# Sources of Asymmetry and the Concept of Nonregularity of n-Dimensional Density Matrices

## Abstract

**:**

**ρ**is parametrized and interpreted in terms of its asymmetry properties through the introduction of a family of components of purity that are invariant with respect to arbitrary rotations of the nD Cartesian reference frame and that are composed of two categories of meaningful parameters of different physical nature: the indices of population asymmetry and the intrinsic coherences. It is found that the components of purity coincide, up to respective simple coefficients, with the intrinsic Stokes parameters, which are also introduced in this work, and that determine two complementary sources of purity, namely the population asymmetry and the correlation asymmetry, whose weighted square average equals the overall degree of purity of

**ρ**. A discriminating decomposition of

**ρ**as a convex sum of three density matrices, viz. the pure, the fully random (maximally mixed) and the discriminating component, is introduced, which allows for the definition of the degree of nonregularity of

**ρ**as the distance from

**ρ**to a density matrix of a system composed of a pure component and a set of 2D, 3D,… and nD maximally mixed components. The chiral properties of a state

**ρ**are analyzed and characterized from its intimate link to the degree of correlation asymmetry. The results presented constitute a generalization to nD systems of those established and exploited for polarization density matrices in a series of previous works.

## 1. Introduction

**v**is fixed), such a state is pure. Nevertheless, in general, uncertainties or fluctuations on the components of

**v**should be considered, and the corresponding mixed state, which necessarily involves a certain increase of symmetry, is represented by the associated density matrix

**ρ**whose elements ${\rho}_{ij}$ are the ensemble averages ${\rho}_{ij}=\langle {v}_{i}^{*}{v}_{j}\rangle $ over the realizations of the components of

**v**.

**ρ**can be considered a density matrix if and only if

**ρ**is a trace-normalized positive semidefinite Hermitian matrix, i.e., $\mathrm{tr}\rho =1$, $\rho ={\rho}^{\u2020}$ (where the dagger indicates conjugate transpose and “tr” stands for the trace), while the real n eigenvalues of

**ρ**are nonnegative. Therefore,

**ρ**can always be expressed as $\rho =U\Lambda {U}^{\u2020}$, where

**U**is a unitary matrix and $\Lambda \equiv \mathrm{diag}\left({\lambda}_{1},\dots ,{\lambda}_{n}\right)$ is the diagonal matrix whose diagonal elements are the eigenvalues of

**ρ**taken in decreasing order $\left({\lambda}_{1}\ge {\lambda}_{2}\ge \dots \ge {\lambda}_{n}\right)$. The diagonalization of

**ρ**leads to its spectral decomposition

**ρ**(which in turn are the vector-columns of

**U**, and represent the pure eigenstates of

**ρ**) and ${L}_{k}$ are diagonal matrices whose only nonzero component is ${l}_{k}=1$. By denoting $r\equiv \mathrm{rank}\rho $ and considering an arbitrary generalized basis of ${\u2102}^{n}$, being composed of (a) a set of r independent unit vector states ${w}_{i}$ generating $\mathrm{range}\rho $ ($\mathrm{range}\rho $ being the subspace of ${\u2102}^{n}$ generated by the r eigenvectors ${u}_{1},\dots ,{u}_{r}$ with corresponding nonzero eigenvalues ${\lambda}_{1},\dots ,{\lambda}_{r}$), and (b) a set of $n-r$ independent unit vector states ${w}_{j}$ generating $\mathrm{ker}\rho $ ($\mathrm{ker}\rho $ being the subspace of ${\u2102}^{n}$ generated by the $n-r$ eigenvectors ${u}_{r+1},\dots ,{u}_{n}$ with corresponding zero eigenvalues ${\lambda}_{r+1}=\dots ={\lambda}_{n}=0$), the density matrix

**ρ**can be expressed as a convex sum of the r pure density matrices $\rho ={w}_{i}\otimes {w}_{i}^{\u2020}$ $\left(i=1,\dots ,r\right)$, through the arbitrary decomposition of

