with $z_j=A_j$ and $z_k=B_k$. Inequality (45) was illustrated for the set of three operators given by the Pauli matrices. In the special case of $N=M$ and $A_j=B_j$, inequality (45) takes the form

$$\mathrm{T}\mathrm{r}\left(X\right)\ge {\displaystyle \frac{2}{\lambda }}\sum _{1\le j<k\le N}\left|G_{jk}\right|\ge {\displaystyle \frac{2}{\lambda }}\sum _{1\le j<k\le N}\left|Y_{jk}\right|.$$

(46)

This inequality is stronger than (43), provided $\lambda <N-1$. It was shown in [71] that such situations do exist. A simple example was the set of three Pauli’s matrices ($N=3$), acting on the mixed quantum state described by the $2\times 2$ density matrix $\rho =diag\left[{cos}^2(\theta /2),{sin}^2(\theta /2)\right]$. Then, $\mathrm{T}\mathrm{r}\left(X\right)=2+{sin}^2\left(\theta \right)$, $\sum \left|Y_{jk}\right|=(1/2)cos\theta$ and $\lambda =1+|cos\theta |<2$. We see that even the improved inequality (46) is rather weak in this example. Other nice trace inequalities are discussed in Section 7.4.

3.1.2. Inequalities for Sums of Uncertainties

A simple inequality for the “uncertainties” $\Delta A\equiv \sqrt{{\sigma }_A}$ and $\Delta B\equiv \sqrt{{\sigma }_B}$ was found in [72]:

$$\Delta A+\Delta B\ge \Delta (A+B).$$

(47)

It is a simple consequence of the Schwarz inequality for the vectors in the Hilbert space $\delta \widehat{A}|\psi \rangle$ and $\delta \widehat{B}|\psi \rangle$. Obviously, it is assumed here that observables A and B (described by means of the Hermitian operators) have the same dimensions, in order that the observable $A+B$ could make sense. Otherwise, some rescaling factors should be used. It was written in [72] that “the physical meaning of the sum uncertainty relation (47) is that if we have an ensemble of quantum systems, then the ignorance in totality is always less than the sum of the individual ignorance.”

Since $\Delta (-A)=\Delta \left(A\right)$, an immediate consequence of (47) is its generalization to the case of N observables and arbitrary real numbers $p_i$ (only positive numbers $p_i$ were considered in [72])

$$\sum _{i=1}^N\left|p_i\right|\Delta A_i\ge \Delta \left(\sum _{i=1}^N{p}_i{A}_i\right).$$

(48)

Then, introducing the operators $\widehat{A}={\sum }_{j=1}^N{p}_j{\widehat{A}}_j$, $\widehat{B}={\sum }_{k=1}^M{q}_k{\widehat{B}}_k$ and combining (48) with the Robertson inequality (4), one can obtain the inequality

$$\left(\sum _{j=1}^N\left|p_j\right|\Delta A_j\right)\left(\sum _{k=1}^M\left|q_k\right|\Delta B_k\right)\ge {\displaystyle \frac{1}{2}}\left|\sum _{j=1}^N\sum _{k=1}^M{p}_j{q}_k〈\left[{\widehat{A}}_j,{\widehat{B}}_k\right]〉\right|.$$

(49)

(Only the special case of $N=M$, $p_i=q_i=1$ and $\left[{\widehat{A}}_j,{\widehat{B}}_k\right]=i{\delta }_{jk}\widehat{C}$ was considered in [72]). In particular, for N coordinates and momenta one obtains [72]

$$\left(\sum _{i=1}^N\Delta x_i\right)\left(\sum _{i=1}^N\Delta p_i\right)\ge N\hslash /2.$$

(50)

This inequality differs from those given in Section 3.1.1, because it contains sums of standard deviations instead of sums of variances.

Several authors [73,74,75,76] derived many inequalities containing certain mixtures of the variances and their square roots (“uncertainties”). The simplest examples are as follows,

$$\sum _{i=1}^N{\left(\Delta A_i\right)}^2\ge {\displaystyle \frac{1}{N-2}}\left\{\sum _{1\le i<j\le N}{\left[\Delta \left(A_i+A_j\right)\right]}^2-{\displaystyle \frac{1}{{(N-1)}^2}}{\left[\sum _{1\le i<j\le N}\Delta \left(A_i+A_j\right)\right]}^2\right\},$$

(51)

$$\sum _{i=1}^N{\left(\Delta A_i\right)}^2\ge {\displaystyle \frac{1}{N}}{\left[\Delta \left(\sum _{i=1}^N{A}_i\right)\right]}^2+{\displaystyle \frac{1}{N^2}}{\left[\sum _{1\le i<j\le N}\Delta \left(A_i-A_j\right)\right]}^2,$$

(52)

$$\sum _{i=1}^N\left(\Delta A_i\right)\ge {\displaystyle \frac{1}{N-2}}\left[\sum _{1\le i<j\le N}\Delta \left(A_i+A_j\right)-\Delta \left(\sum _{i=1}^N{A}_i\right)\right].$$

(53)

These inequalities were illustrated for three spin-1/2 operators: see Section 6.4. Note, however, that Equations (51)–(53) contain no commutators. Therefore, they can hardly be considered as the “uncertainty relations” in the strict sense of these words.

3.1.3. Symplectic Invariants and Resulting Uncertainty Relations

In view of a complicated structure of inequalities expressing multidimensional uncertainty relations, several authors studied possible canonical forms of the $2n\times 2n$ covariance matrices [77,78,79]. If the initial momentum-coordinate vector $\mathbf{q}=\left(p_1,p_2,\dots ,p_n,x_1,x_2,\dots ,x_n\right)$ is linearly transformed as $\mathbf{q}=S{\mathbf{q}}^{\prime }$, then, the covariance matrix Q is related to the transformed matrix $Q^{\prime }$ as $Q=S{Q}^{\prime }\tilde{S}$, where $\tilde{S}$ is the transposed matrix. Since the uncertainty inequalities are determined by the commutator matrix Y, it seems reasonable to use a transformation that does not change matrix $Y=-i(\hslash /2)\mathrm{\Sigma }$. Such transformations, satisfying the condition $S\mathrm{\Sigma }\tilde{S}=\mathrm{\Sigma }$, are called symplectic transformations. In particular, $|detS|=1$, so that $detQ=det{Q}^{\prime }$. The fundamental theorem in this area, proved by Williamson [80] (see also [81,82] for the discussion and simplified proof), tells us that any positive definite symmetrical matrix Q can be transformed by means of symplectic transformations to the canonical diagonal form

$$Q^{\left(can\right)}=diag\left({\kappa }_1,{\kappa }_2,\dots ,{\kappa }_n,{\kappa }_1,{\kappa }_2,\dots ,{\kappa }_n\right)$$

(54)

with positive values ${\kappa }_j$. The uncertainty relation in this formulation is the statement that (see also, e.g., Refs. [83,84] for another normalization of coefficients)

$${\kappa }_j\ge \hslash /2,\mathrm{for} \mathrm{all}j=1,2,\dots ,n.$$

(55)

(A reduction to the diagonal form with identical blocks $Q_x$ and $Q_p$ implies some scaling transformations to arrive at blocks with the same physical dimension).

Returning to the initial $2n\times 2n$ matrix Q, one can obtain the following consequences of (55) [79,85]

$$\mathrm{T}\mathrm{r}\left[{\left(iQ\mathrm{\Sigma }\right)}^{2k}\right]\ge 2^{1-2k}n{\hslash }^{2k},k=1,2,\dots .$$

(56)

(The imaginary unit i was missed in [79]; this misprint was corrected in [85]). In terms of the block matrices (23) we can write

$${\left(iQ\mathrm{\Sigma }\right)}^2=∥\begin{array}{cc}{Q}_p{Q}_x-Q_{px}^2& Q_{px}{Q}_p-Q_p{Q}_{xp}\\ Q_{xp}{Q}_x-Q_x{Q}_{px}& Q_x{Q}_p-Q_{xp}^2\end{array}∥,$$

(57)

so that (56) for $k=1$ has the form

$$\mathrm{T}\mathrm{r}\left(Q_p{Q}_x\right)-\mathrm{T}\mathrm{r}\left(Q_{px}^2\right)\ge n{\hslash }^2/4,$$

(58)

which is different from (36) and (37). If $n=1$, then (58) coincides with (8).

Important inequalities can be obtained if one considers the following polynomial of order $2n$ with respect to an auxiliary parameter $\mu$:

$$\mathcal{D}\left(\mu \right)\equiv det(Q-\mu \mathrm{\Sigma })=\sum _{k=0}^{2n}{\mathcal{D}}_k^{\left(n\right)}{\mu }^k.$$

(59)

This polynomial is invariant with respect to any symplectic transformation. Consequently, each coefficient ${\mathcal{D}}_k^{\left(n\right)}$ is invariant with respect to such transformations, as well. Therefore the coefficients ${\mathcal{D}}_k^{\left(n\right)}$ were named in [49,86] “quantum universal invariants”, because their values are preserved in time during the evolution governed by arbitrary quadratic Hamiltonians.

After the reduction of matrix Q to the canonical diagonal form (54), one can write

$$\mathcal{D}\left(\mu \right)=\prod _{j=1}^n\left({\kappa }_j^2+{\mu }^2\right).$$

(60)

Consequently, $\mathcal{D}\left(\mu \right)=\mathcal{D}(-\mu )$, and the only nonzero universal invariants ${\mathcal{D}}_{2j}^{\left(n\right)}$ can be expressed in terms of the symplectic eigenvalues ${\kappa }_j$ as follows:

$${\mathcal{D}}_0^{\left(n\right)}={\left({\kappa }_1{\kappa }_2\dots {\kappa }_n\right)}^2,$$

(61)

$${\mathcal{D}}_2^{\left(n\right)}={\mathcal{D}}_0^{\left(n\right)}\sum _{j=1}^n{\kappa }_j^{-2},{\mathcal{D}}_4^{\left(n\right)}={\mathcal{D}}_0^{\left(n\right)}\sum _{j<k}{\left({\kappa }_j{\kappa }_k\right)}^{-2},{\mathcal{D}}_6^{\left(n\right)}={\mathcal{D}}_0^{\left(n\right)}\sum _{j<k<l}{\left({\kappa }_j{\kappa }_k{\kappa }_l\right)}^{-2},\dots$$

(62)

$${\mathcal{D}}_{2n-6}^{\left(n\right)}=\sum _{j<k<l}{\left({\kappa }_j{\kappa }_k{\kappa }_l\right)}^2,{\mathcal{D}}_{2n-4}^{\left(n\right)}=\sum _{j<k}{\left({\kappa }_j{\kappa }_k\right)}^2,{\mathcal{D}}_{2n-2}^{\left(n\right)}=\sum _{j=1}^n{\kappa }_j^2.$$

(63)

Obviously, minimal values of all these expressions can be achieved for ${\kappa }_1={\kappa }_2=\cdots ={\kappa }_n=\hslash /2$. Thus, we have the following set of inequalities [79], in addition to the simplest one (22):

$${\mathcal{D}}_2^{\left(n\right)}\ge {\left({\hslash }^2/4\right)}^{n-1}n,{\mathcal{D}}_4^{\left(n\right)}\ge {\left({\hslash }^2/4\right)}^{n-2}n(n-1)/2,$$

(64)

$${\mathcal{D}}_6^{\left(n\right)}\ge {\left({\hslash }^2/4\right)}^{n-3}n(n-1)(n-2)/6,$$

$${\mathcal{D}}_{2j}^{\left(n\right)}\ge {\left({\hslash }^2/4\right)}^{n-j}{\displaystyle \frac{n!}{j!(n-j)!}},$$

(65)

$${\mathcal{D}}_{2n-6}^{\left(n\right)}\ge {\left({\hslash }^2/4\right)}^{3}n(n-1)(n-2)/6,{\mathcal{D}}_{2n-4}^{\left(n\right)}\ge {\left({\hslash }^2/4\right)}^{2}n(n-1)/2,$$

(66)

$${\mathcal{D}}_{2n-2}^{\left(n\right)}\ge \left({\hslash }^2/4\right)n.$$

For $n\ge 2$, a relatively simple expression can be written for the invariant ${\mathcal{D}}_{2n-2}^{\left(n\right)}$ [49,66]:

$${\mathcal{D}}_{2n-2}^{\left(n\right)}=\sum _{i,j=1}^n\left(\overline{p_i{p}_j}·\overline{x_i{x}_j}-\overline{p_i{x}_j}·\overline{x_i{p}_j}\right).$$

(67)

3.2. Inequalities Without Covariances

An obvious disadvantage of inequalities (14)–(17) is that they are rather complicated for $N>2$ observables, because they contain, in addition to N variances $X_{kk}$ and $N(N-1)/2$ mean values of commutators $Y_{jk}$, numerous sums and products of various combinations of $N(N-1)/2$ covariances $X_{jk}$ with $j\ne k$. For example, $detX$ contains 17 different products of covariances if $N=4$ [see Equation (171) in Section 7.1], in addition to 6 different products of mean values of commutators in $detY$. Moreover, inequality (17) seems totally useless if N is an odd number, as soon as $detY=0$ in this case.

One can get rid of all $N(N-1)/2$ covariances, using the scheme proposed in [87] (generalizing the idea put forward in [88] for the special case of $N=3$). Suppose that we know N Hermitian $M\times M$ anticommuting matrices $R_k$ satisfying the relations of the Clifford algebra

$$R_j{R}_k+R_k{R}_j=2I_M{\delta }_{jk},$$

(68)

where $I_M$ is the $M\times M$ unit matrix. Consider the operator $\widehat{f}={\sum }_{k=1}^N{\xi }_k\delta {\widehat{z}}_k{R}_k$, where ${\xi }_k$ are arbitrary real coefficients and ${\widehat{z}}_k$ arbitrary Hermitian operators. It acts in the extended Hilbert space of states $|\mathrm{\Psi }\rangle =|\psi \rangle \otimes |\chi \rangle$, where $|\chi \rangle$ is an auxiliary M-dimensional vector. Then, the condition $\langle \mathrm{\Psi }\left|{\widehat{f}}^{†}\widehat{f}\right|\mathrm{\Psi }\rangle \ge 0$ can be written as the condition of positive semi-definiteness of the Hermitian $M\times M$ matrix

$$F=g{I}_M+i\sum _{j<k}{R}_j{R}_k{y}_{jk},$$

(69)

$$g=\sum _{k=1}^N{\xi }_k^2{X}_{kk},y_{jk}=2{\xi }_j{\xi }_k{Y}_{jk}=-y_{kj}.$$

(70)

The covariances $X_{jk}$ with $j\ne k$ go out due to the anticommutation relations (68).

To perform the scheme, one has to know the explicit form of N anticommuting matrices $R_j$, satisfying the Clifford algebra relations (68). The main technical problem is the dimension of such matrices. According to [89,90,91], this dimension is $2^n\times 2^n$ for $N=2n$ and $N=2n+1$. The special case of $N=3$ (when matrices $R_j$ are three Pauli’s $2\times 2$ matrices) is studied in Section 5.2. The cases of $N=4$ and $N=5$ (four Dirac’s $4\times 4$ matrices) are considered in Section 7.2 and Section 7.3, respectively. Unfortunately, the dimension of matrices $R_j$ rapidly grows with the increase of N. For example, one needs matrices $8\times 8$ for $N=6,7$, matrices $16\times 16$ for $N=8,9$, and so on. Although the structure of such matrices $R_j$ is rather simple [92], the analysis of the positivity conditions for the corresponding matrix (69) is a formidable task for $N>5$. Probably, this can be achieved with the aid of some computer algebra programs.

4. Specific Inequalities for Two Operators

Let us consider some special cases of the general inequalities given above. For two arbitrary (not necessarily Hermitian) operators $\widehat{A}$ and $\widehat{B}$, inequality (14) is equivalent to the following one [45]:

$$〈\delta {\widehat{A}}^{†}\delta \widehat{A}〉〈\delta {\widehat{B}}^{†}\delta \widehat{B}〉\ge {\displaystyle \frac{1}{4}}{\left|〈\delta {\widehat{A}}^{†}\delta \widehat{B}-\delta {\widehat{B}}^{†}\delta \widehat{A}〉\right|}^2+{\displaystyle \frac{1}{4}}{〈\delta {\widehat{A}}^{†}\delta \widehat{B}+\delta {\widehat{B}}^{†}\delta \widehat{A}〉}^2,$$

(71)

$$\delta \widehat{A}=\widehat{A}-\langle \widehat{A}\rangle ,\delta \widehat{B}=\widehat{B}-\langle \widehat{B}\rangle .$$

Actually, inequality (71) is another form of the Cauchy–Schwartz inequality,

$$〈\delta {\widehat{A}}^{†}\delta \widehat{A}〉〈\delta {\widehat{B}}^{†}\delta \widehat{B}〉\ge {\left|〈\delta {\widehat{A}}^{†}\delta \widehat{B}〉\right|}^2,$$

(72)

which can be preferable for non-Hermitian operators, when commutators do not appear automatically.

A simple example of non-Hermitian operators is $\widehat{A}=\widehat{a}$ and $\widehat{B}={\widehat{a}}^{†}$, where $\widehat{a}$ and ${\widehat{a}}^{†}$ are the boson annihilation and creation operators: $[\widehat{a},{\widehat{a}}^{†}]=1$. Then, the following inequality must hold:

$${\delta }_N\left({\delta }_N+1\right)-{\left|{\sigma }_a\right|}^2\ge 0,$$

(73)

$${\sigma }_a=〈{\widehat{a}}^2〉-{〈\widehat{a}〉}^2,{\delta }_N=〈{\widehat{a}}^{†}\widehat{a}〉-{\left|〈\widehat{a}〉\right|}^2.$$

(74)

The inequality

$$\Delta Y_1\Delta Y_2\ge \langle \widehat{N}+1/2\rangle$$

(75)

for operators ${\widehat{Y}}_1=\left({\widehat{a}}^{†2}+{\widehat{a}}^2\right)/2$, ${\widehat{Y}}_2=i\left({\widehat{a}}^{†2}-{\widehat{a}}^2\right)/2$ and $\widehat{N}={\widehat{a}}^{†}\widehat{a}$ was given in paper [93].

For operators $\widehat{a}$ and $\widehat{N}$, with the commutator $\left[\widehat{N},\widehat{a}\right]=-\widehat{a}$, inequality (72) takes the form

$${\sigma }_N{\delta }_N\ge {\left|\langle \widehat{N}\widehat{a}\rangle -\langle \widehat{N}\rangle \langle \widehat{a}\rangle \right|}^2,{\sigma }_N=〈{\left({\widehat{a}}^{†}\widehat{a}\right)}^2〉-{〈{\widehat{a}}^{†}\widehat{a}〉}^2.$$

(76)

The following inequality was derived in paper [94]:

$${\sigma }_N+{\delta }_N\ge \sqrt{\langle \widehat{N}\rangle +3/4}-1.$$

(77)

Two different inequalities were found in paper [95]:

$${\sigma }_N\left({\delta }_N+1/2\right)\ge {\left|〈\widehat{a}〉\right|}^2/4,$$

(78)

$$\left({\sigma }_N+{\displaystyle \frac{1}{4}}\right)\left({\delta }_N+{\displaystyle \frac{1}{2}}\right)\ge {\displaystyle \frac{\langle \widehat{N}\rangle }{4}}+{\displaystyle \frac{1}{8}}.$$

(79)

Inequality (78) cannot be exactly saturated. A more precise (but more complicated) inequality was found in paper [96]. The states saturating inequality (79) were considered in paper [97].

In the case of fermion operators satisfying the relations $\{\widehat{b},{\widehat{b}}^{†}\}=1$ and ${\widehat{b}}^2=0$, one can obtain the following inequality instead of (73):

$${\delta }_N\left(1-{\delta }_N\right)-{\left|\langle \widehat{b}\rangle \right|}^4\ge 0,{\delta }_N=\langle {\widehat{b}}^{†}\widehat{b}\rangle -{\left|\langle \widehat{b}\rangle \right|}^2.$$

(80)

Another form of this inequality is

$$\langle {\widehat{b}}^{†}\widehat{b}\rangle \langle \widehat{b}{\widehat{b}}^{†}\rangle +{\displaystyle |}\langle \widehat{b}\rangle {{\displaystyle |}}^2\langle {\widehat{b}}^{†}\widehat{b}-\widehat{b}{\widehat{b}}^{†}\rangle -2{\left|\langle \widehat{b}\rangle \right|}^4\ge 0.$$

(81)

For Hermitian operators, inequality (71) is nothing but the Schrödinger–Robertson inequality (5). In particular, if $\widehat{A}=\widehat{x}$ and $\widehat{B}=\widehat{p}$, then we have inequality (8).

The following inequality, containing uncertainties instead of their squares, can be found in [98]:

$$\Delta x\Delta p-\mu \left|{\sigma }_{xp}\right|\ge (\hslash /2)\sqrt{1-{\mu }^2},0<\mu <1.$$

(82)

However, this inequality is weaker than the Schrödinger–Robertson inequality (8). This can be seen if one rewrites (82) in terms of the correlation coefficient $r\equiv {\sigma }_{xp}/\left(\Delta x\Delta p\right)$:

$$\Delta x\Delta p\ge {\displaystyle \frac{(\hslash /2)\sqrt{1-{\mu }^2}}{1-\mu \left|r\right|}}.$$

(83)

The ratio of the right-hand side of (83) to the square root of the right-hand side of inequality (9) equals

$$R={\displaystyle \frac{\sqrt{1-{\mu }^2}\sqrt{1-r^2}}{1-\mu \left|r\right|}}.$$

(84)

But one can see that $R<1$ for any values of $\left|r\right|,\mu <1$, as a consequence of inequality ${(\mu -|r\left|\right)}^2\ge 0$. A generalization of inequality (82) found in Ref. [99] has the form

$${\left({\sigma }_x{\sigma }_p\right)}^m-\mu {\sigma }_{xp}^{2m}\ge {(\hslash /2)}^{2m}{\left[1-{\mu }^{1/(1-m)}\right]}^{1-m},1/2\le m<1,0\le \mu <1.$$

(85)

4.1. Three Canonical Observables with a Linear Dependence

A set of three operators, which are not totally independent, was considered in [100]. Let us introduce the operators $\widehat{\tilde{x}}=a\widehat{x}$ and $\widehat{\tilde{p}}=b\widehat{p}$, where positive parameters a and b are chosen in such a way that both new observables, $\tilde{x}$ and $\tilde{p}$, have the same dimension ${\hslash }^{1/2}$. If $ab=1$, then $\left[\widehat{\tilde{x}},\widehat{\tilde{p}}\right]=i\hslash$. Introducing the new operator $\widehat{\xi }=\widehat{\tilde{x}}+\widehat{\tilde{p}}$, we have the commutation relations $\left[\widehat{\tilde{x}},\widehat{\xi }\right]=-\left[\widehat{\tilde{p}},\widehat{\xi }\right]=i\hslash$. Then, multiplying inequalities (3) for three pairs of operators, $\left(\widehat{\tilde{x}},\widehat{\tilde{p}}\right)$, $\left(\widehat{\tilde{x}},\widehat{\xi }\right)$ and $\left(\widehat{\tilde{p}},\widehat{\xi }\right)$, one obtains a trivial consequence ${\sigma }_{\tilde{x}}{\sigma }_{\tilde{p}}{\sigma }_{\xi }\ge {(\hslash /2)}^3$. However, the value ${(\hslash /2)}^3$ cannot be attained for any quantum state, because the states minimizing the uncertainty products for each pair are different. It was proven by Kechrimparis and Weigert in [100] (by means of the direct minimization of the product ${\sigma }_{\tilde{x}}{\sigma }_{\tilde{p}}{\sigma }_{\xi }$ over all normalized pure quantum states) that the correct inequality is

$${\sigma }_{\tilde{x}}{\sigma }_{\tilde{p}}{\sigma }_{\xi }\ge {(\hslash /\sqrt{3})}^3.$$

(86)

However, inequality (86) is not stronger than the Schrödinger–Robertson inequality (9). Indeed, returning to the original variables x and p, we can rewrite (86) as

$${\sigma }_x{\sigma }_p\left(a^2{\sigma }_x+a^{-2}{\sigma }_p+2{\sigma }_{xp}\right)\ge {(\hslash /\sqrt{3})}^3.$$

(87)

Inequality (87) must hold for any positive value of parameter a (with the necessary dimension). Minimizing the left-hand side of (87) with respect to this parameter, we arrive at the inequality

$${\sigma }_x{\sigma }_p\ge ({\hslash }^2/3){\left[2(1+r)\right]}^{-2/3},$$

(88)

where the correlation coefficient r was defined in Equation (9). The ratio of the right-hand side of (88) to the right-hand side of inequality (9) equals ${\left[(16/27)f\left(r\right)\right]}^{1/3}$ with $f\left(r\right)=(1+r){(1-r)}^3$. Since $f^{\prime }\left(r\right)=-2(1+2r){(1-r)}^2$, this ratio attains the maximal value (equal to unity) for $r=-1/2$. Consequently, inequality (88) is weaker than (9). The equality in (86) is attained for ${\sigma }_{\tilde{x}}={\sigma }_{\tilde{p}}={\sigma }_{\xi }=\hslash /\sqrt{3}$. It corresponds to the Gaussian correlated (squeezed) coherent state [100]. We give it the name “KW state”.