**ρ**[4,5,6,7,8].

## 2. nD Stokes Parameters and Bloch Vector

**ρ**are closely linked to the concept of nD Stokes parameters, which are defined as the (real) coefficients ${s}_{k}$ $\left(i=1,\dots ,n-1\right)$ of the expansion of

**ρ**in terms of the basis of Hermitian matrices composed of the identity matrix ${I}_{n}$ together with the ${n}^{2}-1$ generalized Pauli matrices ${\Lambda}_{k}$ (also called generalized Gell-Mann matrices) [9]

**Φ**(coherency matrix) of

**ρ**is $\Phi ={s}_{0}\rho $, with ${s}_{0}=\mathrm{tr}\Phi $, which is equivalent to using the non-normalized form of the Stokes vector. The nD Bloch vector (or coherence vector [9]) associated with

**ρ**is $s={\left({s}_{1},\dots ,{s}_{n-1}\right)}^{\mathrm{T}}$ [9,10,11,12]. Here we have followed the criterion of Byrd and Khaneja [9] for the definition of parameters ${s}_{k}$, in such a manner that the absolute value $\left|s\right|$ of the Bloch vector equals 1 for pure states and, as shown in Section 4, in general coincides with the degree of purity of the state represented by

**ρ**.

## 3. Discriminating Decomposition of a Density Matrix

**ρ**, its spectral decomposition (1) can be rearranged into the characteristic decomposition (also called trivial decomposition) [4]

**ρ**.

**ρ**into a convex composition of the density matrices ${\rho}_{p}$, ${\rho}_{d}$ and ${\rho}_{u}$, which are associated, respectively, with the pure component ${\rho}_{p}$ (which has a single nonzero eigenvalue and is generated by the single eigenstate ${u}_{1}$), the discriminating component ${\rho}_{d}$ (whose peculiar features will be analyzed in Section 9), and the fully random (or unpolarized) component ${\rho}_{u}={I}_{n}/n$, where ${I}_{n}$ is the identity matrix, which corresponds to a maximally mixed state, i.e., to an equiprobable mixture of the n orthonormal eigenstates of

**ρ**. The IPP, which determine the coefficients of the characteristic and discriminating decompositions of

**ρ**is a set of $n-1$ invariant parameters that provide complete quantitative information on the structure of purity of

**ρ**and satisfy the nested inequalities $0\le {P}_{1}\le \dots \le {P}_{k}\le \dots \le {P}_{n-1}\le 1$ [13]. The 3D and 4D formulations of the IPP have proven to be useful to address certain problems in polarization optics [7,14,15].

**ρ**) have a non-neutral effect on the overall purity of

**ρ**.

## 4. Degrees of Purity and Randomness of a Density Matrix

**ρ**is given by [16,17]

**ρ**to a maximally mixed one (fully random) and can also be expressed as follows in terms of the IPP [13],

**ρ**; that is to say, each ${P}_{k}$ leads to a particular contribution $n{P}_{k}^{2}$/[(n − 1)k(k + 1)] to ${P}_{nD}^{2}$, and the lower and upper values of each ${P}_{k}$ are limited by ${P}_{k-1}$ and ${P}_{k+1}$ respectively $\left(0\le {P}_{1}\le \dots \le {P}_{k}\le \dots \le {P}_{n-1}\le 1\right)$. Observe also that, although the number $n-r$ of IPP equal to 1 (${P}_{r}={P}_{r+1}=\dots ={P}_{n-1}=1$) entails that the apparent effective dimensions of the system are r instead of n (i.e., the last $n-r$ eigenvalues of

**ρ**are zero), the apparent excess of dimensions results in a significant contribution to purity. This interesting feature has been studied in Ref. [18] for the particular case of 3D polarization matrices.