5. Inequalities for Three Independent Operators: A General Case

The case of three Hermitian operators ${\widehat{z}}_j$ ($j=1,2,3$) was studied in detail in the frameworks of the general scheme described in Section 3 (but presumably without any knowledge of the Robertson approach) by Synge [101]. (Although only pure states were considered in that paper, it is evident that the same relations are valid for mixed states as well.) The main results of [101] are the inequalities

$$X_{11}{X}_{22}{X}_{33}\ge \sum _{\left(123\right)}{X}_{11}\left(X_{23}^2+Y_{23}^2\right)+2\sum _{\left(123\right)}{X}_{23}{Y}_{31}{Y}_{12}-2X_{12}{X}_{23}{X}_{31},$$

(89)

$$4X_{11}{X}_{22}{X}_{33}\ge \sum _{\left(123\right)}{X}_{11}\left(X_{23}^2+Y_{23}^2\right)-\sum _{\left(123\right)}{X}_{23}{Y}_{31}{Y}_{12}+X_{12}{X}_{23}{X}_{31},$$

(90)

and their consequence (called “the Schrödinger inequality for $n=3$” in [101])

$$3X_{11}{X}_{22}{X}_{33}\ge X_{11}\left(X_{23}^2+Y_{23}^2\right)+X_{22}\left(X_{13}^2+Y_{13}^2\right)+X_{33}\left(X_{12}^2+Y_{12}^2\right).$$

(91)

The meaning of quantities $X_{jk}$ and $Y_{jk}$ is the same as in Equation (13). The symbol ${\sum }_{\left(123\right)}$ designates the sum over all cyclic permutations of the indexes $1\to 2\to 3$. Actually, inequality (89) (which was re-derived also in [57]) is nothing but inequality $detF\ge 0$ (14) for $N=3$. Combining it with Hadamard’s inequality (15), one can see that the right-hand side of (89) must be non-negative:

$$\sum _{\left(123\right)}{X}_{11}\left(X_{23}^2+Y_{23}^2\right)+2\sum _{\left(123\right)}{X}_{23}{Y}_{31}{Y}_{12}-2X_{12}{X}_{23}{X}_{31}\ge 0.$$

(92)

Applying the known inequality for the arithmetic mean and the geometric mean [53],

$$\sum _{k=1}^n{x}_k\ge n{\left(\prod _{k=1}^n{x}_k\right)}^{1/n},x_k>0$$

(93)

with $n=3$ to the right-hand side of (91) one can arrive at the inequality

$$X_{11}{X}_{22}{X}_{33}\ge {\left[\left(X_{23}^2+Y_{23}^2\right)\left(X_{13}^2+Y_{13}^2\right)\left(X_{12}^2+Y_{12}^2\right)\right]}^{1/2}.$$

(94)

However, this is the consequence of three products of the Schrödinger–Robertson inequalities (5) for pairs $\left(z_1,z_2\right)$, $\left(z_1,z_3\right)$ and $\left(z_2,z_3\right)$. One can check that the relations (89)–(91) and (94) become equalities for the KW state described at the end of the preceding section (i.e., the Gaussian correlated squeezed state with ${\sigma }_{\tilde{x}}={\sigma }_{\tilde{p}}={\sigma }_{\xi }=\hslash /\sqrt{3}$ and $r=-1/2$).

A trivial consequence of (94) is the inequality

$$X_{11}{X}_{22}{X}_{33}\ge \left|Y_{12}{Y}_{23}{Y}_{31}\right|,$$

(95)

which can also be obtained from the product of three Robertson’s inequalities (4). Actually, it can be strengthened: see Equation (112) in Section 5.2 Another trivial consequence of (91) is [101]

$$3X_{11}{X}_{22}{X}_{33}\ge X_{11}{Y}_{23}^2+X_{22}{Y}_{13}+X_{33}{Y}_{12}^2.$$

(96)

However, the equality in this relation (called “the Heisenberg inequality for $n=3$” in [101]) cannot be reached for any quantum state. A more precise inequality (111) is given in Section 5.2.

Several inequalities for three arbitrary operators were derived in Ref. [56]. One of the simple examples can be written as follows,

$$det\left(X\right)\ge \langle {\left(\delta {\widehat{z}}_1{Y}_{23}+\delta {\widehat{z}}_2{Y}_{31}+\delta {\widehat{z}}_3{Y}_{12}\right)}^2\rangle .$$

(97)

Some extensions for non-Hermitian operators can be found in the paper [102].

5.1. Two Observables Coupled with a Third One

Inequality (89) has the form $X_{11}{X}_{22}{X}_{33}\ge a{X}_{11}+b{X}_{22}+c$, where coefficients a, b and c do not contain variances $X_{11}$ and $X_{22}$. Moreover a and b are non-negative. Due to the standard arithmetic-geometric inequality (93) with $n=2$, we have $a{X}_{11}+b{X}_{22}\ge 2\sqrt{ab{X}_{11}{X}_{22}}$. This means that $X_{33}{\xi }^2-2\sqrt{ab}\xi -c\ge 0$, where $\xi =\sqrt{X_{11}{X}_{22}}\ge 0$. Consequently, $X_{33}\xi \ge \sqrt{ab}+\sqrt{ab+c{X}_{33}}$. Thus, we arrive at the following generalization of the Schrödinger–Robertson inequality (5) to the case when two observables $z_1$ and $z_2$ are coupled somehow with the third observable $z_3$ [103]:

$$\Delta z_1\Delta z_2\ge \sqrt{G_{12}^2+{\Omega }^2+2\mathrm{\Gamma }}+\Omega .$$

(98)

The following combinations of covariances and mean values of commutators are introduced here:

$$G_{jk}^2=X_{jk}^2+Y_{jk}^2,\Omega =\left|G_{13}{G}_{23}\right|/X_{33},$$

(99)

$$X_{33}\mathrm{\Gamma }=X_{12}\left(Y_{23}{Y}_{31}-X_{23}{X}_{31}\right)+Y_{12}\left(X_{23}{Y}_{31}+Y_{23}{X}_{31}\right).$$

(100)

In particular, if $\left[{\widehat{z}}_1,{\widehat{z}}_3\right]=\left[{\widehat{z}}_2,{\widehat{z}}_3\right]=0$ (for example, $z_1=x$, $z_2=p_x$ and $z_3=y$), but there are correlations between all these observables due to some kinds of entanglement, then

$$\Delta z_1\Delta z_2\ge \sqrt{Y_{12}^2+{\left(X_{12}-X_{13}{X}_{23}/X_{33}\right)}^2}+\left|X_{13}{X}_{23}\right|/X_{33}.$$

(101)

Note that the right-hand side of (101) is sensitive to the signs of covariances. An impossibility of saturation of the uncertainty relations in systems entangled with some other systems was demonstrated in paper [104]. An idea to use some auxiliary operators to increase the accuracy of the product and sum forms of the uncertainty relations was considered in paper [105], with main applications to systems in finite-dimensional Hilbert spaces. In that paper, the starting point was the equality

$$\langle A^{†}A\rangle \langle B^{†}B\rangle ={\displaystyle \frac{1}{4}}{\left|\langle {[A,B]}_{+}\rangle \right|}^2+{\displaystyle \frac{1}{4}}{\left|\langle {[A,B]}_{-}\rangle \right|}^2+\langle C^{†}C\rangle \langle B^{†}B\rangle ,$$

$$C=A-\langle B^{†}A\rangle B/\langle B^{†}B\rangle ,{[A,B]}_{\pm }=A^{†}B\pm B^{†}A.$$

Actually, the quantum state of the subsystem described by two operators ${\widehat{z}}_1$ and ${\widehat{z}}_2$ is not pure, but mixed. Therefore, one can suppose a possible existence of generalizations of inequality (1) containing some degrees of purity (or mixing) of the quantum state. Such inequalities do exist: they are considered in Section 11. Inequalities (98) and (101) show, how the knowledge of some additional information on the quantum state increases the minimal value of the uncertainty product between two observables. The simplest constraint is the nonzero covariance or the correlation coefficient [45]. Uncertainty relations under the constraint of a fixed degree of Gaussianity are considered in Section 11.6.

5.2. Inequalities Without Covariances for Three Operators

Let us see how the general scheme proposed in Section 3.2 works in the case of $N=3$. The main idea is to extend the Hilbert space of quantum states $|\psi \rangle$, considering the tensor products $|\mathrm{\Psi }\rangle =|\psi \rangle \otimes |\chi \rangle$, where $|\chi \rangle$ is an auxiliary spinor. In this extended space, we introduce the operator $\widehat{F}={\sum }_{j=1}^3{\alpha }_j{\sigma }_j\Delta {\widehat{z}}_j$ with $\Delta {\widehat{z}}_j\equiv {\widehat{z}}_j-\langle {\widehat{z}}_j\rangle$, where ${\alpha }_j$ are arbitrary real numbers and ${\sigma }_j$ are the standard $2\times 2$ Pauli matrices. Then, using the properties of the Pauli matrices and performing averaging over the state $|\psi \rangle$, one can write $\langle \mathrm{\Psi }|{\widehat{F}}^{†}\widehat{F}|\mathrm{\Psi }\rangle =\langle \chi \left|\mathcal{A}\right|\chi \rangle$ with the $2\times 2$ Hermitian matrix (here ${\sigma }_0$ is the $2\times 2$ unit matrix)

$$\mathcal{A}=({\alpha }_1^2{X}_{11}+{\alpha }_2^2{X}_{22}+{\alpha }_3^2{X}_{33}){\sigma }_0-2{\sigma }_1{\alpha }_2{\alpha }_3{Y}_{23}-2{\sigma }_2{\alpha }_3{\alpha }_1{Y}_{31}-2{\sigma }_3{\alpha }_1{\alpha }_2{Y}_{12},$$

(102)

where coefficients $X_{jk}$ and $Y_{jk}$ were defined in (13). We see that matrix (102) does not contain the covariances $X_{jk}$ with $j\ne k$. Since $\langle \mathrm{\Psi }\left|{\widehat{F}}^{†}\widehat{F}\right|\mathrm{\Psi }\rangle \ge 0$ for any physical state, matrix (102) must be positive semi-definite. Considering $\langle \chi \left|\mathcal{A}\right|\chi \rangle$ as a bilinear form with respect to components of six-dimensional vector $\mathbf{v}=\left({\alpha }_1{\chi }_1,{\alpha }_2{\chi }_1,{\alpha }_3{\chi }_1,{\alpha }_1{\chi }_2,{\alpha }_2{\chi }_2,{\alpha }_3{\chi }_2\right)$ (where ${\chi }_1$ and ${\chi }_2$ are the complex components of the auxiliary spinor $|\chi \rangle$), one could write $\langle \chi \left|\mathcal{A}\right|\chi \rangle ={\mathbf{v}}^{*}\mathrm{\Phi }\mathbf{v}$ with the following $6\times 6$ Hermitian matrix $\mathrm{\Phi }$:

$$\mathrm{\Phi }=∥\begin{array}{cccccc}{X}_{11}& Y_{21}& 0& 0& 0& i{Y}_{31}\\ Y_{21}& X_{22}& 0& 0& 0& Y_{32}\\ 0& 0& X_{33}& i{Y}_{31}& Y_{32}& 0\\ 0& 0& i{Y}_{13}& X_{11}& Y_{12}& 0\\ 0& 0& Y_{32}& Y_{12}& X_{22}& 0\\ i{Y}_{13}& Y_{32}& 0& 0& 0& X_{33}\end{array}∥.$$

(103)

Then, the condition of positive semi-definiteness of matrix $\mathrm{\Phi }$, namely $det\mathrm{\Phi }\ge 0$, results in the inequality [88]

$$X_{11}{X}_{22}{X}_{33}\ge X_{11}{Y}_{23}^2+X_{22}{Y}_{31}^2+X_{33}{Y}_{12}^2.$$

(104)

Unfortunately, this inequality is not correct, because it is not satisfied, for example, for the operators considered in Section 4.1 in the KW quantum state with

$$X_{11}=X_{22}=X_{33}=\hslash /\sqrt{3},Y_{12}^2=Y_{13}^2=Y_{32}^2={\hslash }^2/4.$$

(105)

A possible origin of mistake based on using matrix (103) is that not all components of vector $\mathbf{v}$ in the bilinear form ${\mathbf{v}}^{*}\mathrm{\Phi }\mathbf{v}$ are independent, since $v_4/v_1=v_5/v_2=v_6/v_3$.

Correct inequalities can be obtained from the condition $det\mathcal{A}\ge 0$, which guarantees that $2\times 2$ matrix $\mathcal{A}$ is positive semi-definite. After simple calculation one can arrive at the inequality [106]

$${\alpha }_1^2{X}_{11}+{\alpha }_2^2{X}_{22}+{\alpha }_3^2{X}_{33}\ge 2{\left[{\left({\alpha }_1{\alpha }_2{Y}_{12}\right)}^2+{\left({\alpha }_2{\alpha }_3{Y}_{23}\right)}^2+{\left({\alpha }_1{\alpha }_3{Y}_{13}\right)}^2\right]}^{1/2},$$

(106)

which must hold for arbitrary real numbers ${\alpha }_1$, ${\alpha }_2$ and ${\alpha }_3$. Looking for the most symmetric relations, let us choose ${\alpha }_1={\alpha }_2={\alpha }_3$. Then, we obtain the inequality

$$X_{11}+X_{22}+X_{33}\ge 2{\left(Y_{12}^2+Y_{23}^2+Y_{13}^2\right)}^{1/2}.$$

(107)

Another proof of relation (107) and further generalizations can be found in Ref. [107].

The following inequality was derived in Refs. [108,109]:

$$X_{11}+X_{22}+X_{33}\ge {\displaystyle \frac{2}{\sqrt{3}}}\left(\left|Y_{12}\right|+\left|Y_{23}\right|+\left|Y_{13}\right|\right).$$

(108)

However, it is weaker than (107), due to the special case of the Cauchy inequality

$$\left(\sum _{k=1}^n{a}_k^2\right)\left(\sum _{k=1}^n{b}_k^2\right)\ge {\left(\sum _{k=1}^n\left|a_k{b}_k\right|\right)}^2$$

(109)

for $b_k=1$ and $n=3$.

The choice ${\alpha }_k^2=X_{kk}^n$ in Equation (106) results in the inequality

$$X_{11}^{n+1}+X_{22}^{n+1}+X_{33}^{n+1}\ge 2{\left[Y_{12}^2{X}_{11}^n{X}_{22}^n+Y_{23}^2{X}_{33}^n{X}_{22}^n+Y_{13}^2{X}_{11}^n{X}_{33}^n\right]}^{1/2}.$$

(110)

Wishing to find an inequality for the triple product $X_{11}{X}_{22}{X}_{33}$, let us choose ${\alpha }_1^2=X_{22}{X}_{33}$, ${\alpha }_2^2=X_{11}{X}_{33}$ and ${\alpha }_3^2=X_{22}{X}_{11}$. Then, the following inequality arises instead of the weak relation (96) and incorrect inequality (104):

$$X_{11}{X}_{22}{X}_{33}\ge {\displaystyle \frac{4}{9}}\left(X_{11}{Y}_{23}^2+X_{22}{Y}_{13}^2+X_{33}{Y}_{12}^2\right).$$

(111)

One can check that inequalities (107)–(111) turn into equalities for the KW quantum state (105). Applying inequality (106) with ${\alpha }_1=Y_{23}$, ${\alpha }_2=Y_{31}$ and ${\alpha }_3=Y_{12}$ to the right-hand side of (111), we obtain the inequality [87,110]

$$X_{11}{X}_{22}{X}_{33}\ge {(4/3)}^{3/2}\left|Y_{12}{Y}_{13}{Y}_{23}\right|,$$

(112)

which is by 50% stronger than (95) (since ${(4/3)}^{3/2}\approx 1.54$).

6. Concrete Families of Three Operators

6.1. Angular Momentum Operators

A natural system of three operators is the set of the angular momentum operators ${\widehat{L}}_x,{\widehat{L}}_y,{\widehat{L}}_z$, satisfying the commutation relations

$$\left[{\widehat{L}}_x,{\widehat{L}}_y\right]=i\hslash {\widehat{L}}_z,\left[{\widehat{L}}_y,{\widehat{L}}_z\right]=i\hslash {\widehat{L}}_x,\left[{\widehat{L}}_z,{\widehat{L}}_x\right]=i\hslash {\widehat{L}}_y.$$

In this case, the inequalities contain the following symmetric combinations of the first and second order moments:

$$\mathcal{L}=L_{xx}{L}_{yy}{L}_{zz},\mathcal{M}=L_{xy}{L}_{yz}{L}_{zx},\mathcal{R}=L_{xx}{L}_{yz}^2+L_{yy}{L}_{zx}^2+L_{zz}{L}_{xy}^2,$$

$$\mathcal{J}=L_{xy}{L}_x{L}_y+L_{yz}{L}_y{L}_z+L_{zx}{L}_z{L}_x,\mathcal{Q}=L_{xx}{L}_x^2+L_{yy}{L}_y^2+L_{zz}{L}_z^2,$$

so that $detF\equiv \mathcal{L}+2\mathcal{M}-\mathcal{R}$. Inequalities following from (14)–(16) were derived in [28,49]. Combining them with some results from [56], one can obtain the following compact inequalities:

$$\mathcal{Q}\ge 2\left|\mathcal{J}\right|,$$

(113)

$$\mathcal{L}\ge {\displaystyle \frac{1}{4}}{\hslash }^2(\mathcal{Q}+2|\mathcal{J}\left|\right)+\mathcal{R}-2\mathcal{M}\ge 0,$$

(114)

$$4\mathcal{L}\ge {\displaystyle \frac{1}{4}}{\hslash }^2(\mathcal{Q}-\mathcal{J})+\mathcal{R}+\mathcal{M}.$$

(115)

Inequality (113) becomes the trivial equality $0=0$ for any eigenstate $|j,m\rangle$ (since $L_{zz}=L_x=L_y=0$ in this case). An equality also happens for the $SU\left(2\right)$ coherent states of the Perelomov type [56]. In these cases, $detF\equiv 0$. Inequality (91) yields

$$3\mathcal{L}\ge \mathcal{R}+{\hslash }^2\mathcal{Q}/4,$$

(116)

whereas inequality (111) means that

$$\mathcal{L}\ge {\hslash }^{2}Q/9.$$

(117)

The non-negativeness of the second order principal minors of the matrix F, corresponding to the angular momentum operators, results in the following generalization of inequality (12):

$$L_{xx}{L}_{yy}-L_{xy}^2\ge {\displaystyle \frac{{\hslash }^2}{4}}{L}_z^2.$$

(118)

Its consequence is

$${\left(L_{xx}{L}_{yy}{L}_{zz}\right)}^2\ge \prod _{\left(xyz\right)}\left(L_{yz}^2+{\displaystyle \frac{{\hslash }^2}{4}}{L}_x^2\right).$$

(119)

It can also be derived from (94). Inequality (107) reads as

$$\langle {\mathbf{L}}^2\rangle -{\langle \mathbf{L}\rangle }^2\ge \hslash \left|\langle \mathbf{L}\rangle \right|\equiv \hslash \sqrt{L_x^2+L_y^2+L_z^2},$$

(120)

whereas inequality (112) takes the form

$$L_{xx}{L}_{yy}{L}_{zz}\ge {\displaystyle \frac{{\hslash }^3}{3\sqrt{3}}}{L}_x{L}_y{L}_z.$$

(121)

The following consequence of the relation (12) was obtained in paper [111]:

$$\left(L_{zz}+{\displaystyle \frac{{\hslash }^2}{4}}\right){\displaystyle \frac{L_{xx}+L_{yy}}{〈{\widehat{L}}_x^2+{\widehat{L}}_y^2〉}}\ge {\displaystyle \frac{{\hslash }^2}{4}}.$$

(122)

6.2. A Charged Particle in a Magnetic Field

It is worth noting that inequalities of Section 6.1 also hold for the covariances of the kinetic momentum components ${\pi }_j=p_j-(e/c)A_j\left(\mathbf{x}\right)$ (which are proportional to the velocity) of a charged particle in a magnetic field, provided the quantities $L_{\alpha \beta }$ are replaced by ${\pi }_{\alpha \beta }$ and $L_{\alpha }$ by $(e/c)\langle B_{\alpha }\rangle$ [where $B_{\alpha }$ is the magnetic induction vector component and $A_j\left(\mathbf{x}\right)$ is the vector potential component], in accordance with commutation relations of the type

$$[{\widehat{\pi }}_x,{\widehat{\pi }}_y]=i\hslash (e/c)B_z(x,y,z).$$

(123)

For example, inequality (120) can be written as

$$\langle {\widehat{\pi }}^2\rangle -{\langle \widehat{\pi }\rangle }^2\ge \hslash \left(\right|e|/c)\sqrt{{\langle B_x\rangle }^2+{\langle B_y\rangle }^2+{\langle B_z\rangle }^2}.$$

(124)

Using Equation (108), the authors of paper [112] obtained the inequality

$$\langle {\widehat{\pi }}^2\rangle -{\langle \widehat{\pi }\rangle }^2\ge {\displaystyle \frac{|\hslash (e/c\left)\right|}{\sqrt{3}}}\left(|\langle B_x\rangle |+|\langle B_y\rangle |+|\langle B_y\rangle |\right),$$

(125)

which is weaker than (124). In addition, the right-hand side of inequality (125) is not invariant with respect to rotation of the coordinate axes in the case of homogeneous magnetic field. In this case, only the specific choice $|B_x|=|B_y|=|B_z|=|\mathbf{B}|/\sqrt{3}$ transforms inequality (125) to the strong inequality (124), which predicts the correct Landau value $\hslash \omega /2$ (where $\omega =\left|e\mathbf{B}\right|/\left(mc\right)$, m being the particle mass) for the minimal energy of a charged particle in the constant magnetic field.

6.3. “Degrees of Uncertainty” of the Angular Momentum

A set of simple inequalities resulting from inequalities like (12) was considered by Delbourgo in paper [61]. He studied the problem of finding quantities, which could characterize “the degree of uncertainty” of the angular momentum in the most adequate manner. Three quantities were considered:

$$\begin{array}{ccc}\hfill {\Delta }^{3}L& =& {\left(L_{xx}{L}_{yy}{L}_{zz}\right)}^{1/2}\equiv {\mathcal{L}}^{1/2},\hfill \end{array}$$

(126)

$$\begin{array}{ccc}\hfill {\Delta }^{2}L& =& {\left(L_{xx}{L}_{yy}+L_{yy}{L}_{zz}+L_{zz}{L}_{xx}\right)}^{1/2},\hfill \end{array}$$

(127)

$$\begin{array}{ccc}\hfill \Delta L& =& {\left(L_{xx}+L_{yy}+L_{zz}\right)}^{1/2}.\hfill \end{array}$$

(128)

They were named, respectively, as “uncertainty volume”, “uncertainty area” and “uncertainty radius”. It was shown that the best characteristics of the uncertainty in the angular momentum is the “uncertainty radius” $\Delta L$. A consequence of (12) is the inequality

$${\Delta }^{2}L\ge {\displaystyle \frac{\hslash }{2}}\left|\langle \widehat{\mathbf{L}}\rangle \right|.$$

(129)

On the other hand, in view of (107) we have

$${(\Delta L)}^2\ge \hslash \left|〈\widehat{\mathbf{L}}〉\right|\equiv \hslash {\left(\sum _{j=1}^3{L}_j^2\right)}^{1/2},$$

(130)

whereas inequality (43) yields the relation [69]

$${(\Delta L)}^2\ge {\displaystyle \frac{\hslash }{2}}\sum _{j=1}^3\left|L_j\right|,$$

(131)

which is weaker than (130). Applying the arithmetic-geometric inequality (93) to the definitions of $\Delta L$ and ${\Delta }^{2}L$, one obtains the relations

$${\Delta }^{2}L\ge \sqrt{3}{\left({\Delta }^{3}L\right)}^{2/3}\equiv \sqrt{3}{\mathcal{L}}^{1/3},{(\Delta L)}^3\ge 3\sqrt{3}{\Delta }^{3}L\equiv 3\sqrt{3}{\mathcal{L}}^{1/2}.$$

(132)

For a given value of the principal quantum number j, the quantity $\Delta L$ is minimal in the state $|jj\rangle$, with ${(\Delta L)}_{min}=\hslash \sqrt{j}$. In this case, we have the equality in (130), whereas the right-hand side of (131) is twice smaller.