**ρ**for the quantity $\mathrm{tr}{\rho}^{2}$ [19]. Maximally mixed states satisfy $\mathrm{tr}{\rho}^{2}=1/n$, so that $1/n\le \mathrm{tr}{\rho}^{2}\le 1$. Nevertheless, an interesting feature of using ${P}_{nD}$ instead of $\mathrm{tr}{\rho}^{2}$ as a measure of the degree of purity is that it takes the more natural limiting value ${P}_{nD}=0$ for maximally mixed states (maximal statistical symmetry). In addition, the overall measure of purity provided by ${P}_{nD}$ can be expressed, as in (9), as a weighted square average of the IPP, which in turn provides detailed and complete quantitative information on the structure of purity of

**ρ**. The parameter ${P}_{nD}$ was first introduced by Samson under the scope of ultra low-frequency magnetic fields [16], and later by Barakat [17] (with a different, but equivalent mathematical expression). ${P}_{nD}$ was also considered implicitly by Byrd and Khaneja [9] as the magnitude of the coherence vector (or Bloch vector) associated with an nD density matrix. The ability of ${P}_{3D}$ to represent the degree of polarization (or degree of polarimetric purity) for electromagnetic waves, as well as some important features, have been studied by Setälä et al. [18,20,21], Luis [22] and by Gil et al. [4,23,24]. Furthermore, ${P}_{4D}$ was independently introduced by Gil and Bernabéu as the depolarization index [25] associated with Mueller matrices representing the transformation of polarization states by the action of a material medium.

## 5. The Intrinsic Density Matrix

**ρ**, it has $n-1$ quantities that are invariant with respect to unitary similarity transformations $V\rho {V}^{\u2020}$(with

**V**unitary). An interesting set of such invariant quantities is that constituted by the IPP [13] because they determine the quantitative (but not qualitative) structure of purity in a hierarchical and meaningful manner. To get a more qualitative view of the information contained in

**ρ**, it is also interesting to explore its invariants with respect to changes of the n-dimensional Cartesian reference frame taken for its representation. Such changes correspond to orthogonal similarity transformations ${Q}^{\mathrm{T}}\rho Q$, so that the density matrix of a given physical system adopts a particular form for each Cartesian coordinate system ${X}_{1},{X}_{2},\dots {X}_{n}$ considered. Among them, the intrinsic reference frame ${X}_{1O},{X}_{2O},\dots {X}_{nO}$ is defined as the one for which the real part $\mathrm{Re}\left({\rho}_{O}\right)$ of the corresponding form ${\rho}_{O}$ of the same state that is represented by

**ρ**with respect to ${X}_{1},{X}_{2},\dots {X}_{n}$ becomes diagonal, $\mathrm{Re}\left({\rho}_{O}\right)=\mathrm{diag}\left({a}_{1},\dots ,{a}_{n}\right)$, with ${a}_{1}\ge {a}_{2}\ge \dots \ge {a}_{n}$. Observe that, from the very definition of the density matrix, the equalities $\mathrm{tr}\rho =\mathrm{tr}{\rho}_{O}=\mathrm{tr}\mathrm{Re}\left({\rho}_{O}\right)={\displaystyle {\sum}_{k=1}^{n}{a}_{k}}=1$ are satisfied. The transformation from

**ρ**to ${\rho}_{O}$ through the rotation of the coordinate system from ${X}_{1},{X}_{2},\dots {X}_{n}$ to ${X}_{1O},{X}_{2O},\dots {X}_{nO}$ is performed by means of the corresponding orthogonal similarity transformation ${\rho}_{O}={Q}_{O}^{\mathrm{T}}\rho {Q}_{O}$, where ${Q}_{O}$ is a proper orthogonal matrix (${Q}_{O}^{\mathrm{T}}={Q}_{O}^{-1}$,$\mathrm{det}{Q}_{O}=+1$). Therefore, the intrinsic density matrix ${\rho}_{O}$ has the peculiar form

**N**, whose components are denoted by $\epsilon {n}_{ij}$ (with $\epsilon =\pm 1$ depending on whether $i+j$ is an odd or even number respectively), is an antisymmetric matrix that encompasses all the imaginary part of ${\rho}_{O}$. The coefficient $1/2$ in the definition of

**N**and the sign coefficient ε in its components have been introduced for the sake of consistency with the components of the spin vector of polarization density matrices [36,37,38].