Various uncertainty relations for the angular momentum operator were extensively studied in papers [16,113,114]. The authors of [113] obtained several inequalities in terms of eigenvalues of the covariance matrix F (named there as “principal variances”). In the most symmetrical form, such inequalities were given in study [114], whose authors considered the uncertainty relations for the SU(2) group, namely, for three Hermitian operators obeying the same commutation relations as angular momentum operators: $\left[{\widehat{S}}_k,{\widehat{S}}_l\right]=i{ϵ}_{klm}{\widehat{S}}_m$ (where ${ϵ}_{klm}$ is the Levi-Civita fully antisymmetric tensor). If ${\lambda }_1$, ${\lambda }_2$, ${\lambda }_3$ are three eigenvalues of the covariance matrix F, and ${\widehat{S}}_0$ is the Casimir operator, the following inequalities hold:

$$0\le {\lambda }_1{\lambda }_2{\lambda }_3\le 〈{\widehat{S}}_0^3{\left({\widehat{S}}_0+2\right)}^3〉/27,$$

(133)

$$〈{\widehat{S}}_0^2〉\le {\lambda }_1{\lambda }_2+{\lambda }_2{\lambda }_3+{\lambda }_3{\lambda }_1\le 〈{\widehat{S}}_0^2{\left({\widehat{S}}_0+2\right)}^2〉/3,$$

(134)

$$2〈{\widehat{S}}_0〉\le {\lambda }_1+{\lambda }_2+{\lambda }_3\le 〈{\widehat{S}}_0\left({\widehat{S}}_0+2\right)〉.$$

(135)

The authors of [16] considered the subspaces with fixed values ${\hslash }^{2}s(s+1)$ of the squared angular momentum operator ${\widehat{\mathbf{L}}}^2$. A lot of inequalities were derived there. One of them, which does not contain the concrete value of parameter s, is as follows (the notation is the same as above, it differs from the original notation in [16]):

$$L_{xx}{L}_{yy}+L_{yy}{L}_{zz}+L_{zz}{L}_{xx}\ge {\displaystyle \frac{{\hslash }^2}{4}}{〈\widehat{\mathbf{L}}〉}^2.$$

(136)

A stronger version of this inequality was also found. If the axes are chosen in such a way that $L_{zz}\ge L_{xx}\ge L_{yy}$, then

$$L_{zz}\left(L_{xx}+L_{yy}\right)\ge {\displaystyle \frac{{\hslash }^2}{4}}{〈\widehat{\mathbf{L}}〉}^2.$$

(137)

6.4. Spin-1/2

In the case of spin-$1/2$ operators, satisfying the anticommutation relation $\{{\widehat{s}}_{\alpha },{\widehat{s}}_{\beta }\}={\delta }_{\alpha \beta }{\hslash }^2/2$, the (co)variances $s_{jk}$ depend on the mean values $s_j$ and $s_k$ only: $s_{jk}={\delta }_{jk}{\hslash }^2/4-s_j{s}_k$, where $s_j\equiv \langle {\widehat{s}}_j\rangle$. Since $s_{jj}\ge 0$, the simplest inequality is $s_j^2\le {\hslash }^2/4$. Summing three such inequalities, one obtains $s_x^2+s_y^2+s_z^2\le 3{\hslash }^2/4$. However, the correct inequality, following from (118), is much stronger:

$$s_x^2+s_y^2+s_z^2\le {\displaystyle \frac{{\hslash }^2}{4}}.$$

(138)

This is a consequence of the noncommutativity of the spin operators.

The inequalities of Section 6.1 in this special case can be expressed in terms of the following symmetrical non-negative combinations of the average values of spin components:

$${\sigma }_2=s_x^2+s_y^2+s_z^2,{\sigma }_3=s_x^2{s}_y^2+s_y^2{s}_z^2+s_z^2{s}_x^2,$$

(139)

$${\sigma }_4=s_x^4+s_y^4+s_z^4,{\sigma }_6={\left(s_x{s}_y{s}_z\right)}^2.$$

(140)

For example, inequality (113) assumes the form $\left({\hslash }^2/4\right){\sigma }_2-{\sigma }_4\ge 2{\sigma }_3$, which is in fact a consequence of (138). The first inequality in (114) can be written as

$${\displaystyle \frac{{\hslash }^4}{16}}-{\displaystyle \frac{{\hslash }^2}{4}}{\sigma }_2\ge {\displaystyle \frac{{\hslash }^2}{4}}{\sigma }_2-{\sigma }_4+2{\sigma }_3\ge 4{\sigma }_3,$$

(141)

so that it can be considered as a generalization of (138). The second inequality (114) leads to the relation

$${\displaystyle \frac{3{\hslash }^2}{4}}{\sigma }_3+{\displaystyle \frac{{\hslash }^4}{16}}{\sigma }_2\ge {\displaystyle \frac{{\hslash }^2}{4}}{\sigma }_4+{\sigma }_6.$$

(142)

Inequality (116) can be rewritten as

$${\displaystyle \frac{3{\hslash }^4}{16}}+2{\sigma }_3+{\sigma }_4\ge {\hslash }^2{\sigma }_2.$$

(143)

The inequality resulting from (115) turns out to be a consequence of inequalities (141) and (143). Inequality (119) becomes

$$\prod _{\left(xyz\right)}\left({\displaystyle \frac{{\hslash }^2}{4}}-s_x^2\right)\ge {\left[\prod _{\left(xyz\right)}\left({\displaystyle \frac{{\hslash }^2}{4}}{s}_x^2+s_y^2{s}_z^2\right)\right]}^{1/2}.$$

(144)

The following inequalities were given in [74] as special cases of inequalities (51)–(53) for three Pauli matrices [remember the connection ${\widehat{s}}_j=(\hslash /2){\sigma }_j$]:

$${(\Delta {\sigma }_x)}^2+{(\Delta {\sigma }_y)}^2+{(\Delta {\sigma }_z)}^2\ge {\displaystyle \frac{1}{3}}{\left[\Delta \left({\sigma }_x+{\sigma }_y+{\sigma }_z\right)\right]}^2+{\displaystyle \frac{1}{9}}{\left[\sum _{1\le i<j\le 3}\Delta \left({\sigma }_i-{\sigma }_j\right)\right]}^2,$$

(145)

$${(\Delta {\sigma }_x)}^2+{(\Delta {\sigma }_y)}^2+{(\Delta {\sigma }_z)}^2\ge {\displaystyle \frac{1}{4}}\sum _{1\le i<j\le 3}{\left[\Delta \left({\sigma }_i-{\sigma }_j\right)\right]}^2,$$

(146)

$${(\Delta {\sigma }_x)}^2+{(\Delta {\sigma }_y)}^2+{(\Delta {\sigma }_z)}^2\ge \sum _{1\le i<j\le 3}{\left[\Delta \left({\sigma }_i+{\sigma }_j\right)\right]}^2-{\displaystyle \frac{1}{4}}{\left[\sum _{1\le i<j\le 3}\Delta \left({\sigma }_i+{\sigma }_j\right)\right]}^2.$$

(147)

6.5. Angular Momentum, Sine and Cosine Operators

Another interesting triple of operators is

$$\widehat{L}\equiv {\widehat{L}}_z=-i\hslash \partial /\partial \phi ,\widehat{C}=cos\phi ,\widehat{S}=sin\phi ,$$

(148)

where $\phi$ is the phase variable. It is well known (see, for example, [115]) that the triple (148) provides one of the simplest solutions to the “phase – angular momentum” problem in quantum mechanics. The commutation relations for the triple (148) are as follows:

$$\left[\widehat{L},\widehat{S}\right]=-ih\widehat{C},\left[\widehat{L},\widehat{C}\right]=ih\widehat{S},\left[\widehat{C},\widehat{S}\right]=0.$$

(149)

An inequality containing operators $\widehat{C}$ and $\widehat{S}$ on an equal footing was given in [115] (see also [30]):

$$\overline{L^2}\left(\overline{C^2}+\overline{S^2}\right)\ge {\displaystyle \frac{{\hslash }^2}{4}}\left[{\left(\overline{C}\right)}^2+{\left(\overline{S}\right)}^2\right].$$

(150)

Since $\overline{C^2}+\overline{S^2}\equiv 1-{\left(\overline{C}\right)}^2-{\left(\overline{S}\right)}^2$, this inequality can be rewritten as

$${\left(\overline{C}\right)}^2+{\left(\overline{S}\right)}^2\le {\displaystyle \frac{\overline{L^2}}{\overline{L^2}+{\hslash }^2/4}}.$$

(151)

The consequence of (89) is the inequality

$$\begin{array}{ccc}\hfill \overline{L^2}·\overline{C^2}·\overline{S^2}& \ge & \overline{C^2}{\left(\overline{LS}\right)}^2+\overline{S^2}{\left(\overline{LC}\right)}^2+\overline{L^2}{\left(\overline{CS}\right)}^2-2\overline{LS}·\overline{SC}·\overline{CL}\hfill \\ & & +{\displaystyle \frac{{\hslash }^2}{4}}\left[\overline{C^2}{\left(\overline{C}\right)}^2+\overline{S^2}{\left(\overline{S}\right)}^2+2\overline{CS}·\overline{C}·\overline{S}\right]\ge 0,\hfill \end{array}$$

(152)

whereas (91) and (94) yield, respectively,

$$3\overline{L^2}·\overline{C^2}·\overline{S^2}\ge \overline{C^2}{\left(\overline{LS}\right)}^2+\overline{S^2}{\left(\overline{LC}\right)}^2+\overline{L^2}{\left(\overline{CS}\right)}^2+{\displaystyle \frac{{\hslash }^2}{4}}\left[\overline{C^2}{\left(\overline{C}\right)}^2+\overline{S^2}{\left(\overline{S}\right)}^2\right],$$

(153)

$$\overline{L^2}·\overline{C^2}·\overline{S^2}\ge \left|\overline{CS}\right|{\left\{\left[{\left(\overline{LS}\right)}^2+{\displaystyle \frac{{\hslash }^2}{4}}{\left(\overline{C}\right)}^2\right]\left[{\left(\overline{LC}\right)}^2+{\displaystyle \frac{{\hslash }^2}{4}}{\left(\overline{S}\right)}^2\right]\right\}}^{1/2}.$$

(154)

Inequality (111) yields

$$\overline{L^2}·\overline{C^2}·\overline{S^2}\ge {\displaystyle \frac{4}{9}}{\hslash }^2\left[\overline{C^2}{\left(\overline{C}\right)}^2+\overline{S^2}{\left(\overline{S}\right)}^2\right].$$

(155)

Introducing the “exponential phase operator” $\widehat{E}=\widehat{C}-i\widehat{S}$, one can express the inequality (4) in the form

$${(\Delta L)}^2\ge {\displaystyle \frac{{\hslash }^2{\left|\langle \widehat{E}\rangle \right|}^2}{4(1-|\langle \widehat{E}\rangle {|}^2)}}.$$

(156)

Some ways of improving inequality (156) in terms of the trace and determinant of the covariance matrix of operators $\widehat{C}$ and $\widehat{S}$ were considered in paper [116]. Inequalities for the sign and cosine operators were also studied in Refs. [117,118].

6.6. Bi-Products of the Coordinate and Momentum in One Dimension

Now let us return to the coordinate and momentum operators in one dimension. In this case, we can introduce three operators,

$${\widehat{R}}_1={\left(\delta \widehat{p}\right)}^2,{\widehat{R}}_2={\left(\delta \widehat{x}\right)}^2,{\widehat{R}}_3={\displaystyle \frac{1}{2}}\left(\delta \widehat{p}\delta \widehat{x}+\delta \widehat{x}\delta \widehat{p}\right),$$

(157)

where $\delta \widehat{p}=\widehat{p}-\langle \widehat{p}\rangle$ and $\delta \widehat{x}=\widehat{x}-\langle \widehat{x}\rangle$. The commutation relations for the triple (157) are

$$\left[{\widehat{R}}_1,{\widehat{R}}_2\right]=-4i\hslash {\widehat{R}}_3,\left[{\widehat{R}}_2,{\widehat{R}}_3\right]=2i\hslash {\widehat{R}}_2,\left[{\widehat{R}}_1,{\widehat{R}}_3\right]=-2i\hslash {\widehat{R}}_1,$$

(158)

so that matrix F has the form

$$F=∥\begin{array}{ccc}{R}_{11}& R_{12}-2i\hslash R_3& R_{13}-i\hslash R_1\\ R_{12}+2i\hslash R_3& R_{22}& R_{23}+i\hslash R_2\\ R_{13}+i\hslash R_1& R_{23}-i\hslash R_2& R_{33}\end{array}∥.$$

(159)

The non-negativeness of the second-order principal minors of matrix F results in the following inequalities:

$$R_{11}{R}_{22}-R_{12}^2\ge 4{\hslash }^2{R}_3^2,R_{11}{R}_{33}-R_{13}^2\ge {\hslash }^2{R}_1^2,R_{22}{R}_{33}-R_{23}^2\ge {\hslash }^2{R}_2^2.$$

(160)

The condition $detF\ge 0$ yields the inequality

$$\begin{array}{ccc}& & R_{11}{R}_{22}{R}_{33}+2R_{12}{R}_{23}{R}_{31}+{\hslash }^2\left(4R_{13}{R}_2{R}_3+4R_{23}{R}_1{R}_3-2R_{12}{R}_1{R}_2\right)\hfill \\ & & -R_{11}\left(R_{23}^2+{\hslash }^2{R}_2^2\right)-R_{22}\left(R_{13}^2+{\hslash }^2{R}_1^2\right)-R_{33}\left(R_{12}^2+4{\hslash }^2{R}_3^2\right)\ge 0.\hfill \end{array}$$

(161)

Similar inequalities for three Hermitian operators, which are generators of the Lie algebra $su(1,1)$, can be found also in [56], where they were illustrated in examples of the SU(1,1) coherent states.

The first inequality in (160) can be represented in a more explicit form if one uses the identity

$$R_{12}-R_{33}\equiv R_3^2-R_1{R}_2-{\displaystyle \frac{3}{4}}{\hslash }^2$$

(162)

and the inequality $R_1{R}_2-R_3^2\ge {\hslash }^2/4$ [coinciding with (8)]. Thus, we get the following chain of inequalities for the double product of the fourth-order “uncertainties”:

$$\begin{array}{ccc}\hfill \left[\overline{{\left(\delta p\right)}^4}-{\sigma }_{pp}^2\right]\left[\overline{{\left(\delta x\right)}^4}-{\sigma }_{xx}^2\right]& \ge & {\hslash }^4+4{\hslash }^2{\sigma }_{px}^2+D_{px}^2-2D_{px}\left({\Delta }_{px}+{\displaystyle \frac{3}{4}}{\hslash }^2\right)\hfill \\ & \ge & {\hslash }^4+6{\hslash }^2{\sigma }_{px}^2+D_{px}^2-2\overline{{\left(\delta p\delta x\right)}^2}\left({\Delta }_{px}+{\displaystyle \frac{3}{4}}{\hslash }^2\right)\hfill \\ & \ge & {\hslash }^4+6{\hslash }^2{\sigma }_{px}^2-2\overline{{\left(\delta p\delta x\right)}^2}\left({\sigma }_{pp}{\sigma }_{xx}+{\displaystyle \frac{3}{4}}{\hslash }^2\right),\hfill \end{array}$$

(163)

where the following short notations are used:

$$\overline{{\left(\delta p\delta x\right)}^2}={\displaystyle \frac{1}{4}}〈{\left[\left(\delta \widehat{p}\delta \widehat{x}+\delta \widehat{x}\delta \widehat{p}\right)\right]}^2〉,D_{px}=\overline{{\left(\delta p\delta x\right)}^2}-{\sigma }_{px}^2,{\Delta }_{px}={\sigma }_{pp}{\sigma }_{xx}-{\sigma }_{px}^2.$$

(164)

Equations (111) and (112) for the triple products assume the following forms:

$$\begin{array}{ccc}& & \left[\langle {\left(\delta p\right)}^4\rangle -{\sigma }_{pp}^2\right]\left[\langle {\left(\delta x\right)}^4\rangle -{\sigma }_{xx}^2\right]\left[\overline{{\left(\delta p\delta x\right)}^2}-{\sigma }_{px}^2\right]\hfill \\ & & \ge {\displaystyle \frac{4}{9}}{\hslash }^2\left\{\left[\langle {\left(\delta p\right)}^4\rangle -{\sigma }_{pp}^2\right]{\sigma }_{xx}^2+\left[\langle {\left(\delta x\right)}^4\rangle -{\sigma }_{xx}^2\right]{\sigma }_{pp}^2+4\left[\overline{{\left(\delta p\delta x\right)}^2}-{\sigma }_{px}^2\right]{\sigma }_{px}^2\right\},\hfill \end{array}$$

(165)

$$\left[\overline{{\left(\delta p\right)}^4}-{\sigma }_{pp}^2\right]\left[\overline{{\left(\delta x\right)}^4}-{\sigma }_{xx}^2\right]\left[\overline{{\left(\delta p\delta x\right)}^2}-{\sigma }_{px}^2\right]\ge {\displaystyle \frac{16{\hslash }^3}{3\sqrt{3}}}{\sigma }_{pp}{\sigma }_{xx}\left|{\sigma }_{px}\right|.$$

(166)

Inequalities containing higher-order statistical moments (up to the sixth order) were considered in papers [119,120,121].

6.7. Fourth-Order Moments in the Gaussian States

The fourth-order statistical moments contained in Equations (163), (165), and (166) can be expressed in terms of covariances in the special case of Gaussian states. For these states, it is known that mean values of symmetrical (or Wigner–Weyl) products [122] of four operators $\widehat{A}$, $\widehat{B}$, $\widehat{C}$ and $\widehat{D}$ (with zero mean values) can be calculated as sums of pair products of their covariances [123]:

$$\begin{array}{ccc}\hfill {\langle ABCD\rangle }_W& \equiv & \int W\left(x,p\right)ABCDdxdp/(2\pi \hslash )\hfill \\ & =& \overline{AB}·\overline{CD}+\overline{AC}·\overline{BD}+\overline{AD}·\overline{BC}.\hfill \end{array}$$

(167)

Here, $A,B,C,D$ can be any of the variables x and p. The symbol ${\langle ABCD\rangle }_W$ means the quantum mechanical mean value of the sum of all different products of operators $\widehat{A},\widehat{B},\widehat{C},\widehat{D}$, taken in all possible orders, divided by the number of terms in the sum. Mean values of concrete products of operators in predefined orders can be expressed in terms of symmetrical mean values with the aid of commutation relations.

The immediate consequences of Equation (167) are the well-known formulas for the fourth-order mean values of the canonical coordinates in the Gaussian states,

$$\langle {\left(\delta x\right)}^4\rangle =3{\sigma }_{xx}^2,\langle {\left(\delta p\right)}^4\rangle =3{\sigma }_{pp}^2.$$

(168)

The following relations hold for the mean values of cross-products (for brevity, we write here x and p instead of $\delta x$ and $\delta p$):

$${\langle x^2{p}^2\rangle }_W={\displaystyle \frac{1}{6}}〈{\widehat{x}}^2{\widehat{p}}^2+{\widehat{p}}^2{\widehat{x}}^2+\widehat{x}\widehat{p}\widehat{x}\widehat{p}+\widehat{p}\widehat{x}\widehat{p}\widehat{x}+\widehat{x}{\widehat{p}}^2\widehat{x}+\widehat{p}{\widehat{x}}^2\widehat{p}〉={\displaystyle \frac{1}{2}}〈{\widehat{x}}^2{\widehat{p}}^2+{\widehat{p}}^2{\widehat{x}}^2〉+{\displaystyle \frac{1}{2}}{\hslash }^2,$$

$$\overline{{\left(\delta p\delta x\right)}^2}={\displaystyle \frac{1}{2}}〈{\left(\delta \widehat{p}\right)}^2{\left(\delta \widehat{x}\right)}^2+{\left(\delta \widehat{x}\right)}^2{\left(\delta \widehat{p}\right)}^2〉+{\displaystyle \frac{3}{4}}{\hslash }^2={\langle {\left(\delta x\right)}^2{\left(\delta p\right)}^2\rangle }_W+{\displaystyle \frac{1}{4}}{\hslash }^2.$$

For the Gaussian states, Equation (167) yields

$${\langle {\left(\delta x\right)}^2{\left(\delta p\right)}^2\rangle }_W={\sigma }_{xx}{\sigma }_{pp}+2{\sigma }_{xp}^2$$

However, inequalities (163), (165) and (166) remain rather complicated even for the Gaussian states. If ${\sigma }_{xp}=0$, inequality (163) can be written as follows,

$$3{\left({\sigma }_{xx}{\sigma }_{pp}\right)}^2+{\hslash }^2{\sigma }_{xx}{\sigma }_{pp}\ge {\displaystyle \frac{5}{16}}{\hslash }^4.$$

(169)

Since ${\sigma }_{xx}{\sigma }_{pp}\ge {\hslash }^2/4$, the left-hand side of (169) cannot be smaller than $7{\hslash }^4/16$. Similarly, inequality (165) with ${\sigma }_{xp}=0$ can be reduced to the weak inequality ${\sigma }_{xx}{\sigma }_{pp}\ge 7{\hslash }^2/36$. These observations indicate that the minimal product of the fourth-order moments $\langle {\left(\delta x\right)}^4\rangle \langle {\left(\delta p\right)}^4\rangle$ cannot be achieved in the Gaussian states: see Section 12.

7. Four and More Operators

The simplest inequality for four Hermitian operators is Robertson’s inequality (17)

$$\begin{array}{ccc}\hfill X_{11}{X}_{22}{X}_{33}{X}_{44}\ge det\left(Y\right)& =& Y_{12}^2{Y}_{34}^2+Y_{13}^2{Y}_{24}^2+Y_{14}^2{Y}_{23}^2\hfill \\ & -& 2\left(Y_{12}{Y}_{13}{Y}_{24}{Y}_{34}+Y_{12}{Y}_{14}{Y}_{23}{Y}_{43}+Y_{13}{Y}_{14}{Y}_{32}{Y}_{42}\right).\hfill \end{array}$$

(170)

The most complete inequality, containing all possible covariances and mean values of commutators, is inequality (14) for the determinant of matrix $F=X+iY$. However, the explicit form of this inequality contains a huge number of terms (all four-component products of 10 different members of matrix X and 6 different members of matrix Y), that presenting them here can hardly be justified (see, e.g., Ref. [56]). Instead of this, let us consider a few special cases.

7.1. Four Quadratures

An important special case is the set of two coordinate and two momentum operators: ${\widehat{z}}_1={\widehat{x}}_1$, ${\widehat{z}}_2={\widehat{x}}_2$, ${\widehat{z}}_3={\widehat{p}}_1$, ${\widehat{z}}_4={\widehat{p}}_2$. Then, the only nonzero mean values of commutators are $Y_{13}=Y_{24}=\hslash /2$ (and their antisymmetric partners), so that $detY={\hslash }^4/16$. The most complete inequality in this case is (22) with $n=2$. The explicit expression for the function $det\left(Q\right)\equiv det\left(X\right)$ was found in Refs. [86,124] (the notation (11) is used here and in other formulas):

$$\begin{array}{ccc}\hfill det\left(Q\right)& =& \left[\left(\overline{p_1^2}\right)\left(\overline{x_1^2}\right)-{\left(\overline{p_1{x}_1}\right)}^2\right]\left[\left(\overline{p_2^2}\right)\left(\overline{x_2^2}\right)-{\left(\overline{p_2{x}_2}\right)}^2\right]\hfill \\ & & +{\left(\overline{p_1{p}_2}\right)}^2{\left(\overline{x_1{x}_2}\right)}^2+{\left(\overline{p_1{x}_2}\right)}^2{\left(\overline{p_2{x}_1}\right)}^2-{\left(\overline{p_1{p}_2}\right)}^2\left(\overline{x_1^2}\right)\left(\overline{x_2^2}\right)\hfill \\ & & -{\left(\overline{x_1{x}_2}\right)}^2\left(\overline{p_1^2}\right)\left(\overline{p_2^2}\right)-{\left(\overline{p_2{x}_1}\right)}^2\left(\overline{p_1^2}\right)\left(\overline{x_2^2}\right)-{\left(\overline{p_1{x}_2}\right)}^2\left(\overline{p_2^2}\right)\left(\overline{x_1^2}\right)\hfill \\ & & +2\left(\overline{p_1{x}_1}\right)\left[\left(\overline{p_1{x}_2}\right)\left(\overline{x_1{x}_2}\right)\left(\overline{p_2^2}\right)+\left(\overline{p_2{x}_1}\right)\left(\overline{p_1{p}_2}\right)\left(\overline{x_2^2}\right)-\left(\overline{p_2{x}_2}\right)\left(\overline{p_1{x}_2}\right)\left(\overline{p_2{x}_1}\right)\right]\hfill \\ & & +2\left(\overline{p_2{x}_2}\right)\left[\left(\overline{p_1{x}_2}\right)\left(\overline{p_1{p}_2}\right)\left(\overline{x_1^2}\right)+\left(\overline{p_2{x}_1}\right)\left(\overline{x_1{x}_2}\right)\left(\overline{p_1^2}\right)-\left(\overline{p_1{x}_1}\right)\left(\overline{p_1{p}_2}\right)\left(\overline{x_1{x}_2}\right)\right]\hfill \\ & & -2\left(\overline{p_1{x}_2}\right)\left(\overline{p_2{x}_1}\right)\left(\overline{p_1{p}_2}\right)\left(\overline{x_1{x}_2}\right)\ge {\hslash }^4/16.\hfill \end{array}$$

(171)

Inequality (66) ${\mathcal{D}}_2^{\left(2\right)}\ge {\hslash }^2/2$ assumes the following explicit form

$$D_2^{\left(2\right)}\equiv \overline{p_x^2}\overline{x^2}-{\left(\overline{x{p}_x}\right)}^2+\overline{p_y^2}\overline{y^2}-{\left(\overline{y{p}_y}\right)}^2+2\overline{xy}\overline{p_x{p}_y}-2\overline{x{p}_y}\overline{y{p}_x}\ge {\hslash }^2/2.$$

(172)

Therefore, Equation (14) gives the inequality which is stronger than (171):

$$detQ\ge {\displaystyle \frac{{\hslash }^2}{4}}\left(D_2^{\left(2\right)}-{\displaystyle \frac{{\hslash }^2}{4}}\right).$$

(173)

An equivalent (but very complicated in the explicit form) inequality, expressing the condition ${\kappa }_j\ge \hslash /2$ [see (55)] for $n=2$, was found in [125]. The non-negativeness of the third-order principal minors of the matrix F leads to the inequality

$$\begin{array}{ccc}& & \left(\overline{p_2^2}\right)\left[\left(\overline{p_1^2}\right)\left(\overline{x_1^2}\right)-{\left(\overline{p_1{x}_1}\right)}^2-{\displaystyle \frac{{\hslash }^2}{4}}\right]+2\left(\overline{p_1{p}_2}\right)\left(\overline{p_2{x}_1}\right)\left(\overline{x_1{p}_1}\right)\hfill \\ & & -{\left(\overline{p_1{p}_2}\right)}^2\left(\overline{x_1^2}\right)-{\left(\overline{p_2{x}_1}\right)}^2\left(\overline{p_1^2}\right)\ge 0.\hfill \end{array}$$

(174)

Three other inequalities can be obtained from (174) through the substitution of the indexes $1\leftrightarrow 2$ or the variables $p\leftrightarrow x$. Several equivalent inequalities can be found in [56,119,121].

R. Simon has derived the following inequality for $2\times 2$ covariance matrices [126]:

$$\begin{array}{ccc}& & det\left({\mathcal{M}}_{11}\right)det\left({\mathcal{M}}_{22}\right)+{\left[det\left({\mathcal{M}}_{12}\right)-{\hslash }^2/4\right]}^2-\mathrm{T}\mathrm{r}\left({\mathcal{M}}_{21}\mathrm{\Sigma }{\mathcal{M}}_{11}\mathrm{\Sigma }{\mathcal{M}}_{12}\mathrm{\Sigma }{\mathcal{M}}_{22}\mathrm{\Sigma }\right)\hfill \\ & & \ge {\hslash }^2\left[det\left({\mathcal{M}}_{11}\right)+det\left({\mathcal{M}}_{22}\right)\right]/4,\hfill \end{array}$$

(175)

$${\mathcal{M}}_{jk}=∥\begin{array}{cc}\overline{x_j{x}_k}& \overline{x_j{p}_k}\\ \overline{p_j{x}_k}& \overline{p_j{p}_k}\end{array}∥,\mathrm{\Sigma }=∥\begin{array}{cc}0& 1\\ -1& 0\end{array}∥.$$

An equivalent inequality containing only determinants of matrices has the form [127]

$$detQ\ge {\hslash }^2\left[det\left({\mathcal{M}}_{11}\right)+det\left({\mathcal{M}}_{22}\right)\right]/4+{\hslash }^{2}det\left({\mathcal{M}}_{12}\right)/2-{\hslash }^4/16.$$

(176)

One can check that inequality (176) coincides exactly with (173). The symplectic eigenvalues of the full $4\times 4$ covariance matrix Q can be expressed in terms of of the universal invariants ${\mathcal{D}}_0$ and ${\mathcal{D}}_2$ as follows [128],

$$2|{\kappa }_{1,2}|=\sqrt{{\mathcal{D}}_2+2\sqrt{{\mathcal{D}}_0}}\pm \sqrt{{\mathcal{D}}_2-2\sqrt{{\mathcal{D}}_0}}.$$

(177)

An inequality containing only two “internal” covariances was derived in [18]:

$$\left(\overline{p_1^2}\right)\left(\overline{x_1^2}\right)\left(\overline{p_2^2}\right)\left(\overline{x_2^2}\right)-{\left(\overline{p_1{x}_1}\right)}^2{\left(\overline{p_2{x}_2}\right)}^2\ge {\hslash }^4/16.$$

(178)

Two inequalities without covariances, derived in [18], look unexpected at first glance:

$$a{\left[\left(\overline{p_1^2}\right)\left(\overline{x_2^2}\right)\right]}^n+b{\left[\left(\overline{p_2^2}\right)\left(\overline{x_1^2}\right)\right]}^n\ge 2\sqrt{ab}{(\hslash /2)}^{2n},a,b>0,$$

(179)

$$\Delta p_1\Delta x_2+\Delta p_2\Delta x_1\ge \hslash .$$

(180)

However, they are consequences of the Heisenberg relation (1) and the inequality $x+y\ge 2\sqrt{xy}$. A nontrivial inequality derived in [18] has the form

$$a\left[\left(\overline{p_1^2}\right)+\left(\overline{x_1^2}\right)\right]+b\left[\left(\overline{p_2^2}\right)+\left(\overline{x_2^2}\right)\right]+c\left[\left(\overline{p_1{p}_2}\right)-\left(\overline{x_1{x}_2}\right)\right]\ge \hslash \sqrt{{(a+b)}^2-c^2},$$

(181)

provided $a,b>0$ and $4ab>c^2$.

For the two-dimensional system of boson annihilation and creation operators, ${\widehat{a}}_1,{\widehat{a}}_2,{\widehat{a}}_1^{†},{\widehat{a}}_2^{†}$, the condition $detF\ge 0$ assumes the following explicit form:

$$\begin{array}{ccc}& & \left({\mathcal{N}}_1{\mathcal{N}}_2-{\left|\overline{a_1^{†}{a}_2}\right|}^2\right)\left({\tilde{\mathcal{N}}}_1{\tilde{\mathcal{N}}}_2-{\left|\overline{a_1^{†}{a}_2}\right|}^2\right)-{\left|\overline{a_1{a}_2}\right|}^2\left({\mathcal{N}}_1{\tilde{\mathcal{N}}}_2+{\mathcal{N}}_2{\tilde{\mathcal{N}}}_1\right)\hfill \\ & & -{\left|\overline{a_1^2}\right|}^2{\mathcal{N}}_2{\tilde{\mathcal{N}}}_2-{\left|\overline{a_2^2}\right|}^2{\mathcal{N}}_1{\tilde{\mathcal{N}}}_1+{\left|{\left(\overline{a_1^2}\right)}^2{\left(\overline{a_2^2}\right)}^2-{\left(\overline{a_1{a}_2}\right)}^2\right|}^2\hfill \\ & & +2Re[\left(2{\mathcal{N}}_1+1\right){\left(\overline{a_2^2}\right)}^{*}\left(\overline{a_1{a}_2}\right)\left(\overline{a_1^{†}{a}_2}\right)+\left(2{\mathcal{N}}_2+1\right){\left(\overline{a_1^2}\right)}^{*}\left(\overline{a_1{a}_2}\right)\left(\overline{a_2^{†}{a}_1}\right)\hfill \\ & & -\left(\overline{a_1^2}\right){\left(\overline{a_2^2}\right)}^{*}{\left(\overline{a_1^{†}{a}_2}\right)}^2-{\left|\overline{a_1{a}_2}\right|}^2\left(\overline{a_1^{†}{a}_2}\right)\left(\overline{a_2^{†}{a}_1}\right)]\ge 0,\hfill \end{array}$$

(182)

where

$${\mathcal{N}}_j=〈{\widehat{a}}_j^{†}{\widehat{a}}_j〉-{\left|〈{\widehat{a}}_j〉\right|}^2,{\tilde{\mathcal{N}}}_j=1+{\mathcal{N}}_j.$$

(183)

The non-negativeness of the second-order principal minors, besides inequalities like (73), leads to the inequalities

$${\mathcal{N}}_1{\mathcal{N}}_2-\left(\overline{a_1^{†}{a}_2}\right)\left(\overline{a_2^{†}{a}_1}\right)\ge 0,{\mathcal{N}}_1{\tilde{\mathcal{N}}}_2-{\left|\overline{a_1{a}_2}\right|}^2\ge 0,$$

(184)

while the non-negativeness of the third-order principal minors results in relations

$${\mathcal{N}}_1{\tilde{\mathcal{N}}}_1{\mathcal{N}}_2-{\mathcal{N}}_1{\left|\overline{a_1{a}_2}\right|}^2-{\tilde{\mathcal{N}}}_1{\left|\overline{a_1^{†}{a}_2}\right|}^2-{\mathcal{N}}_2{\left|\overline{a_1^2}\right|}^2+2Re\left[\left(\overline{a_1^2}\right){\left(\overline{a_1{a}_2}\right)}^{*}\left(\overline{a_1^{†}{a}_2}\right)\right]\ge 0,$$

(185)

$${\mathcal{N}}_1{\tilde{\mathcal{N}}}_1{\tilde{\mathcal{N}}}_2-{\tilde{\mathcal{N}}}_1{\left|\overline{a_1{a}_2}\right|}^2-{\mathcal{N}}_1{\left|\overline{a_1^{†}{a}_2}\right|}^2-{\tilde{\mathcal{N}}}_2{\left|\overline{a_1^2}\right|}^2+2Re\left[\left(\overline{a_1^2}\right){\left(\overline{a_1{a}_2}\right)}^{*}\left(\overline{a_1^{†}{a}_2}\right)\right]\ge 0.$$

(186)

Relations (182)–(186) contain the centered second order moments, defined as in (11). However, they obviously remain valid if one puts formally $\langle \widehat{f}\rangle =\langle \widehat{g}\rangle =0$ in the definition (11).

In the case of fermion (anticommuting) annihilation and creation operators, ${\widehat{b}}_1,{\widehat{b}}_2,{\widehat{b}}_1^{†},{\widehat{b}}_2^{†}$, the inequalities analogous to (182)–(186) are as follows:

$$\begin{array}{ccc}& & \left({\mathcal{F}}_1{\mathcal{F}}_2-{\left|\langle {\widehat{b}}_1^{†}{\widehat{b}}_2\rangle \right|}^2\right)\left({\tilde{\mathcal{F}}}_1{\tilde{\mathcal{F}}}_2-{\left|\langle {\widehat{b}}_1^{†}{\widehat{b}}_2\rangle \right|}^2\right)+{\left|\langle {\widehat{b}}_1{\widehat{b}}_2\rangle \right|}^2\left({\mathcal{F}}_2-{\mathcal{F}}_1\right)\hfill \\ & & +{\left|\langle {\widehat{b}}_1{\widehat{b}}_2\rangle \right|}^4-2{\left|\langle {\widehat{b}}_1{\widehat{b}}_2\rangle \right|}^2{\left|\langle {\widehat{b}}_1^{†}{\widehat{b}}_2\rangle \right|}^2\ge 0,\hfill \end{array}$$

(187)

$${\mathcal{F}}_j=〈{\widehat{b}}_j^{†}{\widehat{b}}_j〉,{\tilde{\mathcal{F}}}_j=1-{\mathcal{F}}_j,$$

(188)

$${\mathcal{F}}_1{\mathcal{F}}_2-{\left|〈{\widehat{b}}_1^{†}{\widehat{b}}_2〉\right|}^2\ge 0,{\mathcal{F}}_1{\tilde{\mathcal{F}}}_2-{\left|〈{\widehat{b}}_1{\widehat{b}}_2〉\right|}^2\ge 0,$$

(189)

$${\mathcal{F}}_1{\tilde{\mathcal{F}}}_1{\mathcal{F}}_2-{\mathcal{F}}_1{\left|〈{\widehat{b}}_1{\widehat{b}}_2〉\right|}^2-{\tilde{\mathcal{F}}}_1{\left|〈{\widehat{b}}_1^{†}{\widehat{b}}_2〉\right|}^2\ge 0,$$

(190)

$${\mathcal{F}}_1{\tilde{\mathcal{F}}}_1{\tilde{\mathcal{F}}}_2-{\mathcal{F}}_1{\left|〈{\widehat{b}}_1^{†}{\widehat{b}}_2〉\right|}^2-{\tilde{\mathcal{F}}}_1{\left|〈{\widehat{b}}_1{\widehat{b}}_2〉\right|}^2\ge 0.$$

(191)

These inequalities have a more simple form than (182)–(186), because they contain only noncentered moments like $\langle \widehat{f}\widehat{g}\rangle$. This happens because many average values of the fermionic operators, such as $\langle {\widehat{b}}^2\rangle$, vanish due to the anticommutation relations.

7.2. Inequalities Without Covariances for Four Operators

A scheme used in Section 5.2 can be generalized to sets of four arbitrary Hermitian operators, if one replaces three $2\times 2$ Pauli’s matrices ${\sigma }_k$ with four Hermitian $4\times 4$ Dirac’s matrices. Let us choose these matrices in the form [106]

$${\gamma }_k=∥\begin{array}{cc}0& {\sigma }_k\\ {\sigma }_k& 0\end{array}∥,k=1,2,3,{\gamma }_4=∥\begin{array}{cc}{I}_2& 0\\ 0& -I_2\end{array}∥,$$

where $I_n$ is the $n\times n$ unit matrix. The following anticommutation and commutation relations hold:

$${\gamma }_j^2=I_4,{\gamma }_m{\gamma }_n+{\gamma }_n{\gamma }_m=0,m\ne n,j,m,n=1,2,3,4,$$

(192)

$${\gamma }_1{\gamma }_2-{\gamma }_2{\gamma }_1=2i∥\begin{array}{cc}{\sigma }_3& 0\\ 0& {\sigma }_3\end{array}∥,{\gamma }_2{\gamma }_3-{\gamma }_3{\gamma }_2=2i∥\begin{array}{cc}{\sigma }_1& 0\\ 0& {\sigma }_1\end{array}∥,$$

$${\gamma }_3{\gamma }_1-{\gamma }_1{\gamma }_3=2i∥\begin{array}{cc}{\sigma }_2& 0\\ 0& {\sigma }_2\end{array}∥,{\gamma }_k{\gamma }_4-{\gamma }_4{\gamma }_k=2∥\begin{array}{cc}0& -{\sigma }_k\\ {\sigma }_k& 0\end{array}∥,k=1,2,3.$$

Let us consider the operator $\widehat{f}={\sum }_{k=1}^4{\xi }_k{\widehat{z}}_k{\gamma }_k$, where ${\xi }_k$ are arbitrary real coefficients and ${\widehat{z}}_k$ arbitrary Hermitian operators. Then, the condition $\langle {\widehat{f}}^{†}\widehat{f}\rangle \ge 0$ can be written as the condition of positive semi-definiteness of the Hermitian matrix $F=∥\begin{array}{cc}A& B^{†}\\ B& A\end{array}∥$, where

$$A=g_{\xi }{I}_2-2\sum _{\left(123\right)}{\xi }_1{\xi }_2{Y}_{12}{\sigma }_3=∥\begin{array}{cc}{g}_{\xi }-2{\xi }_1{\xi }_2{Y}_{12}& -2{\xi }_2{\xi }_3{Y}_{23}+2i{\xi }_3{\xi }_1{Y}_{31}\\ -2{\xi }_2{\xi }_3{Y}_{23}-2i{\xi }_3{\xi }_1{Y}_{31}& g_{\xi }+2{\xi }_1{\xi }_2{Y}_{12}\end{array}∥,$$

$$g_{\xi }=\sum _{k=1}^4{\xi }_k^2{X}_{kk},$$

$$B=2i\sum _{j=1}^3{\xi }_j{\xi }_4{Y}_{j4}{\sigma }_j=2∥\begin{array}{cc}i{\xi }_3{\xi }_4{Y}_{34}& i{\xi }_1{\xi }_4{Y}_{14}+{\xi }_2{\xi }_4{Y}_{24}\\ i{\xi }_1{\xi }_4{Y}_{14}-{\xi }_2{\xi }_4{Y}_{24}& -i{\xi }_3{\xi }_4{Y}_{34}\end{array}∥.$$

The covariances $X_{jk}$ with $j\ne k$ do not appear due to the anticommutation relations (192). The positivity condition containing all variances $X_{kk}$ and mean values of commutators $Y_{jk}$ is $detF\ge 0$. After some algebra, it can be written in the following compact form:

$$detF={\left(g_{\xi }^2-4V_{\xi }\right)}^2-64{\left({\xi }_1{\xi }_2{\xi }_3{\xi }_4\right)}^2{\mathrm{\Lambda }}^2\ge 0,$$

(193)

$$V_{\xi }=\sum _{j<k}{\left({\xi }_j{\xi }_k{Y}_{jk}\right)}^2,\mathrm{\Lambda }=\left|Y_{12}{Y}_{34}+Y_{23}{Y}_{14}+Y_{31}{Y}_{24}\right|.$$

(194)

Note that $\mathrm{\Lambda }$ is invariant with respect to the ordering of indexes, due to the property $Y_{jk}=-Y_{kj}$.

Taking all ${\xi }_k=1$ (this means that the dimensions of operators ${\widehat{z}}_k$ should be made equal by means of some scaling transformations), we get the inequality

$$\left|{\left(\sum _{k=1}^4{X}_{kk}\right)}^2-4\sum _{j<k}{Y}_{jk}^2\right|\ge 8\left|Y_{12}{Y}_{34}+Y_{23}{Y}_{14}+Y_{31}{Y}_{24}\right|.$$

(195)

Its simplified (weaker) version is

$${\left(\sum _{k=1}^4{X}_{kk}\right)}^2\ge 4\sum _{j<k}{Y}_{jk}^2.$$

(196)

For the (renormalized) coordinate and momentum operators, inequality (195) assumes the form

$${\left[{\left(\overline{x_1^2}+\overline{p_1^2}+\overline{x_2^2}+\overline{p_2^2}\right)}^2-2{\hslash }^2\right]}^2\ge 4{\hslash }^4.$$

However, this result is a simple consequence of the standard uncertainty relation (3).

Choosing

$${\xi }_1^2=X_{22}{X}_{33}{X}_{44},{\xi }_2^2=X_{11}{X}_{33}{X}_{44},{\xi }_3^2=X_{22}{X}_{11}{X}_{44},{\xi }_4^2=X_{22}{X}_{11}{X}_{33},$$

one can transform inequality (193) to the form

$${\left(4P-\mathrm{\Psi }\right)}^2\ge 4P{\mathrm{\Lambda }}^2,P=X_{11}{X}_{22}{X}_{33}{X}_{44},$$

(197)

$$\mathrm{\Psi }=Y_{12}^2{X}_{33}{X}_{44}+Y_{13}^2{X}_{22}{X}_{44}+Y_{14}^2{X}_{33}{X}_{22}+Y_{23}^2{X}_{11}{X}_{44}+Y_{24}^2{X}_{33}{X}_{11}+Y_{34}^2{X}_{11}{X}_{22}.$$

(198)

Resolving inequality (197) with respect to variable P, one gets the inequality

$$8P\ge 2\mathrm{\Psi }+{\mathrm{\Lambda }}^2+\mathrm{\Lambda }\sqrt{4\mathrm{\Psi }+{\mathrm{\Lambda }}^2}.$$

(199)

(Another solution, $8P\le 2\mathrm{\Psi }+{\mathrm{\Lambda }}^2-\mathrm{\Lambda }\sqrt{4\mathrm{\Psi }+{\mathrm{\Lambda }}^2}$, is unphysical). In the case of coordinate and momentum operators, we have $\mathrm{\Psi }={\hslash }^2\left(\overline{x_1^2}\overline{p_1^2}+\overline{x_2^2}\overline{p_2^2}\right)/4$ and $\mathrm{\Lambda }={\hslash }^2/4$, so that (199) turns into the equality for all minimum uncertainty states (with $\overline{x_1^2}\overline{p_1^2}=\overline{x_2^2}\overline{p_2^2}={\hslash }^2/4$). Note that

$$\mathrm{\Psi }\ge {\mathrm{\Psi }}_{*}=2\left(Y_{12}^2{Y}_{34}^2+Y_{23}^2{Y}_{14}^2+Y_{31}^2{Y}_{24}^2\right).$$

(200)

Therefore, one can get rid of variances in the right-hand side of (199), replacing $\mathrm{\Psi }$ with ${\mathrm{\Psi }}_{*}$.

7.3. Five Operators

Several inequalities for five operators were derived in Ref. [87]. The scheme was the same as in the preceding subsection, but the set of four gamma-matrices was extended by adding the matrix ${\gamma }_5={\gamma }_1{\gamma }_2{\gamma }_3{\gamma }_4$. The simplest result was the inequality

$$\sum _{k=1}^5{X}_{kk}\ge 2{\left[\sum _{j<k}{Y}_{jk}^2\right]}^{1/2}.$$

(201)

More strong (and more involved) inequalities have the form

$${\left(\sum _{k=1}^5{\xi }_k^2{X}_{kk}\right)}^2\ge \sum _{j<k}{y}_{jk}^2+2{\left(\sum _{k=1}^5{u}_k^2\right)}^{1/2},$$

(202)

where $y_{jk}=2{\xi }_j{\xi }_k{Y}_{jk}$ with arbitrary real numbers ${\xi }_k$. Coefficients $u_k$ are as follows,

$$u_1=y_{23}{y}_{45}+y_{34}{y}_{25}+y_{42}{y}_{35},u_2=y_{13}{y}_{45}+y_{34}{y}_{15}+y_{41}{y}_{35},$$

$$u_3=y_{12}{y}_{45}+y_{24}{y}_{15}+y_{41}{y}_{25},u_4=y_{12}{y}_{35}+y_{23}{y}_{15}+y_{31}{y}_{25},$$

$$u_5=y_{12}{y}_{34}+y_{23}{y}_{14}+y_{31}{y}_{24}.$$

The inequality for the product ${\mathrm{\Pi }}_5={\left({\prod }_{k=1}^5{X}_{kk}\right)}^{1/2}$ has the same structure as (199), where each term is a complicated combination of products of coefficients $Y_{jk}$ [87].

7.4. Compact Sum and Product Inequalities for N Operators

In view of rather complicated structures of the most strong inequalities for N variables, demonstrated in the preceding sections, it seems reasonable to find more simple (although apparently not the best) relations. A good example is the inequality

$$\sum _{k=1}^N{X}_{kk}\ge 2{\left[\sum _{j<k}{Y}_{jk}^2\right]}^{1/2}.$$

(203)

It was proved initially in [129] for N observables which are linear combinations of the canonical coordinate and momentum operators. In Refs. [87,106], it was derived for three, four, and five operators: see Equations (107), (196) and (201). The proof in the general case was given in Ref. [130]. Although inequality (203) is not the strongest one for $N\ge 4$, it has very nice and compact matrix form:

$${\left[\mathrm{T}\mathrm{r}\left(X\right)\right]}^2\ge 2\mathrm{T}\mathrm{r}\left(Y\tilde{Y}\right).$$

(204)

The simplest product inequality was obtained in Ref. [130] for $N=2K$:

$$\prod _{k=1}^{2K}{X}_{kk}\ge {\displaystyle \frac{1}{{(K!)}^2}}\sum Y_{j_1{j}_2}^2{Y}_{j_3{j}_4}^2\dots Y_{j_{2K-1},j_{2K}}^2,$$

(205)

where the sum is taken over all permutations of indexes with $j_{2k-1}<j_{2k}$. A similar (but more complicated) inequality holds for $N=2K+1$. The following inequality was derived in paper [110]:

$$\prod _{k=1}^N{X}_{kk}\ge {\left({\displaystyle \frac{1}{\sqrt{3}}}\right)}^N{\left(2\prod _{j<l}\left|Y_{jl}\right|\right)}^{{\displaystyle \frac{2}{N-1}}}.$$

(206)

Specific uncertainty relations in finite-dimensional Hilbert spaces were considered, e.g., in papers [117,118,131,132,133,134,135,136,137,138,139,140]. In particular, various inequalities for products of variances of non-commuting unitary operators (e.g., obeying the relation $\widehat{U}\widehat{V}=e^{i\varphi }\widehat{V}\widehat{U}$) were found in Refs. [141,142,143,144,145,146,147]. Several inequalities containing probabilities in the qudit states were derived and interpreted in the paper [148].