**ρ**with respect to arbitrary n-dimensional rotations of the Cartesian coordinate system is equal to the number ${n}^{2}-1$ of real variables of

**ρ**(recall that it is Hermitian and trace-normalized) minus the number $n\left(n-1\right)/2$ of parameters (angles) associated with an n×n orthogonal matrix, which results in $l=-1+n\left(n+1\right)/2$, i.e., the n intrinsic populations ${a}_{i}$ $\left(i=1,\dots ,n\right)$, with the restriction ${\sum}_{i=1}^{n}{a}_{i}}=1$, plus the $n\left(n-1\right)/2$ intrinsic coherences ${n}_{ij}$ $\left(i=1,\dots ,n;ji\right)$ (hereafter denoted as IC). Note that, as usual when dealing with density matrices of quantum mechanical systems, the terms populations and coherences are used in this work to refer to the diagonal and off-diagonal elements of

**ρ**respectively [39], while the adjective intrinsic is used for quantities derived from the intrinsic density matrix. It is remarkable that in polarization optics $\left(n=3\right)$ the pseudovector ${\left(-{n}_{23},{n}_{13},-{n}_{12}\right)}^{T}$ defined from the elements of

**N**is precisely the spin density vector of the state represented by the corresponding 3×3 polarization density matrix

**ρ**[36,37]. Nevertheless, the fact that the number $n\left(n-1\right)/2$ of intrinsic coherences equals the dimensions n is not a general property but it is a genuine feature of three-dimensional systems. This is the reason why the generalization of the concept of spin density vector, defined for the case $n=3$, to density matrices with dimensions $n>3$ is not possible in a consistent manner.

**ρ**has the peculiar feature that the Stokes parameters corresponding to the symmetric (nondiagonal) generators ${\mathrm{W}}_{kl}$ vanish, and therefore the intrinsic Bloch vector contains no more than $l=-1+n\left(n+1\right)/2$ nonzero Stokes parameters.

**ρ**is positive semidefinite, and despite the fact that it is characterized by the nonnegativity of its four eigenvalues, the nonnegativity of its principal minors implies certain restrictions on the values of ${a}_{i}$ and ${n}_{ij}$, among which we can mention $4{a}_{i}{a}_{j}\ge {n}_{ij}^{2}$ and $4{a}_{i}{a}_{j}{a}_{k}\ge {a}_{i}{n}_{i}^{2}+{a}_{j}{n}_{j}^{2}+{a}_{k}{n}_{k}^{2}$, which will be useful in further analyses.

## 6. Population and Correlation Asymmetries. Intrinsic Stokes Parameters

**ρ**, we introduce the $n-1$ indices of population asymmetry (hereafter IP) defined from the n intrinsic populations of

**ρ**in the following manner.

**ρ**. A state for which all the intrinsic populations are equal ${a}_{1}={a}_{2}=\dots ={a}_{n}$ (full population symmetry, i.e., $\mathrm{Re}\rho ={I}_{n}/n$) is characterized by ${M}_{1}={M}_{2}=\dots $$={M}_{n-1}=0$, while the maximal population asymmetry is reached when ${a}_{1}=1$, ${a}_{2}=\dots ={a}_{n}=0$, i.e., ${M}_{1}={M}_{2}=\dots $$={M}_{n-1}=1$.

**ρ**. Therefore, the CP constitutes a proper complete set of l rotational invariants of

**ρ**. To go deeper into the physical significance of the CP, let us consider the expansion of the intrinsic density matrix ${\rho}_{O}$ in terms of the SU(n) generators ${\Lambda}_{k}$ introduced in Section 2, which adopts the form

**ρ**) are given by the set composed of the $n-1$ intrinsic population-Stokes parameters

**S**, while the intrinsic correlation-Stokes parameters are arranged in its upper off-diagonal part. Obviously, when the non-normalized version $\Phi =\left(\mathrm{tr}\Phi \right)\rho $ of the coherency matrix is considered, then the corresponding (non-normalized) Stokes parameters matrix is given by $\left(\mathrm{tr}\Phi \right)T$. Observe also that ${P}_{nD}={\Vert S\Vert}_{2}^{2}-1$.