8. Inequalities Based on the Coordinate Probability Density Alone

An interesting generalization of the HKW inequality (3) was given by Peña-Auerbach and Cetto [149]. To derive it, one should use the polar decomposition of the wave function,

$$\psi \left(x\right)=\sqrt{\rho \left(x\right)}exp\left[i\phi \left(x\right)\right],{\int }_{-\infty }^{\infty }\rho \left(x\right)dx=1,Im\left(\rho \right)=Im\left(\phi \right)=0.$$

(207)

The mean value of the momentum operator equals $\langle \widehat{p}\rangle =\hslash {\int }_{-\infty }^{\infty }{\phi }^{\prime }\left(x\right)\rho \left(x\right)dx$, where the prime means the derivative. On the other hand, the variance ${\sigma }_p$ can be written as a sum of two non-negative terms: ${\sigma }_p={\sigma }^{\left(\rho \right)}+{\sigma }^{\left(\phi \right)}$, where

$${\sigma }^{\left(\rho \right)}={\hslash }^2{\int }_{-\infty }^{\infty }dx{\displaystyle \frac{{\rho }^{\prime 2}}{4\rho }}={\hslash }^2{\int }_{-\infty }^{\infty }dx{\left(d\right|\psi |/dx)}^2\equiv {\hslash }^2{F}_{\psi },$$

(208)

$${\sigma }^{\left(\phi \right)}={\hslash }^2{\int }_{-\infty }^{\infty }dx{\phi }^{\prime 2}\rho -{\langle \widehat{p}\rangle }^2.$$

(209)

Then, the Schwarz inequality (34) results in the following relations:

$${\sigma }_x{\sigma }^{\left(\rho \right)}\ge {\displaystyle \frac{{\hslash }^2}{4}}{\left|{\int }_{-\infty }^{\infty }dx(x-\langle x\rangle ){\rho }^{\prime }\right|}^2={\displaystyle \frac{{\hslash }^2}{4}}.$$

(210)

The meaning of inequality (210) is quite clear: the coordinate probability density of the state with a small coordinate variance cannot vary too slowly. Note that inequality (210) is stronger than (3), as soon as ${\sigma }_p\ge {\sigma }^{\left(\rho \right)}$. Moreover, it is stronger than the Schrödinger–Robertson inequality (8). To show this, let us consider quantum states with $\langle \widehat{x}\rangle =\langle \widehat{p}\rangle =0$. Then, ${\sigma }_{xp}={\int }_{-\infty }^{\infty }dx{\phi }^{\prime }\rho$, so that

$${\sigma }_x{\sigma }_p-{\sigma }_{px}^2={\sigma }_x{\sigma }^{\left(\rho \right)}+{\sigma }_x{\sigma }^{\left(\phi \right)}-{\sigma }_{px}^2\ge {\sigma }_x{\sigma }^{\left(\rho \right)},$$

(211)

because

$${\sigma }_x{\sigma }^{\left(\phi \right)}-{\sigma }_{px}^2={\hslash }^2\left\{{\int }_{-\infty }^{\infty }dx{x}^2\rho {\int }_{-\infty }^{\infty }dx{\phi }^{\prime 2}\rho -{\left[{\int }_{-\infty }^{\infty }dxx{\phi }^{\prime }\rho \right]}^2\right\}\ge 0,$$

(212)

due to the Schwartz inequality. Inequalities (8) and (210) are equivalent for the Gaussian wave functions of the form $Nexp\left(-a{x}^2+ib{x}^2\right)$ (with real constant coefficients $a>0$ and b). For such states, (8) and (210) become strict equalities. But taking the function $\psi \left(x\right)=Nexp\left(-a{x}^2+ic{x}^4\right)$, we still have ${\sigma }_x{\sigma }^{\left(\rho \right)}={\hslash }^2/4$, whereas ${\sigma }_x{\sigma }_p-{\sigma }_{px}^2=\left({\hslash }^2/4\right)\left(1+6c^2/a^4\right)$.

The physical meaning of inequality (210) was discussed, e.g., in Refs. [149,150,151,152,153,154,155,156,157,158] in the frameworks of the so-called stochastic mechanics. A straightforward generalization of inequality (210) to the case of n spatial dimensions was derived by Skála and Kapsa [159,160,161]. Introducing in addition to the coordinate variance matrix $Q_x$ the matrix $Q^{\left(\rho \right)}$ with elements

$$Q_{jk}^{\left(\rho \right)}=\int {\displaystyle \frac{1}{4\rho }}{\displaystyle \frac{{\partial }^2\rho }{\partial x_j\partial x_k}}d\mathbf{x},$$

one has the following enforcement of inequality (24) ($I_n$ means the $n\times n$ unit matrix):

$$det\left(Q_x{Q}^{\left(\rho \right)}-I_n{\hslash }^2/4\right)\ge 0.$$

(213)

8.1. Connections with the “Fisher Information” and Generalized Mean Values

An interesting interpretation of inequality (210) was given by Hall [162]. He called ${\sigma }^{\left(\rho \right)}$ as the variation of “nonclassical” part of the momentum and interpreted the quantity

$$\delta x={\left[{\int }_{-\infty }^{\infty }dx\rho \left(x\right){\left({\displaystyle \frac{dln\rho \left(x\right)}{dx}}\right)}^2\right]}^{-1/2}\equiv {\left[{\int }_{-\infty }^{\infty }dx{\rho }^{\prime 2}/\rho \right]}^{-1/2}$$

(214)

as the “Fisher length”. Then, obviously ${\left(\delta x\right)}^2{\sigma }^{\left(\rho \right)}\equiv {\hslash }^2/4$, so that the HKW inequality $\Delta x\Delta p\ge \hslash /2$ is the consequence of the Cramér–Rao inequality [163] $\Delta x\ge \delta x$. The importance of the quantity $F_{\psi }$ introduced in Equation (208) as a useful measure of the probability density localization was emphasized by Stam [164], who derived the inequality equivalent to (210) and noticed that $F_{\psi }$ is the special case of a more general concept of the Fisher information [165] related to the probability density $\rho \left(x\right)$. Since that time, various forms of the “uncertainty relations” in terms of the quantum Fisher information attracted high attention, especially for the past few decades: see, e.g., examples in papers [166,167,168,169,170,171,172,173] and references therein. However, this area needs a separate review.

For example, an attractive possibility to improve the Robertson inequality (17) for N operators when N is odd number was announced, e.g., in papers [174,175,176]. The first main feature of the new approach is the replacement of commutators $\left[{\widehat{z}}_m,{\widehat{z}}_n\right]$ in the definition of elements of matrix Y (13) with the products of commutators $\left[{\widehat{z}}_m,\widehat{\rho }\right]\left[{\widehat{z}}_n,\widehat{\rho }\right]$, where $\widehat{\rho }$ is the statistical operator (density matrix) of the quantum state. The second new feature is the replacement of the standard definition of mean value (2) with a more complicated construction

$${\langle \widehat{A},\widehat{B}\rangle }_{\widehat{\rho },f}=\mathrm{T}\mathrm{r}\left\{m_f{\left(\widehat{\rho }\widehat{A},\widehat{A}\widehat{\rho }\right)}^{-1}\widehat{B}\right\},m_f(\widehat{C},\widehat{D})={\widehat{C}}^{1/2}f\left({\widehat{C}}^{-1/2}\widehat{D}{\widehat{C}}^{-1/2}\right){\widehat{C}}^{1/2},$$

where function $f\left(x\right)$ belongs to the specific family of “symmetric normalized operator monotone functions”, satisfying the conditions

$$f\left(1\right)=1,xf(1/x)=f\left(x\right),f\left(x\right)\le f\left(y\right)ifx\le y.$$

The usual case corresponds to $f\left(x\right)=(x+1)/2$. Other examples considered in the literature are

$$f\left(x\right)=2x/(x+1),f\left(x\right)={\left(1+\sqrt{x}\right)}^2,\dots$$

However, it is unclear whether such complicated mathematical constructions have any physical sense, as soon as no explicit examples (e.g., for three angular momentum operators and for pure quantum states described by wave functions) were demonstrated.

9. State-Extended and State-Independent UR

9.1. Trifonov’s Inequalities

In the preceding sections, all mean values were calculated with respect to some given single quantum state (pure or mixed). Trifonov [69,177,178,179] has obtained generalizations of URs, containing average values of observables with respect to different quantum states. He described an extension of Robertson’s scheme to arbitrary numbers of N observables and M states. The main result was the inequality [178]

$$C_r^{\left(N\right)}\left(\sum _{m=1}^M{X}^{\left(m\right)}\right)\ge C_r^{\left(N\right)}\left(\sum _{m=1}^M{Y}^{\left(m\right)}\right).$$

(215)

We use here the same notation (13) for the covariance matrix X and commutator matrix Y, adding superscripts $\left(m\right)$ to indicate the state (pure or mixed) used for calculating the respective average values. The characteristic coefficients $C_r^{\left(N\right)}\left(X\right)$ were defined in Equation (20). In the special case of $N=M=2$, one has the following generalization of inequality (5):

$${\displaystyle \frac{1}{2}}\left[X_{11}^{\left(1\right)}{X}_{22}^{\left(2\right)}+X_{11}^{\left(2\right)}{X}_{22}^{\left(1\right)}\right]-\left|X_{12}^{\left(1\right)}{X}_{12}^{\left(2\right)}\right|\ge Y_{12}^{\left(1\right)}{Y}_{12}^{\left(2\right)}.$$

(216)

Another simple inequality is [69]

$${(\Delta x)}^{\left(1\right)}{(\Delta p)}^{\left(2\right)}+{(\Delta x)}^{\left(2\right)}{(\Delta p)}^{\left(1\right)}\ge \hslash .$$

(217)

More involved inequalities can be found in the paper [69].

9.2. Maccone–Pati Inequalities and Their Generalizations

Maccone and Pati [180] proved the inequality, which can be written in our notation as

$${\sigma }_A+{\sigma }_B\ge 2\left|Y_{AB}\right|+{\left|〈\psi \left|\widehat{A}-i{S}_{AB}\widehat{B}\right|{\psi }^{\perp }〉\right|}^2,S_{AB}=sign\left(Y_{AB}\right).$$

(218)

Here, $|{\psi }^{\perp }\rangle$ is any pure state, orthogonal to the given pure state $|\psi \rangle$ (one can replace $\widehat{A}$ and $\widehat{B}$ with $\delta \widehat{A}$ and $\delta \widehat{B}$ in this and all other similar formulas). The average values are calculated with respect to the state $|\psi \rangle$. Obvious assumptions are that observables A and B have the same dimensions (otherwise they should be rescaled somehow), and that the vectors $|{\psi }^{\perp }\rangle$ and $|\psi \rangle$ are normalized: $\langle {\psi }^{\perp }|{\psi }^{\perp }\rangle =\langle \psi |\psi \rangle =1$.

There are several prescriptions, how to construct a normalized orthogonal state $|{\psi }^{\perp }\rangle$ [181]. One of them, used in [180], is as follows [182]:

$$\left|{\psi }_O^{\perp }\right.〉=\delta \widehat{O}|\psi \rangle /\Delta O,$$

(219)

where $\widehat{O}$ can be an arbitrary operator (see also [183]). Another possibility, used in [184,185], is

$$\left|{\psi }_{\chi }^{\perp }\right.〉=\left(|\chi \rangle -|\psi \rangle \langle \psi |\chi \rangle \right){\left(1-{\left|\langle \psi |\chi \rangle \right|}^2\right)}^{-1/2},$$

(220)

where $|\chi \rangle$ is an arbitrary quantum state.

The simplest proof of (218) is the consequence of the identity $\parallel \delta \widehat{A}\mp i\delta \widehat{B}\parallel \equiv {\sigma }_A+{\sigma }_B\mp i〈\left[\widehat{A},\widehat{B}\right]〉$ and the Schwarz inequality in the form

$${\left|〈\psi \left|\widehat{A}\pm i\widehat{B}\right|{\psi }^{\perp }〉\right|}^2={\left|〈\psi \left|\widehat{A}\pm i\widehat{B}-\langle A\pm iB\rangle \right|{\psi }^{\perp }〉\right|}^2={\left|〈\psi \left|\delta \widehat{A}\pm i\delta \widehat{B}\right|{\psi }^{\perp }〉\right|}^2\le \parallel \delta \widehat{A}\mp i\delta \widehat{B}\parallel .$$

Another inequality derived in [180] reads

$${\sigma }_A+{\sigma }_B\ge {\displaystyle \frac{1}{2}}{\left|〈{\psi }_{A+B}^{\perp }\left|\widehat{A}+\widehat{B}\right|\psi 〉\right|}^2.$$

(221)

But after putting the explicit form of vector $\left|{\psi }_{A+B}^{\perp }\right.〉$, given by Equation (219), in the right-hand side of (221), one can see that this inequality is reduced to the trivial statement ${\sigma }_A+{\sigma }_B\ge 2{\sigma }_{AB}$.

Some advantages of inequalities (218) and (221) over (4) were illustrated in [180] for operators ${\widehat{L}}_x$ and ${\widehat{L}}_y$ and various states with the orbital quantum number $l=1$. The optimal choice of orthogonal states, saturating inequalities (218) and (221), was also considered in [180]. Interpretations of inequalities (218) and (221), as well as their generalizations to systems in finite-dimensional Hilbert spaces and applications to qubits were given in [132]. It was mentioned in [180] that inequality (218) could be generalized to include the covariance term ${\sigma }_{AB}$. This was performed explicitly in [108]. In our notation, the inequality derived there can be written as follows:

$${\sigma }_A+{\sigma }_B\ge 2G_{AB}+{\left|〈\psi \left|\widehat{A}-{\displaystyle \frac{{\sigma }_{AB}+i{Y}_{AB}}{G_{AB}}}\widehat{B}\right|{\psi }^{\perp }〉\right|}^2,$$

(222)

where $G_{AB}$ was defined in (5) and $Y_{AB}$ was defined in (13), so that $G_{AB}\equiv \left|{\sigma }_{AB}+i{Y}_{AB}\right|$.

Maccone and Pati [180] also gave the inequality

$$\Delta A\Delta B\ge {\displaystyle \frac{1}{2}}\left|〈\left[\widehat{A},\widehat{B}\right]〉\right|{\left(1-{\displaystyle \frac{1}{2}}{\left|〈\psi \left|{\displaystyle \frac{\widehat{A}}{\Delta A}}-i{S}_{AB}{\displaystyle \frac{\widehat{B}}{\Delta B}}\right|{\psi }^{\perp }〉\right|}^2\right)}^{-1}.$$

(223)

It was generalized in [108]:

$$\Delta A\Delta B\ge G_{AB}{\left(1-{\displaystyle \frac{1}{2}}{\left|〈\psi \left|{\displaystyle \frac{\widehat{A}}{\Delta A}}-{\displaystyle \frac{{\sigma }_{AB}+i{Y}_{AB}}{G_{AB}}}{\displaystyle \frac{\widehat{B}}{\Delta B}}\right|{\psi }^{\perp }〉\right|}^2\right)}^{-1}.$$

(224)

However, a deeper analysis shows that inequalities (223) and (224) are not stronger than the original inequalities (4) and (5). This can be seen if one takes, for example, the orthogonal vector $\left|{\psi }^{\perp }\right.〉$ in the form $\left|{\psi }_B^{\perp }\right.〉$. Then, straightforward calculations show that inequality (224) can be written as $\eta \le 1-{(\eta -1)}^2/2$, where $\eta =G_{AB}/(\Delta A\Delta B)$. But this inequality is reduced to ${\eta }^2\le 1$, which is exactly inequality (5). Similar conclusions were made, e.g., in Ref. [186].

A generalization of inequality (218) to the case of three observables, derived in [108], has the following form in our notation:

$${\sigma }_A+{\sigma }_B+{\sigma }_C\ge {\displaystyle \frac{1}{3}}{\left|〈\psi \left|\widehat{D}\right|{\psi }_D^{\perp }〉\right|}^2+{\displaystyle \frac{2}{3}}{\left|〈\psi \left|{\widehat{D}}_{\zeta }\right|{\psi }^{\perp }〉\right|}^2+{\displaystyle \frac{2}{\sqrt{3}}}\left|Y_{AB}+Y_{BC}+Y_{CA}\right|,$$

(225)

where $\zeta =exp(\pm 2i\pi /3)$, $\widehat{D}=\widehat{A}+\widehat{B}+\widehat{C}$ and ${\widehat{D}}_{\zeta }=\widehat{A}+\zeta \widehat{B}+{\zeta }^2\widehat{C}$. Obviously, each operator in the right-hand side (in particular, $\widehat{D}$) can be replaced by the shifted operator, like $\delta \widehat{D}$, without any changes. Therefore, the first term in the right-hand side of (225) is nothing but ${\sigma }_D/3$. Calculating ${\sigma }_D$ explicitly and suppressing the non-negative term with ${\widehat{D}}_{\zeta }$, one can arrive at the symmetric inequality

$${\sigma }_A+{\sigma }_B+{\sigma }_C\ge {\sigma }_{AB}+{\sigma }_{BC}+{\sigma }_{CA}+\sqrt{3}\left|Y_{AB}+Y_{BC}+Y_{CA}\right|.$$

(226)

This relation turns into exact equality for the KW states (105).

9.3. “Weighted-like”, “Tighten”, “Reverse” and “Improved” Uncertainty Relations

Generalizations/modifications of inequalities containing orthogonal states and some auxiliary parameters were named “weighted uncertainty relations” in [187,188]. For example [187],

$$(1+\lambda ){\sigma }_A+\left(1+{\lambda }^{-1}\right){\sigma }_B\ge 4\left|Y_{AB}\right|+{\left|\langle \psi |\widehat{A}-i\widehat{B}|{\psi }_1^{\perp }\rangle \right|}^2+{\lambda }^{-1}{\left|\langle \psi |\lambda \widehat{A}-i\widehat{B}|{\psi }_2^{\perp }\rangle \right|}^2,$$

(227)

for any positive parameter $\lambda$ and all normalized states $|{\psi }_1^{\perp }\rangle$ and $|{\psi }_2^{\perp }\rangle$ orthogonal to $|\psi \rangle$. Another example is the inequality [188]

$$\Delta A\Delta B\ge {\displaystyle \frac{\left|〈\left[\widehat{A},\widehat{B}\right]〉\right|\sqrt{\lambda }}{1+\lambda -{\left|〈\psi \left|\frac{\widehat{A}}{\Delta A}-i{S}_{AB}\sqrt{\lambda }\frac{\widehat{B}}{\Delta B}\right|{\psi }^{\perp }〉\right|}^2}}$$

(228)

for any unit vector $|{\psi }^{\perp }\rangle$ perpendicular to $|\psi \rangle$ and arbitrary parameter $\lambda >0$. It is easy to find the maximal value of the right-hand side of (228) with respect to parameter $\lambda$. Then, the following inequality arises:

$$\Delta A\Delta B\ge {\displaystyle \frac{\left|Y_{AB}\right|}{\sqrt{\left(1-{\left|{\chi }_A\right|}^2\right)\left(1-{\left|{\chi }_B\right|}^2\right)}+S_{AB}Im\left({\chi }_A{\chi }_B^{*}\right)}},$$

(229)

where ${\chi }_A=〈\psi \left|\left(\widehat{A}/\Delta A\right)\right|{\psi }^{\perp }〉$ and ${\chi }_B=〈\psi \left|\left(\widehat{B}/\Delta B\right)\right|{\psi }^{\perp }〉$. However, it seems that inequality (229) is not better than (5). For example, if $\left|{\psi }^{\perp }\right.〉=\left|{\psi }_B^{\perp }\right.〉$, then ${\chi }_B=1$ and ${\chi }_A=\left({\sigma }_{AB}+i{Y}_{AB}\right)/(\Delta A\Delta B)$, so that the right-hand side of (229) equals $\Delta A\Delta B$. Further generalizations were given in the paper [189].

Some complicated inequalities seem “more tightened” for two incompatible observables were derived and illustrated for the ${\widehat{L}}_x$ and ${\widehat{L}}_y$ components of the spin-1/2 and spin-1 operators in paper [190]. The “reverse” uncertainty relations derived in [190] give “upper limits” on the product and the sum of two variances. The simplest inequality has the form

$$\sqrt{{\sigma }_A}+\sqrt{{\sigma }_B}\le \sqrt{{\displaystyle \frac{2{\sigma }_{A-B}}{1-r_{AB}}}},$$

(230)

where $r_{AB}$ is the correlation coefficient defined in Equation (9). However, calculating the squares of the left-hand and right-hand sides of (230), one can arrive at the trivial inequality ${\sigma }_A+{\sigma }_B\ge 2\sqrt{{\sigma }_A{\sigma }_B}$. Multidimensional versions of “reverse” uncertainty relations were considered in paper [191]. The “quantum-control-assisted reverse uncertainty relation” was constructed in Ref. [192].

Tight (when inequalities can be saturated) variance-based sum-uncertainty relations for sets of operators belonging to various algebras (in particular, the Weyl–Heisenberg algebra, special unitary algebras up to rank 4, and semisimple compact algebras) have been obtained in the paper [193]. It was shown that lower bounds in those relations depend only on the irreducible representation assumed to be carried by the Hilbert space of the state of the system. Uncertainty relations that give lower bounds to the sum of variances of two or more observables, both bounded and unbounded, were considered in the paper [194]. The inclusion of higher-order statistical moments (such as “skewness” and “kurtosis”) in the modified sum uncertainty relations was suggested in paper [195]. The concept of “uncertainty regions” was the subject of Ref. [138]. State-independent URs were considered in paper [196].

The following modification (“improvement”) of inequality (5) was obtained in paper [197] for pure quantum states:

$${\sigma }_A{\sigma }_B\ge G_{AB}^2/{\mu }^2\left({\theta }_{AB}\right),$$

(231)

where the positive coefficient $\mu \left({\theta }_{AB}\right)$ depends on the “angle” between the vectors $\delta \widehat{A}$ and $\delta \widehat{B}$ in the Hilbert space:

$$\mu \left({\theta }_{AB}\right)={\int }_0^1\left|sexp\left(i{\theta }_{AB}\right)+(1-s)exp(-i{\theta }_{AB})\right|ds,$$

(232)

$$cos\left({\theta }_{AB}\right)={\displaystyle \frac{|\langle \delta \widehat{A}\delta \widehat{B}\rangle |}{\sqrt{{\sigma }_A{\sigma }_B}}}\equiv {\displaystyle \frac{G_{AB}}{\sqrt{{\sigma }_A{\sigma }_B}}}.$$

(233)

However, the practical value of inequality (231) seems rather limited, because one has to know the variances ${\sigma }_A$ and ${\sigma }_B$ beforehand. The integral (232) can be reduced to the form (writing here $\theta \equiv {\theta }_{AB}$)

$$\mu \left(\theta \right)={\int }_0^{1}dy\sqrt{{cos}^2\left(\theta \right)+y^2{sin}^2\left(\theta \right)}.$$

Using the known indefinite integral

$$\int udx={\displaystyle \frac{1}{2}}\left[xu+(a/b)ln(xb+u)\right],u=\sqrt{a+b^2{x}^2},$$

one can obtain the explicit form (assuming $0<{\theta }_{AB}<\pi /2$)

$$\mu \left({\theta }_{AB}\right)={\displaystyle \frac{1}{2}}+{\displaystyle \frac{{cos}^2\left({\theta }_{AB}\right)}{2sin\left({\theta }_{AB}\right)}}ln\left[{\displaystyle \frac{1+sin\left({\theta }_{AB}\right)}{cos\left({\theta }_{AB}\right)}}\right].$$

(234)

This complicated expression confirms that inequality (231) can hardly be useful in physical applications, especially taking into account that $\mu \left({\theta }_{AB}\right)\ge 1/2$ even for ${\theta }_{AB}\to \pi /2$, when $G_{AB}/\sqrt{{\sigma }_A{\sigma }_B}\ll 1$.

10. Bargmann–Faris Inequalities and Their Generalizations

Interesting multidimensional extensions of inequality (3) were found by Bargmann [198] and generalized by Faris [199]. Here we follow mainly the study [199]. The starting point is the inequality for arbitrary operators $\widehat{A}$ and $\widehat{B}$

$$\langle {\widehat{A}}^{†}\widehat{A}\rangle +\langle {\widehat{B}}^{†}\widehat{B}\rangle \ge i\langle {\widehat{A}}^{†}\widehat{B}-{\widehat{B}}^{†}\widehat{A}\rangle ,$$

(235)

which is nothing but the transformed obvious inequality $\langle {(\widehat{A}-i\widehat{B})}^{†}(\widehat{A}-i\widehat{B})\rangle \ge 0$. Replacing $\widehat{A}$ by $\widehat{A}{\langle {\widehat{A}}^{†}\widehat{A}\rangle }^{-1/2}$ and $\widehat{B}$ by $\widehat{B}{\langle {\widehat{B}}^{†}\widehat{B}\rangle }^{-1/2}$ we obtain a special case of inequality (71)

$$2{\left[\langle {\widehat{A}}^{†}\widehat{A}\rangle \langle {\widehat{B}}^{†}\widehat{B}\rangle \right]}^{1/2}\ge i\langle {\widehat{A}}^{†}\widehat{B}-{\widehat{B}}^{†}\widehat{A}\rangle .$$

(236)

Another useful inequality follows from the Schwarz inequality

$${\left[\langle {\widehat{A}}^{†}\widehat{A}\rangle \langle {\widehat{B}}^{†}\widehat{B}\rangle \right]}^{1/2}\ge \left|\langle {\widehat{A}}^{†}\widehat{B}\rangle \right|,$$

(237)

if one assumes that $\widehat{A}={\widehat{Y}}^{1/2}$ and $\widehat{B}={\widehat{Y}}^{-1/2}$, where $\widehat{Y}$ is a positively definite operator:

$$\langle \widehat{Y}\rangle \langle {\widehat{Y}}^{-1}\rangle \ge 1.$$

(238)

To obtain the required multidimensional generalization of inequality (3), let us separate the angular and radial parts of the operator ${\widehat{\mathbf{p}}}^2=-{\hslash }^2\Delta$, introducing the angular momentum projection operators ${\widehat{L}}_{jk}={\widehat{x}}_j{\widehat{p}}_k-{\widehat{x}}_k{\widehat{p}}_j$ in the n-dimensional case. Each operator ${\widehat{L}}_{jk}$ commutes with the operators ${\widehat{\mathbf{x}}}^2={\widehat{x}}_1^2+{\widehat{x}}_2^2+\dots +{\widehat{x}}_n^2\equiv {\widehat{r}}^2$ and ${\widehat{\mathbf{p}}}^2$, and the same is true for the operator ${\widehat{\mathbf{L}}}^2={\sum }_{j<k}{\widehat{L}}_{jk}^2$. Now let us consider the chain of relations

$${\widehat{r}}^{-2}{\widehat{\mathbf{L}}}^2={\displaystyle \frac{1}{2}}\sum _{j,k}{\widehat{L}}_{jk}{\widehat{r}}^{-2}{\widehat{L}}_{jk}=\sum _{j,k}\left({\widehat{p}}_k{\widehat{x}}_j{\widehat{\mathbf{x}}}^{-2}{\widehat{x}}_j{\widehat{p}}_k-{\widehat{p}}_k{\widehat{x}}_j{\widehat{r}}^{-2}{\widehat{x}}_k{\widehat{p}}_j\right)={\widehat{\mathbf{p}}}^2-\left(\widehat{\mathbf{p}}\widehat{\mathbf{x}}\right){\widehat{r}}^{-2}\left(\widehat{\mathbf{x}}\widehat{\mathbf{p}}\right).$$

(239)

The next step is to move the operator ${\widehat{r}}^{-2}$ in the second term on the right-hand side of the last equality to the left of the operator $\widehat{\mathbf{p}}$ with the aid of relations $\left(\widehat{\mathbf{p}}\widehat{\mathbf{x}}\right){\widehat{r}}^{-2}={\widehat{r}}^{-2}\left(\widehat{\mathbf{p}}\widehat{\mathbf{x}}+2i\hslash \right)$ and $\left[\left(\widehat{\mathbf{p}}\widehat{\mathbf{x}}\right),\left(\widehat{\mathbf{x}}\widehat{\mathbf{p}}\right)\right]=0$. Then, after some transformations, one can obtain the identity

$$\left(\widehat{\mathbf{p}}\widehat{\mathbf{x}}\right){\widehat{r}}^{-2}\left(\widehat{\mathbf{x}}\widehat{\mathbf{p}}\right)={\widehat{r}}^{-2}\left(\widehat{\mathbf{x}}\widehat{\mathbf{p}}\right){\widehat{r}}^2\left(\widehat{\mathbf{p}}\widehat{\mathbf{x}}\right){\widehat{r}}^{-2},$$

or in another form

$$\left[\left(\widehat{\mathbf{p}}\widehat{\mathbf{x}}\right){\widehat{r}}^{-1}\right]\left[{\widehat{r}}^{-1}\left(\widehat{\mathbf{x}}\widehat{\mathbf{p}}\right)\right]=\left[{\widehat{r}}^{-2}\left(\widehat{\mathbf{x}}\widehat{\mathbf{p}}\right)\widehat{r}\right]\left[\widehat{r}\left(\widehat{\mathbf{p}}\widehat{\mathbf{x}}\right){\widehat{r}}^{-2}\right].$$

Introducing the operators ${\widehat{A}}_1={\widehat{r}}^{-1}\left(\widehat{\mathbf{x}}\widehat{\mathbf{p}}\right)$ and ${\widehat{A}}_2=\widehat{r}\left(\widehat{\mathbf{p}}\widehat{\mathbf{x}}\right){\widehat{r}}^{-2}$, one can rewrite (239) as

$${\widehat{\mathbf{p}}}^2={\widehat{r}}^{-2}{\widehat{\mathbf{L}}}^2+{\widehat{A}}_1^{†}{\widehat{A}}_1={\widehat{r}}^{-2}{\widehat{\mathbf{L}}}^2+{\widehat{A}}_2^{†}{\widehat{A}}_2.$$

(240)

Therefore, the radial part of the operator ${\widehat{\mathbf{p}}}^2$ can be expressed either as ${\widehat{A}}_1^{†}{\widehat{A}}_1$ or as ${\widehat{A}}_2^{†}{\widehat{A}}_2$. Consequently,

$$0\le \langle {\widehat{A}}_1^{†}{\widehat{A}}_1\rangle =\langle {\widehat{A}}_2^{†}{\widehat{A}}_2\rangle \le \langle {\widehat{\mathbf{p}}}^2\rangle ,$$

(241)

because $\langle {\widehat{\mathbf{p}}}^2\rangle \ge 0$ and $\langle {\widehat{r}}^{-2}{\widehat{\mathbf{L}}}^2\rangle \ge 0$. Finally, one should use inequality (236), assuming $\widehat{A}={\widehat{A}}_1$ or $\widehat{A}={\widehat{A}}_2$ and $\widehat{B}=\phi \left(r\right)$, where $\phi \left(r\right)$ is an arbitrary real sufficiently smooth function without singularities (except perhaps at the point $r=0$). The result is the inequality

$$\hslash \langle \partial \phi /\partial r+{\nu }_k\phi \left(r\right)/r\rangle \le 2{\left[\langle {\phi }^2\left(r\right)\rangle \langle {\widehat{\mathbf{p}}}^2-{\widehat{r}}^{-2}{\widehat{\mathbf{L}}}^2\rangle \right]}^{1/2}\le 2{\left[\langle {\phi }^2\left(r\right)\rangle \langle {\widehat{\mathbf{p}}}^2\rangle \right]}^{1/2},$$

(242)

where the value ${\nu }_1=n-1$ corresponds to the choice $\widehat{A}={\widehat{A}}_1$ and the value ${\nu }_2=3-n$ corresponds to $\widehat{A}={\widehat{A}}_2$. In the one-dimensional case ($n=1$), inequality (242) can be derived also from inequality (4), if one takes $\widehat{A}=-i\hslash \partial /\partial r$. This is due to the fact that operator ${\widehat{A}}_1$ is reduced to the Hermitian operator $-i\hslash \partial /\partial r$ for $n=1$. On the other hand, the operator ${\widehat{A}}_2$ becomes Hermitian for $n=3$, when it assumes the form $-i\hslash r^{-1}(\partial /\partial r)r$. Indeed, the chain of relations

$$\langle \phi |{\widehat{A}}_2|\psi \rangle \sim \int r^{2}dr{\phi }^{*}\left(r\right)\left[{\widehat{A}}_2\psi \left(r\right)\right]=-i\hslash \int dr\left(r{\phi }^{*}\right){\displaystyle \frac{\partial }{\partial r}}\left(r\psi \right)=i\hslash \int dr\left(r\psi \right){\displaystyle \frac{\partial }{\partial r}}\left(r{\phi }^{*}\right)$$

proves that $\langle \phi |{\widehat{A}}_2|\psi \rangle =\langle \psi |{\widehat{A}}_2{|\phi \rangle }^{*}$. Therefore, inequality (242) with $\nu ={\nu }_2=0$ is the consequence of inequality (4) in the three-dimensional case as well. Note, however, that inequalities which are obtained directly from the standard inequality (4) turn out weaker [in both the cases and for any non-negative function $\phi \left(r\right)$] than the inequality

$$\hslash \langle \partial \phi /\partial r+2\phi \left(r\right)/r\rangle \le 2{\left[\langle {\phi }^2\left(r\right)\rangle \langle {\widehat{\mathbf{p}}}^2\rangle \right]}^{1/2},$$

(243)

which follows from (242) if one takes $\nu ={\nu }_2$ for $n=1$ and $\nu ={\nu }_1$ for $n=3$. Moreover, the inequality (242) with $\nu ={\nu }_1=n-1$ is always stronger than a similar one with $\nu ={\nu }_2$ for any non-negative function $\phi \left(r\right)$, provided $n\ge 2$.

Assuming $\phi \left(r\right)=r^{1+\alpha }$ in (242), we obtain the set of inequalities [199]

$$\hslash (1+\alpha +{\nu }_k)\langle r^{\alpha }\rangle \le 2{\left[\langle r^{2+2\alpha }\rangle \langle {\widehat{\mathbf{p}}}^2-{\widehat{r}}^{-2}{\widehat{\mathbf{L}}}^2\rangle \right]}^{1/2}\le 2{\left[\langle r^{2+2\alpha }\rangle \langle {\widehat{\mathbf{p}}}^2\rangle \right]}^{1/2}.$$

(244)

In the study [198], inequality (244) with $\nu ={\nu }_1=n-1$ (since only the case of $n\ge 2$ was considered there) was derived immediately from inequality (236) with the aid of the substitutions $\widehat{B}=r^{1+\alpha }$ and $\widehat{A}=-i\hslash r^{-1}(\mathbf{x}\nabla )$, using the obvious relation $|r^{-1}{\mathbf{x}\nabla \psi |}^2\le {|\nabla \psi |}^2$.

Let $\alpha =0$. Then, choosing $\nu ={\nu }_1$ we again arrive at the special case of inequality (37)

$${\left[\langle r^2\rangle \langle {\widehat{\mathbf{p}}}^2\rangle \right]}^{1/2}\ge n\hslash /2.$$

(245)

On the other hand, the choice $\nu ={\nu }_2$ leads to the inequality ${\left[\langle r^2\rangle \langle {\widehat{\mathbf{p}}}^2\rangle \right]}^{1/2}\ge \hslash (4-n)/2$, which is weaker than (245) if $n\ge 2$. But for $n=1$ we obtain the strange inequality

$${\left[\langle x^2\rangle \langle {\widehat{p}}^2\rangle \right]}^{1/2}\ge 3\hslash /2,$$

(246)

which is obviously invalid, e.g., for the oscillator ground state. The solution to this “paradox” is that in deriving inequality (242) we used singular operators, such as ${\widehat{r}}^{-1}$ and ${\widehat{r}}^{-2}$, which need careful handling. Particularly, if $\psi \in L^2$, then $r^{-1}\psi$ must also be normalized. It was shown in study [199] that for $n\ge 2$ inequality (242) holds for all wave functions which are admissible from the physical point of view (provided the function $\phi \left(r\right)$ is not too singular at the origin), since in this case the singularities of the operators ${\widehat{r}}^{-1}$ and ${\widehat{r}}^{-2}$ are compensated in the integrals giving various average values by the factor $dV\sim r^{n-1}dr$, due to the finiteness of the wave functions near the origin. In the one-dimensional case, inequality (242) is valid for restricted classes of functions, for example, for functions behaving like $r^{\delta }$ (with $\delta \ge 1$) when $r\to 0$. In particular, inequality (246) holds for any superposition of the one-dimensional oscillator excited eigenstates with odd quantum numbers, and it becomes an equality for the first excited state. This was noticed for the first time, perhaps, by Husimi [200]. The inequality $\Delta x\Delta p\ge 3\hslash /2$ was proven for odd functions $\psi \left(x\right)$ in one dimension in study [201].

Another interesting case is $\alpha =-2$, when $1+\alpha +{\nu }_k=\pm (n-2)$. Then, the consequence of (244) is the inequality

$$\langle {\widehat{\mathbf{p}}}^2\rangle \ge \langle {\widehat{\mathbf{p}}}^2-{\widehat{r}}^{-2}{\widehat{\mathbf{L}}}^2\rangle \ge {\displaystyle \frac{{\hslash }^2}{4}}{(n-2)}^2\langle r^{-2}\rangle .$$

(247)

If $\alpha =-1$, Equation (244) leads to the inequalities (for restricted classes of functions)

$$\langle {\widehat{\mathbf{p}}}^2\rangle \ge \langle {\widehat{\mathbf{p}}}^2-{\widehat{r}}^{-2}{\widehat{\mathbf{L}}}^2\rangle \ge {\left[{\displaystyle \frac{\hslash }{2}}(n-1)\langle r^{-1}\rangle \right]}^2,n\ge 2,$$

(248)

$$\langle {\widehat{p}}^2\rangle \ge {\hslash }^2{\left[\langle |x{|}^{-1}\rangle \right]}^2.$$

(249)

Inequalities (245), (247) and (248) (or their special cases in three dimensions) were also derived, using different methods, in papers [202,203,204]. A more general inequality in the D-dimensional space was discussed in Ref. [205] (here $a,b>0$ and $\hslash =1$):

$${\langle {\widehat{r}}^a\rangle }^{{\displaystyle \frac{2}{a}}}{\langle {\widehat{p}}^b\rangle }^{{\displaystyle \frac{2}{b}}}\ge \left({\displaystyle \frac{e{D}^{\frac{2}{a}}{\left[\mathrm{\Gamma }(1+D/2)\right]}^{\frac{2}{D}}}{{\left(ae\right)}^{\frac{2}{a}}{\left[\mathrm{\Gamma }(1+D/a)\right]}^{\frac{2}{D}}}}\right)\left({\displaystyle \frac{e{D}^{\frac{2}{b}}{\left[\mathrm{\Gamma }(1+D/2)\right]}^{\frac{2}{D}}}{{\left(be\right)}^{\frac{2}{b}}{\left[\mathrm{\Gamma }(1+D/b)\right]}^{\frac{2}{D}}}}\right).$$

(250)

More strict values of the right-hand side in (250) and some equalities for the products of three and four moments of r and p of different orders were found in Ref. [206].

The following inequality was proven for arbitrary real functions of several variables [207]:

$$\langle {\widehat{\mathbf{p}}}^2\rangle \langle {\phi }^2\left(\mathbf{x}\right)\rangle \ge \left({\hslash }^2/4\right){\langle \nabla \phi \left(\mathbf{x}\right)\rangle }^2.$$

(251)

A generalization of relation (251), which is similar to (242) but holds for arbitrary functions of n variables (not only for the functions possessing a spherical symmetry), was proven in study [208]:

$$\langle {\widehat{\mathbf{p}}}^2\rangle \langle {\phi }^2\left(\mathbf{x}\right)\rangle \ge ({\hslash }^2/4){〈r^{-1}\left[\mathbf{x}\nabla \phi \left(\mathbf{x}\right)+(n-1)\phi \left(\mathbf{x}\right)\right]〉}^2.$$

(252)

The following inequality holds for quantum states with a fixed value of the (hyper)angular momentum quantum number l in the n-dimensional space [208]:

$$\langle {\widehat{\mathbf{p}}}^2\rangle \langle {\phi }^2\left(r\right)\rangle \ge ({\hslash }^2/4){〈\partial \phi /\partial r+(n-1+2l)\phi \left(r\right)/r〉}^2.$$

(253)

Its special case is the inequality [198]

$$\langle {\widehat{\mathbf{p}}}^2\rangle \langle r^{2+2\alpha }\rangle \ge ({\hslash }^2/4){\left[(n+\alpha +2l)\langle r^{\alpha }\rangle \right]}^2.$$

(254)

The case of $\alpha =0$ was also considered in Ref. [209]:

$$\langle {\widehat{\mathbf{p}}}^2\rangle \langle r^2\rangle \ge ({\hslash }^2/4){(n+2l)}^2.$$

(255)

Taking $\alpha =-2$, one obtains the following analog of (247):

$$\langle {\widehat{\mathbf{p}}}^2\rangle \ge {\displaystyle \frac{{\hslash }^2}{4}}{(n-2+2l)}^2\langle r^{-2}\rangle .$$

(256)

The following inequalities were derived in [210] for the states with fixed angular momentum quantum numbers in two and three dimensions:

$$\Delta r\Delta p\ge \left\{\begin{array}{cc}\hslash \left(\right|m|+1)\hfill & \mathrm{in}2D\hfill \\ \hslash (l+3/2)\hfill & \mathrm{in}3D\hfill \end{array}\right..$$

(257)

The equality in these relations is attained for the wave packets with the modified Gaussian radial part of the form $\psi \left(r\right)\sim r^{l}exp(-\lambda r^2)$. A sharpened version of inequality (257) in three dimensions was found in Ref. [211] for the wave functions which can be not eigenstates of the angular momentum:

$$\Delta r\Delta p\ge {\displaystyle \frac{3\hslash }{2}}+{\displaystyle \frac{\hslash L_{inv}^3}{2}}{\displaystyle \frac{\sqrt{L_{inv}^2+4L_{inv}^4+4R_{inv}}-L_{inv}}{R_{inv}+L_{inv}^4}},$$

(258)

where

$${\widehat{\mathbf{L}}}_{inv}=\left(\widehat{\mathbf{r}}-\langle \widehat{\mathbf{r}}\rangle \right)\times \left(\widehat{\mathbf{p}}-\langle \widehat{\mathbf{p}}\rangle \right),L_{inv}^2=\langle {\widehat{L}}_{inv}^2\rangle /{\hslash }^2,R_{inv}=〈{\left({\widehat{L}}_{inv}^2-\langle {\widehat{L}}_{inv}^2\rangle \right)}^2〉/{\hslash }^4.$$

Note that all relations of this section remain valid if one makes the replacement of vectors $\mathbf{r}\leftrightarrow \mathbf{p}$.

11. Uncertainty Relations for Mixed and Non-Gaussian States

Frequently (e.g., in many text books), the preparation uncertainty relations are considered for the pure quantum states only. The validity of (3) and (4) for mixed quantum states, described by means of the statistical operator (density matrix) $\widehat{\rho }$ was established for the first time apparently by Mandelstam and Tamm [9]. Various proofs can be found also, e.g., in Refs. [29,30,43,45,201,212]. All inequalities expressing uncertainty relations in terms of variances are valid both for pure and mixed quantum states. But there is a difference between these two kinds of states. Namely, the strict inequality always occurs for mixed states in relation (3), since the minimum of its left-hand side is achieved only for Gaussian wave packets, which belong to the class of pure states [213]. Similarly, the equality in relation (5) in general holds only for pure states [45]. For example, in the case of an equilibrium state of a harmonic oscillator with frequency $\omega$ at temperature T, the uncertainty product equals ($k_B$ is the Boltzmann constant)

$${\sigma }_p{\sigma }_x={\left[{\displaystyle \frac{\hslash }{2}}coth\left({\displaystyle \frac{\hslash \omega }{2k_{B}T}}\right)\right]}^2$$

(259)

In the high-temperature case, $k_{B}T\gg \hslash \omega$, the right-hand side of (259) is so high that inequality (3) becomes practically useless. Therefore, the following problem arises naturally: to find generalizations of inequality (3), which would contain some extra dependence on parameters characterizing the “degree of purity” of a quantum state, in such a way that generalized relations could turn into an equality (perhaps, approximate) even for highly mixed states.

11.1. Inequalities Containing the “Second-Order Purity”

The simplest parameter characterizing the “purity” of a quantum state is [214]

$$\mu =\mathrm{T}\mathrm{r}{\widehat{\rho }}^2.$$

(260)

Remember that ${\widehat{\rho }}^2=\widehat{\rho }$ for pure states, so that $\mu =1$ due to the normalization condition $\mathrm{T}\mathrm{r}\widehat{\rho }=1$, whereas ${\widehat{\rho }}^2\ne \widehat{\rho }$ and $0<\mu <1$ for mixed states. It is known that for any quantum state described by means of a Gaussian density matrix or Wigner function (or some other quasiprobability distribution), the following equality holds for systems with one degree of freedom (see, e.g., [215]):

$$\sqrt{{\sigma }_{pp}{\sigma }_{xx}-{\sigma }_{xp}^2}={\displaystyle \frac{\hslash }{2\mu }}.$$

(261)

Consequently, one can look for a generalized “purity bounded uncertainty relation” in the form

$$\sqrt{{\sigma }_{pp}{\sigma }_{xx}-{\sigma }_{xp}^2}\ge {\displaystyle \frac{\hslash }{2}}\mathrm{\Phi }\left(\mu \right),$$

(262)

where $\mathrm{\Phi }\left(\mu \right)$ is a monotonic function of $\mu$, satisfying the relations $\mathrm{\Phi }\left(1\right)=1\le \mathrm{\Phi }\left(\mu \right)\le {\mu }^{-1}$ for $0<\mu \le 1$. Actually, it is sufficient to find function $\mathrm{\Phi }\left(\mu \right)$, considering a subfamily of states with zero covariance, taking into account that the quantity $\mathcal{D}\equiv {\sigma }_{pp}{\sigma }_{xx}-{\sigma }_{xp}^2$ is invariant with respect to arbitrary linear canonical transformations of operators $\widehat{x}$ and $\widehat{p}$ [49,86,215].

The first explicit expression for the function $\mathrm{\Phi }\left(\mu \right)$ in the form ${\mathrm{\Phi }}_B\left(\mu \right)=8/\left(9\mu \right)$ was found by Bastiaans [216] in the context of the problem of partially coherent light beams, where parameter $\mu$ had the meaning of the degree of space coherence. However, function ${\mathrm{\Phi }}_B\left(\mu \right)$ does not satisfy the condition $\mathrm{\Phi }\left(1\right)=1$, i.e., using this function, one arrives at the inequality, which is weaker than (3) for pure quantum states. In addition, the lower limit of the uncertainty product $4\hslash /\left(9\mu \right)$ cannot be achieved for any quantum state. Actually, ${\mathrm{\Phi }}_B\left(\mu \right)$ is an asymptotical form of the exact expression found for the first time in [28].

To obtain this exact expression we use the property of any statistical operator $\widehat{\rho }$ to possess a diagonal expansion over some complete orthogonal set of pure states labeled by an integral index m:

$$\widehat{\rho }=\sum _m{\rho }_m|m\rangle \langle m|,\sum {\rho }_m=1,\langle m|n\rangle ={\delta }_{mn}.$$

(263)

Let us order this set in accordance with the inequalities

$$1\ge {\rho }_0\ge {\rho }_1\ge \dots \ge {\rho }_m\ge {\rho }_{m+1}\ge \dots \ge 0,$$

(264)

and introduce an auxiliary function $E\left(\xi \right)={\sigma }_{pp}/\xi +\xi {\sigma }_{xx}$, which can be considered as twice the mean energy of some harmonic oscillator with a unit frequency. The eigenstates $|\tilde{n}\rangle$ of this auxiliary oscillator also form a complete orthonormalized set of states. The coefficients of the expansion of one basis over another form a unitary matrix:

$$|m\rangle =\sum _n{a}_{mn}\left(\xi \right)|\tilde{n}\rangle ,\sum _n{a}_{mn}{a}_{kn}^{*}={\delta }_{mk}.$$

(265)

Bastiaans [216] proved the inequality

$$E\left(\xi \right)/\hslash =\sum _{mn}{\rho }_m{\left|a_{mn}\left(\xi \right)\right|}^2(2n+1)\ge \sum _m{\rho }_m(2m+1)\equiv f\left({\rho }_0,{\rho }_1,\dots \right),$$

(266)

which is essentially based on the orthogonality condition (265) and the ordering condition (264). Since the right-hand side of (266) does not depend anymore on parameter $\xi$, the minimization of $E\left(\xi \right)$ with respect to $\xi$ results in the inequality $\sqrt{{\sigma }_{pp}{\sigma }_{xx}}\ge (\hslash /2)f\left({\rho }_0,{\rho }_1,\dots \right)$. Consequently, one has to find the minimum of function $f\left({\rho }_0,{\rho }_1,\dots \right)$ under the additional constraints

$$\sum {\rho }_m=1,\sum {\rho }_m^2=\mu .$$

(267)

Following [217], let us consider the function $\tilde{f}=f+\sum \left({\lambda }_2{\rho }_m^2-{\lambda }_1{\rho }_m\right)$, where ${\lambda }_1$ and ${\lambda }_2$ are the Lagrange multipliers. The minimum of this function is achieved for linearly decreasing with index m values ${\rho }_m=({\lambda }_1-1-2m)/\left(2{\lambda }_2\right)$ (notice that the ordering condition (264) is satisfied). The values of the Lagrange multipliers ${\lambda }_1$ and ${\lambda }_2$ can be easily found from equations (267). Finally, it appears convenient to parametrize the optimal set of coefficients as

$${\rho }_m={\displaystyle \frac{2(M+\gamma -m)}{(M+1)(M+2\gamma )}},$$

(268)

where M is the maximal integer until which the summations should be performed in Equations (266)–(267), and $\gamma$ is confined in the interval $0\le \gamma \le 1$. The relations between parameters M, $\gamma$, $\mu$ are as follows,

$$2\gamma ={\left\{{\displaystyle \frac{M(M+2)}{3\left[\mu \right(M+1)-1]}}\right\}}^{1/2}-M,\mu ={\displaystyle \frac{2M(2M+1)+12\gamma (M+\gamma )}{3(M+1){(M+2\gamma )}^2}}.$$

(269)

For a given value of the purity coeffient $\mu \le 1$, the integer M must satisfy two compatible constraints:

$${\mu }^{-1}-1\le M\le {\displaystyle \frac{4-3\mu +\sqrt{16+9{\mu }^2}}{6\mu }}.$$

(270)

For given M and $\mu$, the minimal value of function $f\left({\rho }_0,{\rho }_1,\dots \right)$ equals

$$f_{*}(\mu ;M)=1+M-{\left\{{\displaystyle \frac{M}{3}}(M+1)(M+2)\left[\mu -{\displaystyle \frac{1}{M+1}}\right]\right\}}^{1/2}.$$

(271)

Then, the function $\mathrm{\Phi }\left(\mu \right)$ can be obtained from (271), if one chooses the proper value of integer M. For example, if $1-\mu \ll 1$ (a slightly mixed state), then the only possible value is $M=1$. Consequently, for the values of $\mu$ close to unity, function $\mathrm{\Phi }\left(\mu \right)$ takes the form

$${\mathrm{\Phi }}_1\left(\mu \right)=2-\sqrt{2\mu -1}.$$

(272)

If $\epsilon \equiv 1-\mu \ll 1$, then ${\mathrm{\Phi }}_1(1-\epsilon )=1+\epsilon +{\epsilon }^2/2+\dots$. This value is obviously less than ${\mu }^{-1}=1+\epsilon +{\epsilon }^2+\dots$ (although the relative difference does not exceed 10% in the interval $0.5<\mu <1$, and it equals zero at the ends of this interval).