**ρ**can be grouped into three kinds of parameters of different nature, (a) the $n\left(n-1\right)/2$ angles determining the orthogonal matrix ${Q}_{O}$ that performs the rotation transformation from the intrinsic reference frame to the actual one; (b) the $n-1$ IP, ${M}_{k}$, or equivalently, the intrinsic population-Stokes parameters, and (c) the $n\left(n-1\right)/2$ intrinsic coherences, ${n}_{ij}$, or equivalently, the intrinsic correlation-Stokes parameters.

## 7. Structure of Purity of a Density Matrix

**ρ**is invariant under unitary similarity transformations (hence it is also invariant under orthogonal similarity transformations), and can be expressed as follows in terms of the CP

- ${P}_{p}=0$ when the state has full population symmetry, ${a}_{1}={a}_{2}=\dots ={a}_{n}=1/n$, so that the intrinsic density matrix takes the form ${\rho}_{O}={I}_{n}/n+\left(i/2\right)N$. Note that ${P}_{p}=0$ does not necessarily imply that ${P}_{nD}=0$, i.e., full population symmetry is compatible with a certain degree of correlation asymmetry ${P}_{c}\le \sqrt{2\left(n-1\right)/n}$.
- ${P}_{p}=1$ when the state has full population asymmetry ${a}_{1}=1$, ${a}_{2}=\dots ={a}_{n}=0$, i.e., ${\rho}_{O}=\mathrm{diag}\left(1,0,\dots ,0\right)$, which in turn implies ${P}_{nD}=1$ and ${P}_{c}=0$.
- ${P}_{c}=0$ when the state lacks correlation asymmetry, in which case ${\rho}_{O}=\mathrm{diag}\left({a}_{1},{a}_{2},\dots ,{a}_{n}\right)$ and all asymmetry is originated by ${P}_{p}$. The complete interval $0\le {P}_{nD}\le 1$ of values for ${P}_{nD}$ are achievable, ${P}_{nD}$ depending on the relative values of the intrinsic populations.
- ${P}_{c}=1$ corresponds to pure states $\left({P}_{nD}=1\right)$ with maximal correlation asymmetry, in which case ${\rho}_{O}$ necessarily adopts the form,$${\mathsf{\rho}}_{Oc}=\frac{1}{2}\left(\begin{array}{ccccc}1& \mp \phantom{\rule{0.166667em}{0ex}}i& 0& \dots & 0\\ \pm \phantom{\rule{0.166667em}{0ex}}i& 1& 0& \dots & 0\\ 0& 0& 0& \dots & 0\\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0& 0& 0& 0& 0\end{array}\right),$$
- Pure states are characterized by $1={P}_{nD}^{2}={P}_{p}^{2}+n{P}_{c}^{2}/\left[2\left(n-1\right)\right]$, where the full purity is reached through the balanced contributions of the degrees of population and correlation asymmetry, showing that the concept of purity of a state is identified with such a composition of asymmetries, while, as expected, the symmetry appears as a result of the randomness. An analysis of these features for the case $n=3$ can be found in [38]. For a pure state (${P}_{nD}=1$, i.e., ${P}_{1}={P}_{2}=\cdots ={P}_{n-1}=1$), ${\rho}_{O}$ has the generic form$${\mathsf{\rho}}_{Op}=\frac{1}{2}\left(\begin{array}{ccccc}1+cos2\chi & -\phantom{\rule{0.166667em}{0ex}}i\phantom{\rule{0.166667em}{0ex}}sin2\chi & 0& \dots & 0\\ \phantom{\rule{0.166667em}{0ex}}i\phantom{\rule{0.166667em}{0ex}}sin2\chi & 1-cos2\chi & 0& \dots & 0\\ 0& 0& 0& \dots & 0\\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0& 0& 0& 0& 0\end{array}\right),\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\left[\begin{array}{c}-\pi /4\le \chi \le \pi /4\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\\ {M}_{1}=cos2\chi ,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{P}_{c}=sin2\chi \end{array}\right],$$

**ρ**, all the eigenstates ${u}_{k}$ have well-defined and intrinsic respective handedness, so that it can properly be said that

**ρ**involves intrinsic chiral properties associated with those of its eigenstates. For a given pure state ${u}_{k}$ its chirality vanishes when $\chi =0$, while from an overall point of view,

**ρ**carries non-vanishing chirality as long as at least one of the eigenstates ${u}_{k}$ has nonzero spin (i.e., ${\chi}_{k}\ne 0$, or, equivalently ${P}_{c}\left({u}_{k}\right)\ne 0$). The features of the ellipticity angles and the associated spin vectors of generic sets $\left({u}_{1},{u}_{2},{u}_{3}\right)$ of orthonormal states for 3D polarization density matrices have been studied in [40].