Function (272) is well defined for $\mu >1/2$. However, this function gives the minimal possible value of the product $\sqrt{{\sigma }_{pp}{\sigma }_{xx}}$ only in the interval $1\ge \mu \ge 5/9$, because at point ${\mu }_2=5/9$ the value $M=2$ becomes admissible in accordance with (270), and the new expression for $\mathrm{\Phi }\left(\mu \right)$ emerges:

$${\mathrm{\Phi }}_2\left(\mu \right)=3-\sqrt{8(\mu -1/3)}.$$

(273)

At the point ${\mu }_2$ both functions, (272) and (273), coincide: ${\mathrm{\Phi }}_1(5/9)={\mathrm{\Phi }}_2(5/9)=5/3$. Moreover, their first derivatives with respect to $\mu$ coincide at this point, too. But for $\mu <5/9$, we have ${\mathrm{\Phi }}_1\left(\mu \right)>{\mathrm{\Phi }}_2\left(\mu \right)$. It is easy to verify that for the given value of purity $\mu$, the minimal value of function (271) is achieved for the maximal admissible value of integer M, because the derivative of function (271) with respect to M equals $-\infty$ at $M={\mu }^{-1}-1$, which means that this function decreases with increase of M.

Thus, we arrive at the set of functions ${\mathrm{\Phi }}_k\left(\mu \right)\equiv f_{*}(\mu ;M=k)$, representing the minimizing function $\mathrm{\Phi }\left(\mu \right)$ in the intervals ${\mu }_k\ge \mu \ge {\mu }_{k+1}$, where the boundary points ${\mu }_k$ are determined from the condition that (270) becomes an equality for $M=k$:

$${\mu }_3={\displaystyle \frac{7}{18}},{\mu }_4={\displaystyle \frac{3}{10}},{\mu }_5={\displaystyle \frac{11}{45}},{\mu }_6={\displaystyle \frac{13}{63}},\dots ,{\mu }_k={\displaystyle \frac{2(2k+1)}{3k(k+1)}}={\displaystyle \frac{4}{3k}}\left[1-{\displaystyle \frac{1}{2(k+1)}}\right].$$

In particular,

$${\mathrm{\Phi }}_3\left(\mu \right)=4-\sqrt{20(\mu -1/4)},{\mathrm{\Phi }}_4\left(\mu \right)=5-\sqrt{40(\mu -1/5)},{\mathrm{\Phi }}_5\left(\mu \right)=6-\sqrt{70(\mu -1/6)}.$$

For $\mu ={\mu }_k$, these functions assume the values

$$\mathrm{\Phi }\left({\mu }_k\right)={\displaystyle \frac{1+2k}{3}}={\displaystyle \frac{4+\sqrt{16+9{\mu }_k^2}}{9{\mu }_k}}.$$

(274)

The first derivatives of functions ${\mathrm{\Phi }}_k\left(\mu \right)$ and ${\mathrm{\Phi }}_{k+1}\left(\mu \right)$ also coincide at the boundary points ${\mu }_{k+1}$. But their derivatives of the second and higher orders are different at these points.

We see that the explicit analytical form of function $\mathrm{\Phi }\left(\mu \right)$ turns out different for different segments of the interval $0<\mu \le 1$. Therefore, sometimes it could be convenient to use a simple interpolation expression, which is obtained by means of replacing an integer M in (271) by its maximal admissible value (270) (even if this value is not integral):

$$\tilde{\mathrm{\Phi }}\left(\mu \right)={\displaystyle \frac{4+\sqrt{16+9{\mu }^2}}{9\mu }}.$$

(275)

The functions $\tilde{\mathrm{\Phi }}\left(\mu \right)$ and $\mathrm{\Phi }\left(\mu \right)$ coincide at points ${\mu }_k$; in some interval immediately to the right from these points one has $\tilde{\mathrm{\Phi }}\left(\mu \right)>\mathrm{\Phi }\left(\mu \right)$, whereas immediately to the left of these points $\tilde{\mathrm{\Phi }}\left(\mu \right)<\mathrm{\Phi }\left(\mu \right)$. However, the difference between the exact and approximate values does not exceed $0.02$ even for the values of $\mu$ close to unity. Moreover, for $\mu \to 0$, this difference becomes less than ${\mu }^2/64$. For $\mu \ll 1$ the following asymptotical formula holds:

$$\tilde{\mathrm{\Phi }}\left(\mu \right)={\displaystyle \frac{8}{9\mu }}\left(1+{\displaystyle \frac{9}{64}}{\mu }^2+\dots \right),$$

(276)

and $\left|\mathrm{\Phi }\right(\mu )-8/(9\mu \left)\right|<0.01$ for $\mu \le 0,25\approx {\mu }_5$. Actually, the “degree of purity” of a quantum state can be characterized by means of many other quantities different from $\mu =\mathrm{T}\mathrm{r}{\widehat{\rho }}^2$, for example

$${\mu }^{\left(r\right)}={\left[\mathrm{T}\mathrm{r}\left({\widehat{\rho }}^{1+1/r}\right)\right]}^r,r>0.$$

(277)

The related generalizations were studied in Ref. [217].

11.2. Inequalities for Modified “Uncertainties”

From the mathematical point of view, variance uncertainty relations for coordinate and momentum in the case of pure quantum states are consequences of some mutual properties of the Fourier pairs $\psi \left(x\right)$ and $\phi \left(k\right)$, where $\psi \left(x\right)$ is a complex wave function describing the state in the coordinate picture and $\phi \left(k\right)$ is its counterpart in the momentum representation. On the other hand, mixed states are described in terms of density matrices $\rho (x,x^{\prime })$ or Wigner functions $W(x,p)$, which have twice more arguments and obey certain constraints, such as $\rho (x,x^{\prime })={\left[\rho (x^{\prime },x)\right]}^{*}$ or $ImW=0$. Furthermore, the normalization rules and the relations with average values of operators are different for $\rho (x,x^{\prime })$ or $W(x,p)$ and $\psi \left(x\right)$. Therefore, it is impossible to transfer literally to mixed states the same approaches that work for pure states, considering $\rho (x,x^{\prime })$ or $W(x,p)$ simply as “effective two-dimensional wave functions”. Some modifications are necessary, including a redefinition of “uncertainties” and “mean values”.

Usually, the average values entering the definition of quantum variances are calculated with the aid of the statistical operator $\widehat{\rho }$ as

$$\langle \widehat{A}\rangle =\mathrm{T}\mathrm{r}\left(\widehat{A}\widehat{\rho }\right).$$

(278)

Remembering that ${\widehat{\rho }}^n=\widehat{\rho }$ for pure states, but ${\widehat{\rho }}^n\ne \widehat{\rho }$ for mixed ones, there is a possibility to change the definition (278), replacing operator $\widehat{\rho }$ in (278) by ${\widehat{\rho }}^n$ or, in the simplest case, by ${\widehat{\rho }}^2$. This idea was formulated in connection with another problem in [215,218].

Chountasis and Vourdas [219] used a similar idea to find a generalization of uncertainty relations for mixed states. Considering the Wigner function

$$W(x,p)=\int \rho (x+X/2,x-X/2)exp(-iXp)dX$$

(279)

and its Fourier transform (known as the Weyl function or ambiguity function [220])

$$\begin{array}{ccc}\hfill \tilde{W}(X,P)& =& \int W(x,p)exp[-i(Px-Xp)]{\displaystyle \frac{dxdp}{2\pi }}\hfill \end{array}$$

(280)

$$\begin{array}{ccc}& =& \int \rho (x+X/2,x-X/2)exp(-ixP)dx,\hfill \end{array}$$

(281)

they have introduced two kinds of “uncertainties”:

$$\delta x\equiv {\left[\langle \langle x^2\rangle \rangle -{\langle \langle x\rangle \rangle }^2\right]}^{1/2},\delta X\equiv {\langle \langle X^2\rangle \rangle }^{1/2},$$

(282)

where the modified “mean values” are defined as follows,

$$\langle \langle x\rangle \rangle \equiv \int x{\left[W(x,p)\right]}^2{\displaystyle \frac{dxdp}{2\pi }}=\mathrm{T}\mathrm{r}\left[\widehat{x}{\widehat{\rho }}^2\right],$$

(283)

$$\langle \langle x^2\rangle \rangle \equiv \int x^2{\left[W(x,p)\right]}^2{\displaystyle \frac{dxdp}{2\pi }}={\displaystyle \frac{1}{2}}\mathrm{T}\mathrm{r}\left[{\widehat{x}}^2{\widehat{\rho }}^2\right]+{\displaystyle \frac{1}{2}}\mathrm{T}\mathrm{r}\left[\widehat{x}\widehat{\rho }\widehat{x}\widehat{\rho }\right],$$

(284)

$$\langle \langle X^2\rangle \rangle \equiv \int X^2{\left|\tilde{W}(X,P)\right|}^2{\displaystyle \frac{dXdP}{2\pi }}=2\mathrm{T}\mathrm{r}\left[{\widehat{x}}^2{\widehat{\rho }}^2\right]-2\mathrm{T}\mathrm{r}\left[\widehat{x}\widehat{\rho }\widehat{x}\widehat{\rho }\right]$$

(285)

(and similar definitions for p and P; note that $\langle \langle X\rangle \rangle =\langle \langle P\rangle \rangle =0$ identically). The generalized relations found in [219] read

$$\delta X\delta p\ge {\displaystyle \frac{1}{2}}\mathrm{T}\mathrm{r}{\widehat{\rho }}^2,\delta x\delta P\ge {\displaystyle \frac{1}{2}}\mathrm{T}\mathrm{r}{\widehat{\rho }}^2,\mathrm{T}\mathrm{r}{\widehat{\rho }}^2=\int {\left[W(x,p)\right]}^2{\displaystyle \frac{dxdp}{2\pi }}.$$

(286)

The equality signs in (286) are achieved for the Gaussian thermal states.

Ponomarenko and Wolf [221] have defined two complementary “generalized variances” of any operator $\widehat{A}$ in the following way:

$${\langle {(\Delta \widehat{A})}^2\rangle }_{\pm }\equiv \pm \mathrm{T}\mathrm{r}\left({[\Delta \widehat{A},\widehat{\rho }]}_{\pm }^2\right)$$

(287)

where $\Delta \widehat{A}=\widehat{A}-\mathrm{T}\mathrm{r}\left(\widehat{A}\widehat{\rho }\right)$ and ${[,]}_{+}$ stands for the anticommutator. In the basis of eigenstates $|a\rangle$ of operator $\widehat{A}$, the generalized variances can be expressed as [221]

$${\langle {(\Delta \widehat{A})}^2\rangle }_{\pm }=\sum _{a,a^{\prime }}{(a\pm a^{\prime })}^2{\left|\langle a\left|\widehat{\rho }\right|a^{\prime }\rangle \right|}^2.$$

(288)

The authors of [221] derived the inequality

$${\langle {(\Delta \widehat{A})}^2\rangle }_{-}{\langle {(\Delta \widehat{B})}^2\rangle }_{+}\ge {\left|\mathrm{T}\mathrm{r}\left({[\widehat{A},\widehat{B}]}_{-}{\widehat{\rho }}^2\right)\right|}^2$$

(289)

and proved that it is minimized on Gaussian (squeezed thermal) states. However, using (288) one can check that for Gaussian states (without covariance, for simplicity), the generalized variances are related to the standard ones as follows (here $\mu \equiv \mathrm{T}\mathrm{r}{\widehat{\rho }}^2$): ${\langle {(\Delta \widehat{x})}^2\rangle }_{+}=2{\sigma }_x\mu$, ${\langle {(\Delta \widehat{x})}^2\rangle }_{-}=2{\sigma }_x{\mu }^3$ (and similar relations for the momentum). Therefore, for these states, relation (289) is reduced to the known identity (261).

Several inequalities for mixed states were derived in paper [212]. But they seem to be rather weak, because one of their simplest versions for the coordinate and momentum operators has the form ${\sigma }_x{\sigma }_p\ge {(\hslash /2)}^2\mathrm{T}\mathrm{r}\left({\widehat{\rho }}^2\right)$.

11.3. Generalizations to Several Dimensions

Generalizations of the purity bounded uncertainty relations to the multidimensional case were found by Karelin and Lazaruk [222,223]. They looked for inequalities for the product $\Delta \mathbf{x}\Delta \mathbf{p}$, where $\mathbf{x},\mathbf{p}$ are s-dimensional vectors and

$${(\Delta \mathbf{x})}^2\equiv {\displaystyle \frac{1}{s}}\sum _{i=1}^s\mathrm{T}\mathrm{r}\left({\widehat{x}}_i^2\widehat{\rho }\right),{(\Delta \mathbf{p})}^2\equiv {\displaystyle \frac{1}{s}}\sum _{i=1}^s\mathrm{T}\mathrm{r}\left({\widehat{p}}_i^2\widehat{\rho }\right).$$

It is supposed that the average values of $\mathbf{x}$ and $\mathbf{p}$ equal zero. The inequality derived in [222] reads

$${(\Delta \mathbf{x}\Delta \mathbf{p})}^s\ge {\left({\displaystyle \frac{\hslash }{2}}\right)}^s{\displaystyle \frac{C\left(s\right)}{\mu }},C\left(s\right)={\displaystyle \frac{2^{s+1}(s+1)!}{{(s+2)}^{s+1}}},\mu =\mathrm{T}\mathrm{r}{\widehat{\rho }}^2,$$

$$C\left(1\right)={\displaystyle \frac{4}{9}},C(s\to \infty )\sim {\left({\displaystyle \frac{2}{e}}\right)}^{s+2}\sqrt{{\displaystyle \frac{\pi s}{2}}}.$$

More strict (but more complicated) inequalities, generalizing those considered in Section 11.1, were considered in study [223]. In particular, the following interpolating form was given:

$$\Delta \mathbf{x}\Delta \mathbf{p}\ge {\displaystyle \frac{\hslash }{2}}{\displaystyle \frac{s+2L\left(\mu \right)}{s+2}},$$

where $L\left(\mu \right)$ is a root of the transcendental equation [$\mathrm{\Gamma }\left(z\right)$ is Euler’s gamma-function]

$$\mu ={\displaystyle \frac{(s+2L)(s+1)!\mathrm{\Gamma }\left(L\right)}{(s+2)\mathrm{\Gamma }(L+s+1)}}.$$

11.4. Inequalities Containing the “Skew Information”

The following inequality was found for arbitrary observables in paper [224]:

$${\sigma }_A{\sigma }_B-{\sigma }_{AB}^2\ge {\displaystyle \frac{1}{4}}{\left|\langle [\widehat{A},\widehat{B}]\rangle \right|}^2\left[1+max\left({\displaystyle \frac{R_A}{1-R_A}},{\displaystyle \frac{R_B}{1-R_B}}\right)\right],$$

(290)

$$R_A\equiv {\left[\mathrm{T}\mathrm{r}\left(\delta \widehat{A}{\widehat{\rho }}^{1/2}\delta \widehat{A}{\widehat{\rho }}^{1/2}\right)/{\sigma }_A\right]}^2.$$

It was shown that the equality sign in (290) takes place for the $x-p$ variables in the thermal equalibrium state of the harmonic oscillator.

Luo [225] noticed that the variance ${\sigma }_A$ of the observable A in the mixed state $\widehat{\rho }$ “is a hybrid of both classical mixing and quantum uncertainty”. To remove the “classical mixing” part, he suggested to use, instead of ${\sigma }_A$, the quantity

$$U_A=\sqrt{{\sigma }_A^2-{\left({\sigma }_A-I_A\right)}^2},$$

(291)

where

$$I_A=-{\displaystyle \frac{1}{2}}\mathrm{T}\mathrm{r}\left({[{\widehat{\rho }}^{1/2},\widehat{A}]}^2\right)=\mathrm{T}\mathrm{r}\left(\widehat{\rho }{\widehat{A}}^2\right)-\mathrm{T}\mathrm{r}\left({\widehat{\rho }}^{1/2}\widehat{A}{\widehat{\rho }}^{1/2}\widehat{A}\right)$$

(292)

is the so called “skew-information” introduced by Wigner and Yanase [226] (see also [227] about the properties of this quantity). For pure quantum states with $\widehat{\rho }=|\psi \rangle \langle \psi |$, the quantity $I_A$ coincides with the variance ${\sigma }_A$. The inequality proved in [225], which can be considered as a modified version of (4), has the form

$$U_A{U}_B\ge {\displaystyle \frac{1}{4}}{\left|\langle [\widehat{A},\widehat{B}]\rangle \right|}^2.$$

(293)

Obviously, any operator like $\widehat{A}$ can be replaced everywhere with its zero mean value partner $\widehat{A}-\langle \widehat{A}\rangle$. A modified version of (5) was given in paper [228]:

$$U_A{U}_B-{\left|Re\left\{\mathrm{C}\mathrm{o}\mathrm{r}\mathrm{r}\left(\widehat{A},\widehat{B}\right)\right\}\right|}^2\ge {\displaystyle \frac{1}{4}}{\left|\langle [\widehat{A},\widehat{B}]\rangle \right|}^2,$$

(294)

$$\mathrm{C}\mathrm{o}\mathrm{r}\mathrm{r}\left(\widehat{A},\widehat{B}\right)=\mathrm{T}r\left(\widehat{\rho }{\widehat{A}}^{†}\widehat{B}\right)-\mathrm{T}\mathrm{r}\left({\widehat{\rho }}^{1/2}{\widehat{A}}^{†}{\widehat{\rho }}^{1/2}\widehat{B}\right).$$

(295)

Further generalizations were made in [229,230]:

$$U_A^{\left(\alpha \right)}{U}_B^{\left(\alpha \right)}-{\left|Re\left\{{\mathrm{C}\mathrm{o}\mathrm{r}\mathrm{r}}_{\alpha }\left(\widehat{A},\widehat{B}\right)\right\}\right|}^2\ge \alpha (1-\alpha ){\left|\langle [\widehat{A},\widehat{B}]\rangle \right|}^2,0\le \alpha \le 1.$$

(296)

Here, the Wigner–Yanase skew information in the definition of $U_A^{\left(\alpha \right)}$ is replaced with the “Wigner–Yanase–Dyson skew information”

$$I_A^{\left(\alpha \right)}=-{\displaystyle \frac{1}{2}}\mathrm{T}\mathrm{r}\left([{\widehat{\rho }}^{\alpha },\widehat{A}][{\widehat{\rho }}^{1-\alpha },\widehat{A}]\right)=\mathrm{T}\mathrm{r}\left(\widehat{\rho }{\widehat{A}}^2\right)-\mathrm{T}\mathrm{r}\left({\widehat{\rho }}^{\alpha }\widehat{A}{\widehat{\rho }}^{1-\alpha }\widehat{A}\right).$$

(297)

Similarly,

$${\mathrm{C}\mathrm{o}\mathrm{r}\mathrm{r}}_{\alpha }\left(\widehat{A},\widehat{B}\right)=\mathrm{T}\mathrm{r}\left(\widehat{\rho }{\widehat{A}}^{†}\widehat{B}\right)-\mathrm{T}\mathrm{r}\left({\widehat{\rho }}^{\alpha }{\widehat{A}}^{†}{\widehat{\rho }}^{1-\alpha }\widehat{B}\right).$$

(298)

Inequalities (293), (296) and some of their extensions were also proven in papers [231,232]. More complicated inequalities based on the concept of skew information can be found in papers [233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252].

11.5. Simple Examples

To illustrate some of relations presented in this section, let us consider the following mixture of the first Fock states of a harmonic oscillator:

$$\widehat{\rho }={\rho }_0|0\rangle \langle 0|+{\rho }_1|1\rangle \langle 1|,{\widehat{\rho }}^{1/2}={\rho }_0^{1/2}|0\rangle \langle 0|+{\rho }_1^{1/2}|1\rangle \langle 1|,{\rho }_k\ge 0,{\rho }_0+{\rho }_1=1.$$

(299)

Taking $\widehat{A}=\widehat{x}$ and $\widehat{B}=\widehat{p}$ in dimensionless units, one obtains $U_x=U_p={\displaystyle \frac{1}{2}}\sqrt{1+8{\rho }_1^2}.$ Hence, inequality (293) does not provide any nontrivial information in this case. Moreover, since $\mathrm{C}\mathrm{o}\mathrm{r}\mathrm{r}\left(\widehat{x},\widehat{p}\right)=i/2$ for all values of ${\rho }_0$ and ${\rho }_1$, inequality (294) turns out as useless as (293).

On the other hand, ${\sigma }_x={\sigma }_p={\displaystyle \frac{1}{2}}(1+2{\rho }_1)$ in the state (299). The purity of this state equals $\mu =1-2{\rho }_0{\rho }_1$. Then, inequality (262) with function $\mathrm{\Phi }\left(\mu \right)$ given by Equation (272) assumes the form

$$1+2{\rho }_1\ge 2-\sqrt{1-4{\rho }_1+4{\rho }_1^2},$$

with the equality achieved for ${\rho }_1=1/3$.

Another example is the thermal state,

$$\widehat{\rho }=(1-z)\sum _{n=0}^{\infty }{z}^n|n\rangle \langle n|,{\widehat{\rho }}^{1/2}={(1-z)}^{1/2}\sum _{n=0}^{\infty }{z}^{n/2}|n\rangle \langle n|,0\le z<1.$$

(300)

Then,

$${\sigma }_x={\sigma }_p={\displaystyle \frac{1+z}{2(1-z)}},{\sigma }_x-I_x={\displaystyle \frac{z^{1/2}}{1-z}}.$$

Consequently, $U_x=U_p=1/2$. This remarkable result means that the quantum part of fluctuations in the thermal state does not depend on temperature, i.e., high values of variances ${\sigma }_x$ and ${\sigma }_p$ in the high-temperature states have the classical nature. In particular, thermal states minimize the inequality (293) for any temperature.

11.6. Purity- and Gaussianity-Constrained Uncertainty Relations

The minimum of the uncertainty product ${\sigma }_x{\sigma }_p$ is attained in pure Gaussian states. The role of purity $\mu =\mathrm{T}\mathrm{r}\left({\widehat{\rho }}^2\right)$ was considered in Section 11.1. And what can happen if the quantum state is not Gaussian? For example, the uncertainty product is very big in the highly excited harmonic oscillator eigenstates $|n\rangle$: ${\sigma }_x{\sigma }_p={[\hslash (n+1/2)]}^2$. Some answers to this question were given in papers [253,254,255].

There exist many measures of “Gaussianity” or “non-Gaussianity” of a quantum state described by the statistical operator $\widehat{\rho }$ [256,257,258,259,260,261,262]. The authors of [253] used the parameter

$$g={\displaystyle \frac{\mathrm{T}\mathrm{r}\left(\widehat{\rho }{\widehat{\rho }}_G\right)}{\mathrm{T}\mathrm{r}\left({\widehat{\rho }}_G^2\right)}},$$

(301)

where ${\widehat{\rho }}_G$ is the Gaussian statistical operator with the same first- and second-order statistical moments as the operator $\widehat{\rho }$. This parameter is restricted to the interval $2/e\le g\le 2$. The goal was to find function $\mathcal{F}\left(g\right)$ in the inequality

$$\sqrt{\mathcal{D}}\equiv \sqrt{{\sigma }_x{\sigma }_p-{\sigma }_{xp}^2}\ge \hslash \mathcal{F}\left(g\right).$$

(302)

Explicit analytical expressions were found for the interval $3/4\le g<2$:

$$\mathcal{F}\left(g\right)=\left\{\begin{array}{cc}{\displaystyle \frac{g}{2(2-g)}}\hfill & \mathrm{for}1\le g<2,\hfill \\ {\displaystyle \frac{2-g+2\sqrt{1-g}}{2g}}\hfill & \mathrm{for}3/4\le g\le 1.\hfill \end{array}\right.$$

(303)

No analytical expressions for $\mathcal{F}\left(g\right)$ were obtained for the interval $2/e<g<3/4$. Unfortunately, almost all oscillator number states $|n\rangle$ belong to this “tiny” interval: $3/4-2/e\approx 0.014$. Indeed, the operator ${\widehat{\rho }}_G$ for the state ${\widehat{\rho }}_n=|n\rangle \langle n|$ corresponds to the Gaussian thermal state

$${\widehat{\rho }}_G=(1-\xi )\sum _{k=0}^{\infty }{\xi }^k|k\rangle \langle k|,\xi ={\displaystyle \frac{n}{n+1}}.$$

(304)

Hence,

$$\mathrm{T}\mathrm{r}\left({\widehat{\rho }}_G^2\right)={\displaystyle \frac{1-\xi }{1+\xi }},\mathrm{T}\mathrm{r}\left({\widehat{\rho }}_n{\widehat{\rho }}_G\right)=\langle n|{\widehat{\rho }}_G|n\rangle =(1-\xi ){\xi }^n={\displaystyle \frac{n^n}{{(1+n)}^{1+n}}}.$$

As a consequence, the dependence of values $g_n$ on the numbers n is extremely weak for $n\ge 1$:

$$g_n={\displaystyle \frac{(1+2n)n^n}{{(1+n)}^{1+n}}},g_1=0.75,g_2\approx 0.741,g_3\approx 0.738,\dots g_{\infty }\approx 0.73576.$$

(305)

On the other hand, numerical plots in paper [253] show very strong dependence $\mathcal{F}\left(g\right)$ for $g<3/4$. It was shown that the oscillator eigenstates saturate the inequlity (302) with function $\mathcal{F}\left(g\right)$ found numerically. Moreover, this numeric function $\mathcal{F}\left(g\right)$ turns out discontinuous at the points $g_n$.

Using the expansion

$$g_n\approx {\displaystyle \frac{2}{e}}\left(1+{\displaystyle \frac{1}{24N^2}}\right),N=n+1/2\gg 1,$$

one can suggest [by analogy with Equation (275)] the following approximate interpolation formula:

$$\tilde{\mathcal{F}}\left(g\right)\approx {\left[12(eg-2)\right]}^{-1/2},0<eg-2\ll 1.$$

(306)

The purity parameter $\mu$ was taken into account in paper [254]:

$$\sqrt{\mathcal{D}}\ge {\displaystyle \frac{\hslash g}{2\mu (2-g)}},1<g<2.$$

(307)

The inequality for $2/e<g<1$ and $\mu <1$ was given in a complicated approximate form, which is certainly not the best possible.

The authors of study [255] used the measure of “non-Gaussianity” in the form

$$\mathcal{N}\left(\widehat{\rho }\right)=-2ln\left[F\left(\widehat{\rho },{\widehat{\rho }}_G\right)\right]=-2ln\mathrm{T}\mathrm{r}\left(\sqrt{\sqrt{\widehat{\rho }}{\widehat{\rho }}_G\sqrt{\widehat{\rho }}}\right),$$

(308)

where the fidelity $F\left(\widehat{\rho },{\widehat{\rho }}_G\right)$ characterizes a distance between a state $\widehat{\rho }$ and its reference Gaussian state ${\widehat{\rho }}_G$. The following inequality was found:

$$\sqrt{\mathcal{D}}\ge \hslash H^{-1}\left[S\left(\widehat{\rho }\right)+\mathcal{N}\left(\widehat{\rho }\right)\right],$$

(309)

where $S\left(\widehat{\rho }\right)=-\mathrm{T}\mathrm{r}\left[\widehat{\rho }ln\left(\widehat{\rho }\right)\right]$ is the standard quantum entropy. $H^{-1}\left[y\right]$ is the inverse of the monotonic increasing function

$$H\left(x\right)=(x+1/2)ln(x+1/2)-(x-1/2)ln(x-1/2),x\ge 1/2.$$

A weaker but readily computable uncertainty relation has the form

$$\sqrt{\mathcal{D}}\ge \hslash H^{-1}\left[-ln\left(\mu \right)+{\mathcal{N}}_g\left(\widehat{\rho }\right)\right],$$

(310)

where

$${\mathcal{N}}_g\left(\widehat{\rho }\right)=-ln\left[\mathrm{T}\mathrm{r}\left(\widehat{\rho }{\widehat{\rho }}_G\right)+\sqrt{1-\mathrm{T}\mathrm{r}\left({\widehat{\rho }}^2\right)}\sqrt{1-\mathrm{T}\mathrm{r}\left({\widehat{\rho }}_G^2\right)}\right].$$

(311)

Note that $S\left(\widehat{\rho }\right)\ge -ln\left(\mu \right)$ and $\mathcal{N}\left(\widehat{\rho }\right)\ge {\mathcal{N}}_g\left(\widehat{\rho }\right)$. Moreover, $\mathcal{N}={\mathcal{N}}_g$ for any pure quantum state $\widehat{\rho }=|\psi \rangle \langle \psi |$. Unpleasant feature of inequalities (309) and (310) is that function $H^{-1}\left[y\right]$ can be calculated only numerically in the general case. However, a simple analytical form can be found in the most interesting case of highly excited Fock states with $n\gg 1$, when $y={\mathcal{N}}_g\approx ln\left(n\right)+1+1/\left(2n\right)\gg 1$. Indeed, for $x\gg 1$ one has $H\left(x\right)=ln\left(x\right)+1+\mathcal{O}\left(x^{-2}\right)$ and $H^{-1}\left(y\right)\approx e^{y-1}$. Then, the right-hand sides of inequalities (309) and (310) become equal to $\hslash (n+1/2)$, meaning that highly excited Fock states saturate exactly both “non-Gaussianity bounded” inequalities.

12. Inequalities for Higher Moments

Higher-order moments of quadratures or annihilation/creation operators are useful tools in different problems of quantum optics [93,263,264,265,266,267,268,269]. Therefore, finding limitations on their products is an important task.

A special case of the standard uncertainty relation (3) is the inequality (we use the variable $k=p/\hslash$ instead of p in this section)

$$\langle {\widehat{x}}^2\rangle \langle {\widehat{k}}^2\rangle \ge {\displaystyle \frac{1}{4}}={\mathcal{P}}_2.$$

(312)

Introducing the notation ${\mathrm{\Pi }}^{\left(n\right)}\equiv \langle |\widehat{x}{|}^n\rangle \langle |\widehat{k}{|}^n\rangle$, it seems natural to look for the minimal value of this product, ${\mathcal{P}}_n=\mathrm{m}\mathrm{i}\mathrm{n}\left\{{\mathrm{\Pi }}^{\left(n\right)}\right\}$, for an arbitrary exponent n. Trivial estimations can be made with the aid of (312) and the inequality $\langle {\widehat{A}}^2\rangle \ge {\langle \widehat{A}\rangle }^2$. In particular ${\mathcal{P}}_4\ge 4^{-2}$ and ${\mathcal{P}}_8\ge 4^{-4}$. Some other estimations can be performed with the aid of relations given in Section 10. For example, inequality (244) in one dimension with $\alpha =2$ and $\nu ={\nu }_1=0$ takes the form $3\langle {\widehat{x}}^2\rangle \le 2{\left[\langle {\widehat{x}}^6\rangle \langle {\widehat{k}}^2\rangle \right]}^{1/2}$. It is evident that this inequality remains valid if one interchanges the coordinates and momenta. Multiplying both parts of two inequalities of this form, one arrives at the estimations ${\mathcal{P}}_6\ge 81/64$ and ${\mathcal{P}}_{12}\ge {\mathcal{P}}_6^2\ge 9^4/4^6$. Similarly, assuming $\alpha =4$ and $\alpha =6$ in (244), one obtains the inequalities

$$\langle {\widehat{k}}^{10}\rangle \langle {\widehat{x}}^{10}\rangle \langle {\widehat{x}}^2\rangle \langle {\widehat{k}}^2\rangle \ge {(5/2)}^4{\left[\langle {\widehat{x}}^4\rangle \langle {\widehat{k}}^4\rangle \right]}^2\langle {\widehat{k}}^{14}\rangle \langle {\widehat{x}}^{14}\rangle \langle {\widehat{x}}^2\rangle \langle {\widehat{k}}^2\rangle \ge {(7/2)}^4{\left[\langle {\widehat{x}}^6\rangle \langle {\widehat{k}}^6\rangle \right]}^2,$$

which result in the estimations ${\mathcal{P}}_{10}\ge 5^4/4^5$ and ${\mathcal{P}}_{14}\ge 7^4\times 9^4/4^7$. The estimations of this kind can be called “variance bounds” for the ${\mathcal{P}}_n$. They can be improved in an evident way on the basis if inequality (9) (i.e., by means of introducing a correlation coefficient) or of inequality (262) (i.e., using the “degree of purity” of the state), although such bounds can be rather far from the best ones.

Note that inequality (244) is by no means trivial for odd positive values of the exponent $\alpha$, since the quantity r equals $\left|x\right|>0$ in the one-dimensional case. Nonetheless, relations (240) and (242) are still valid when $r=\left|x\right|$. In particular, for $\alpha =1$ we have the inequalities

$${\langle \left|x\right|\rangle }^2\le \langle {\widehat{x}}^4\rangle \langle {\widehat{k}}^2\rangle ,{\left(\langle \left|x\right|\rangle \langle \left|k\right|\rangle \right)}^2\le \langle {\widehat{x}}^4\rangle \langle {\widehat{k}}^4\rangle \langle {\widehat{x}}^2\rangle \langle {\widehat{k}}^2\rangle .$$

(313)

The first of these inequalities assumes the form $2/\pi <3/4$ for the ground state of an oscillator.

Other lower bounds for the products of the higher moments of the wave function $\psi \left(x\right)$ and its Fourier transform $\phi \left(k\right)$ were obtained in studies [270,271] in connection with the so called “entropic uncertainty relations” (briefly discussed at the end of this review). The following inequality can be extracted from those papers (see [28] for more details):

$$M_{a,q}\left[\psi \right]M_{b,r}\left[\phi \right]\ge {\displaystyle \frac{e\pi }{A\left(q\right)A\left(r\right)}},$$

(314)

where

$$M_{a,q}\left[\psi \right]={\left[{\int |x-a|}^q{\left|\psi \left(x\right)\right|}^{2}dx\right]}^{1/q},A\left(q\right)=2\mathrm{\Gamma }(1/q){\left[e{q}^{1-q}\right]}^{1/q}.$$

(315)

The consequence of (314) is the inequality for the product of moments of the same order (let us call it the “entropic bound”):

$${\langle \left|k\right|}^q{\rangle \langle \left|x\right|}^q{\rangle \ge (\pi e/4)}^q{q}^{2(q-1)}{e}^{-2}{\left[\mathrm{\Gamma }(1/q)\right]}^{-2q}.$$

(316)

The right-hand side of (316) equals $1/4$ for $q=2$.

Many related inequalities were given in the paper by Brizuela [272], although without proofs for most of them. Numerical evaluations, based on some variational schemes, were performed by Lynch and Mavromatis (LM) in paper [273]. Four evaluations of the value of ${\mathcal{P}}_n$ are compared with the vacuum product ${\mathrm{\Pi }}_{vac}^{\left(2m\right)}={\left[2^{-m}(2m-1)!!\right]}^2$ in

Table 1. Different evaluations of the minimal higher order products.

n	${\mathbf{\Pi }}_{\mathit{vac}}^{\left(\mathit{n}\right)}$	${\mathcal{P}}_{\mathit{n}}^{\mathit{var}}$	${\mathcal{P}}_{\mathit{n}}^{\mathit{ent}}$	${\mathcal{P}}_{\mathit{n}}^{\mathit{Briz}}$	${\mathcal{P}}_{\mathit{n}}^{\mathit{LM}}$
2	0.25	0.25	0.25	0.25	0.25
4	9/16	1/16	0.3857	3/8	0.4878
6	225/64	81/64	0.88	81/64	2.18
8	43.07	1/256	2.38	9/4	17.18
10	872.1	625/1024	7.22	225/16
12	26,381	1.6	23.4
14	1,114,592	961.5	80

.

We see that the “variance” bounds are very bad for $n=4$ and $n=8$. On the other hand, the variance bounds are better than the entropic ones for $n=6$ and especially $n=14$. Nonetheless, unfortunately, all known analytic bounds are significantly lower than the numeric evaluations.

Probably, the most interesting inequalities are those relating to the moments of the fourth order. The left-hand side of the second-order uncertainty relation (3) is minimal for the Gaussian states. But the fourth-order moments in these states are given by the formula $\langle {\widehat{x}}^4\rangle =3{\langle {\widehat{x}}^2\rangle }^2$. Therefore, the minimal product of the fourth-order moments in the case of Gaussian states equals ${\mathrm{\Pi }}_{vac}^{\left(4\right)}=\langle {\widehat{k}}^4\rangle {\langle {\widehat{x}}^4\rangle }_{vac}=9/16$, which is appreciably greater than the analytic bounds given in Table 1. Thus, the question arises whether states possessing the property $\langle {\widehat{k}}^4\rangle \langle {\widehat{x}}^4\rangle <9/16$ really exist? The answer is positive. It was shown in [28] that ${\mathrm{\Pi }}^{\left(4\right)}\approx 0.4901$ for the superposition of two Fock states $\sqrt{(1+\delta )/2}|0\rangle -\sqrt{(1-\delta )/2}|4\rangle$ with $\delta =\sqrt{150/151}$.

The best known minimal value ${\mathrm{\Pi }}_{min}^{\left(4\right)}={\left(0.6984\right)}^2=0.4878$ was found in [273] by applying the variational approach to the states of the form ${\sum }_{n=0}^K{c}_n|4n\rangle$ with $K=6$. The same result was confirmed in [274]. Practically the same minimal value ${\mathrm{\Pi }}_{min}^{\left(4\right)}\approx 0.49$ was found in papers [275,276,277,278] for the superposition of four coherent states with equal amplitudes and phases shifted by $\pi /2$,

$${|\psi \rangle }_{4\alpha }=B\left(|\alpha \rangle +|i\alpha \rangle +|-\alpha \rangle +|-i\alpha \rangle \right)=\mathcal{N}\sum _{n=0}^{\infty }{\displaystyle \frac{{\alpha }^{4n}}{\sqrt{\left(4n\right)!}}}|4n\rangle ,$$

(317)

with $\alpha \approx 0.67$ (here, B and $\mathcal{N}$ are the normalization factors). Properties of states (317) were studied for a long time from different points of view. These states were named “orthogonal-even coherent states” [275], “four-photon states” [279], “pair cat states” [280], “compass states” [281], “four-headed cat states” [282]. However, finding accurate analytical bounds on the products of the higher order moments is still a challenge.

13. Concluding Remarks About Other Families of “Uncertainty Relations”

Actually, the variance-based uncertainty relations described in this concise review are only “the visible part” of the great “uncertainty iceberg” that grew over the past several decades. The “invisible part” (which is, probably, even more interesting) includes several great areas listed below. For each area, I provide a very short description and a few references to the initial publications and some reviews, as soon as complete reference lists are of the same length or even bigger than that in this review.

In brief, the uncertainty principle in its “preparation” form states that a nonzero function $\psi \left(x\right)$ and its Fourier transform $\phi \left(k\right)$ cannot be sharply localized simultaneously. The variances are the simplest measures of localization. However, in many cases, such measures are not quite adequate. Consider, for example, a superposition of two well-localized functions separated by a big distance:

$$\psi \left(x\right)=N\left\{exp\left[-{(x-a)}^2/b^2\right]+exp\left[-{(x+a)}^2/b^2\right]\right\},\left|b\right|\ll \left|a\right|.$$

(318)

Although both peaks can be very narrow if $\left|b\right|\ll 1$ (in dimensionless units), the variance ${\sigma }_x$ can be made as large as desired, simply by increasing the distance $2\left|a\right|$ between the peaks. The obvious disadvantage of inequality (3) is that it does not forbid the possibility of the existence of a narrow hump of the function ${\left|\phi \left(k\right)\right|}^2$, when the small variance of ${\sigma }_p$ would be compensated for by a large value of the variance ${\sigma }_x$. In fact, this is impossible. This impossibility can be proven if the measures of localization are chosen in the form of the coordinate and momentum “entropies”,

$$S_x=-{\int \left|\psi \left(x\right)\right|}^2{ln\left(\right|\psi \left(x\right)|}^2)dx,S_k=-{\int \left|\phi \left(k\right)\right|}^2{ln\left(\right|\phi \left(k\right)|}^2)dk.$$

(319)

If the function ${\left|\psi \left(x\right)\right|}^2$ is localized in some small domain, its values inside this domain are large, and the corresponding logarithms are positive. As a result $S_x<0$, since the contribution to the integral of the points outside the localization domain are suppressed by small values of ${\left|\psi \left(x\right)\right|}^2$ in those points. If the function ${\left|\phi \left(k\right)\right|}^2$ was also highly localized, then the relation $S_k<0$ would be fulfilled, as well, resulting in the inequality $S_x+S_k<0$. However, the correct inequality reads

$$S_x+S_k\ge ln\left(\pi e\right)>0.$$

(320)

The history of this “entropic uncertainty relation” and its generalizations began in 1957 [270], thirty years after Heisenberg’s relation (1). By now, more than 500 papers have been published on this subject, according to the Web of Science database. Some reviews can be found in Refs. [283,284,285,286,287,288,289,290,291,292,293,294].

Another type of “entropic” inequalities was suggested in study [295]. In that paper, two noncommuting Hermitian operators $\widehat{A}$ and $\widehat{B}$, possessing discrete spectra and complete orthonormalized sets of eigenvectors $\left\{\right|a\rangle \}$ and $\left\{\right|b\rangle \}$, were considered. The entropies $S^A$ and $S^B$ for an arbitrary state $|\psi \rangle$ were defined as follows:

$$S^A=-\sum _a{\left|\langle a|\psi \rangle \right|}^2{ln\left(\right|\langle a|\psi \rangle |}^2),S^B=-\sum _b{\left|\langle b|\psi \rangle \right|}^2{ln\left(\right|\langle a|\psi \rangle |}^2).$$

(321)

It was proved that these quantities satisfy the inequality

$$S_A\left(\right|\psi \rangle )+S_B\left(\right|\psi \rangle )\ge 2ln\left({\displaystyle \frac{2}{1+\mathrm{s}\mathrm{u}\mathrm{p}\left\{\right|\langle a|b\rangle |\}}}\right),$$

(322)

where the supremum is taken with respect to all possible values of the scalar product of vectors $|a\rangle$ and $|b\rangle$. The main advantage of inequality (322) over (4) is that the right-hand side of (322), unlike (4), does not depend on the state $|\psi \rangle$ of the system: it is determined (although implicitly) only by the operators $\widehat{A}$ and $\widehat{B}$. This line of research was continued in papers [296,297,298]. For more recent studies, one may consult Refs. [299,300,301,302].

Several inequalities related to the simultaneous localization of the wave functions and their Fourier transforms have been found by Hardy in 1933 [303]. These inequalities, demonstrating that the Gaussian functions are the “best” in a certain sense, gave rise to numerous studies on the “Hardy uncertainty principle”: see, e.g., papers [304,305,306] and references therein.

The first results concerning the local properties of functions and their Fourier transforms were obtained by Fuchs in 1954 [307], followed by H.J. Landau and Pollak in 1961 [308]. Many impressive inequalities were found in 1978 by Faris [199], who proposed the name “local uncertainty relations”. To understand their significance, let us suppose that, for example, the function $\phi \left(p\right)$ possesses a sharp maximum, so that the value of $\Delta p$ is small. Using re1ation (1), one can conclude that the value of $\Delta x$ must be large. However, the large value of $\Delta x$ would not contradict the assumption that the function $\psi \left(x\right)$ has two sharp “humps” located at a great distance from each other. The essence of the “local uncertainty relations” consists in the statement that in reality not only is the value of $\Delta x$ great when $\Delta p$ is small, but the probability density ${\left|\psi \left(x\right)\right|}^2$ is always small (so that two sharp “humps” cannot exist). To have an idea, I bring here one of the simplest examples:

$${\left|\psi \left(y\right)\right|}^2\le \Delta p/\hslash ,-\infty <y<\infty .$$

(323)

Many other impressive results in this direction were obtained by several authors in 1980s [309,310]. In particular, Hilgevoord and Uffink [311,312] introduced the concepts of the “total width” of the wave function and its “mean peak width”, having derived several inequalities connecting them. For more recent publications, one may consult [313,314].

An apparent similarity between the formula ${\widehat{L}}_z=-i\hslash \partial /\partial \phi$ for the component of the angular momentum operator and the formula $\widehat{p}=-i\hslash \partial /\partial x$ for the linear momentum operator suggests an idea of the existence of the inequality $\Delta L_z\Delta \phi \ge \hslash /2$. However, this relation is incorrect (although it can be met in some simplified textbooks). The well known counterexamples are the eigenstates of ${\widehat{L}}_z$ with $\psi \left(\phi \right)\sim exp\left(im\phi \right)$ and $\Delta L_z=0$. The origin of difficulties is the non-Hermiticity of the operator $\widehat{\phi }$ understood as the simple multiplication operator: if $\psi \left(\phi \right)$ is a periodic function, then $\phi \psi \left(\phi \right)$ is obviously non-periodic. The problem of the correct description of phase or angle in quantum mechanics, including various forms of “uncertainty relations” as a special case, was the subject of numerous publications: see a few examples [115,116,315,316,317,318,319,320,321,322,323,324,325,326].

The energy–time uncertainty relation

$$\Delta E\Delta T\gtrsim h$$

(324)

is one of the most famous and at the same time most controversial formulas of quantum theory. It was introduced by Heisenberg [1] together with his coordinate–momentum uncertainty relation (1). The importance of both relations for the interpretation of quantum mechanics was emphasized by Bohr [4]. However, the further destiny of relations (1) and (324) turned out quite different, because the physical and mathematical meanings of inequality (324) appeared to be much less clear than that of (1) (and less clear than Heisenberg, Bohr, and other creators of quantum mechanics thought initially). The main reason is that, in fact, there are several quite different physical problems where relations like (324) can arise, and in each concrete case, the meaning of the quantities standing on the left-hand side proved to be different. This was clearly demonstrated for the first time by Mandelstam and Tamm [9], and many authors arrived at the same conclusions later [327,328,329,330,331,332,333]. A similar old area of studies is uncertainty relations for real signals in radiophysics and optics [333,334,335,336,337,338,339,340,341,342,343], i.e., the exact meaning of the relation $\Delta \omega \Delta t\gtrsim 1$. This area continues to attract the attention of many researchers [344,345,346,347,348,349].

One more area includes specific uncertainty relations in relativistic quantum mechanics and modifications for curved spaces in connection with the problem of localization of relativistic particles [350,351,352]. In particular, many researchers considered the consequences of the modified canonical commutation relations in the form $\left[\widehat{x},\widehat{p}\right]=i\hslash \left(1+\alpha {\widehat{x}}^2+\beta {\widehat{p}}^2\right)$, known under the name “generalized (gravitational) uncertainty principle (GUP)”. Such modifications result in the existence of the minimal coordinate length. The total number of related publications exceeds several hundred. Few examples can be found in Refs. [353,354,355,356,357,358,359]. A more recent wide area is related to the so-called “thermodynamic uncertainty relations” [360,361,362,363]. Applications of the uncertainty relations include: evaluations of the energy eigenvalues [198,199,200,364,365], the detection and measures of entanglement or separability [83,84,104,111,255,366,367,368,369,370,371], various families of “coherent”, “squeezed”, “intelligent” and other “nonclassical” states [45,370,372,373,374]. Measurement URs and their experimental verifications belong to another wide field of research [15,23,24,25,26,27,30,375,376,377,378]. Brief discussions of some subjects mentioned above can be found in the mini-review [379]. However, detailed reviews still await their authors.

The main conclusion that can be derived from the present review is that the area of uncertainty relations is still quite active almost one hundred years since the first publications. The history shows that new nontrivial results appeared (quite unexpectedly) almost every decade. Hence, one may suppose that new interesting results will be discovered in a not so far away future.