**ρ**also become zero.

## 8. The Concept of Nonregularity of a Density Matrix

**ρ**is formulated as a convex composition of three density matrices, namely (1) the pure component ${\rho}_{p}$, whose intrinsic form ${\rho}_{pO}$ has been described in Equation (22); (2) the symmetric component ${\rho}_{u}={I}_{n}/n$ (also called fully random, or unpolarized component), and (3) the discriminating component ${\rho}_{d}$. In general, ${\rho}_{d}$ has a complicated structure, and it can have different interesting forms. When ${\rho}_{d}$ is a real matrix, then it can be expressed as a weighted sum of density matrices, each one corresponding to a respective equiprobable mixture of a number of, 2, 3,… and n-1 mutually orthogonal pure states with zero spin

**Q**is orthogonal. Observe that the fact that ${\rho}_{d}$ is a real matrix, (hence it is symmetric and can be diagonalized through the orthogonal matrix

**Q**) does not imply that the matrix

**U**that diagonalizes

**ρ**is necessarily a real matrix; that is, there are cases for which there is degeneracy for certain eigenvalues of ${\rho}_{d}$ so that it can be diagonalized either by means of

**U**(in general, complex-valued) or

**Q**(real-valued). In summary, from Equation (7), ${\rho}_{d}$ is in general complex-valued (in which case

**U**is not real-valued), so that we can distinguish between regular states, defined as those where either ${P}_{1}={P}_{n-1}$ (in which case the coefficient of ${\rho}_{d}$ in the discriminating decomposition (7) vanishes), or ${\rho}_{d}$ is a real matrix, and nonregular states, for which ${P}_{1}\ne {P}_{n-1}$ and $\mathrm{Im}{\rho}_{d}\ne 0$ [7,41]. Thus, nonregularity occurs when the correlation asymmetry of ${\rho}_{d}$ is nonzero, ${P}_{c}\left({\rho}_{d}\right)>0$, while regularity appears as the limiting situation where ${P}_{c}\left({\rho}_{d}\right)=0$ (or, alternatively, where the equality ${P}_{1}={P}_{n-1}$ is satisfied). Consequently, the maximal achievable value for ${P}_{c}\left({\rho}_{d}\right)$ should be inspected in order to define a generalized degree of nonregularity (note that the version for $n=3$ was already defined for density polarization matrices in previous work [41]).

**ρ**). The off-diagonal elements of the intrinsic form ${\rho}_{dO}=\mathrm{diag}\left({a}_{d1},\dots ,{a}_{dn}\right)+\left(i/2\right){N}_{dO}$ are purely imaginary and their absolute values, together with the populations ${a}_{dk}$, are limited by the constraints of nonnegativity of all principal minors of ${\rho}_{dO}$, in such a manner that the maximization of ${P}_{c}\left({\rho}_{dO}\right)=\left(1/\sqrt{2}\right){\Vert {N}_{dO}\Vert}_{2}$ requires that ${\rho}_{dO}$ has the form

**ρ**is a nonregular state, with a maximum ${P}_{c}\left({\rho}_{d}\right)=1/2$ for states with maximal nonregularity, hereafter called perfect nonregular states.

**ρ**as

**ρ**is that of ${\rho}_{d}$ but scaled by the coefficient ${P}_{n-1}-{P}_{1}$, and therefore

**ρ**represents a perfect nonregular state $\left({P}_{N}\left(\rho \right)=1\right)$ when

**ρ**itself has the form of a discriminating density matrix $\rho ={\rho}_{d}$ with ${P}_{N}\left({\rho}_{d}\right)=1$.

**ρ**involves necessarily certain chirality associated with the eigenstates of

**ρ**, and by noting that the existence of nonzero correlation asymmetry of the discriminating component ${\rho}_{d}$ of

**ρ**implies nonregularity, we find an interesting and subtle link between the concepts of chirality and nonregularity of a given

**ρ**. Observe also that the complete interval of values $0\le {P}_{c}\left(\rho \right)\le 1$ is achievable for regular states; in particular ${P}_{c}\left(\rho \right)=0$ when

**ρ**is a real matrix (hence regular) and ${P}_{c}\left(\rho \right)=1$ when

**ρ**represents a circularly polarized pure state. Thus, ${P}_{c}\left(\rho \right)$ may be interpreted as a measure of the degree of chirality of the state

**ρ**, while the degree of chirality ${P}_{c}\left({\rho}_{d}\right)$ of the discriminating component determines the degree of nonregularity of

**ρ**.

## 9. Discussion and Conclusions

**ρ**is exploited in order to define a set of quantities (the components of purity -CP- of the state) that provide complete information on the rotational invariant properties associated with

**ρ**in a hierarchical and meaningful manner. In fact, it is found that the CP coincide, up to respective simple coefficients, with the n-dimensional Stokes parameters of the state in its intrinsic representation. These results generalize the obtainment of the six intrinsic Stokes parameters for 3D polarization states, whose physical interpretation is as simple as the intensity, the degree of linear polarization, the degree of directionality and the three intrinsic components of the spin density vector of the state [6,37]. Thus, any polarization density matrix can be conceived as the intrinsic one (linked in a very simple way to the intrinsic Stokes parameters) and a spatial rotation of the Cartesian reference frame (three angular parameters).

**ρ**representing an n-dimensional mixed state, which depends on up to ${n}^{2}-1$ free parameters, the situation is more involved than for 2D or 3D density matrices and the generalization is not straightforward; in fact, as described in Section 5, when $n>3$ the imaginary parts of the off-diagonal elements of

**ρ**cannot be longer interpreted as the components of the spin density vector.

**ρ**with respect to unitary similarity transformations $U\rho {U}^{\u2020}$, a parametrization based on the $n-1$ indices of purity (IPP) was defined in previous work [13]. The IPP, ${P}_{k}$ $\left(k=1,\dots ,n-1\right)$ are constrained by the nested inequalities $0\le {P}_{1}\le \dots \le {P}_{n-1}$ and provide, in an optimal and hierarchical manner, quantitative (but not qualitative) information on the structure of purity of

**ρ**. In fact, the degree of purity ${P}_{nD}$ (which represents the degree of statistical asymmetry) can be obtained as a weighted square average of the IPP.

**ρ**through the introduction of the concept of intrinsic density matrix, which leads to the definition of a number of l (with $l=\left[n\left(n+1\right)/2\right]-1$) invariants of

**ρ**with respect to arbitrary rotations of the n-dimensional Cartesian reference frame, i.e., with respect to orthogonal similarity transformations $Q\rho {Q}^{\mathrm{T}}$. In this case, the total number l of invariants defined in this work and called the components of purity (CP) of

**ρ**, can be grouped into two meaningful sets, namely (a) the $n-1$ indices of population asymmetry (IP), ${M}_{k}$ $\left(k=1,\dots ,n-1\right)$, (or, equivalently, the intrinsic population-Stokes parameters) which are constrained by the nested inequalities $0\le {M}_{1}\le \dots \le {M}_{n-1}\le 1$ and provide, in an optimal and hierarchical manner, complete information on the structure of population asymmetry of

**ρ**, and (b) the $n\left(n-1\right)/2$ intrinsic coherences (IC), ${n}_{ij}$ $\left(i,j=1,\dots ,n;ij\right)$, (or, equivalently, the intrinsic correlation-Stokes parameters) that provide complete information on the correlation asymmetry of

**ρ**.

**ρ**into a convex sum (or incoherent superposition) of three density matrices, namely (1) a pure one, (2) a maximally mixed one, and (3) a discriminating one that holds certain critical properties of

**ρ**and leads to the definition of the degree of nonregularity, which is determined by the degree of correlation asymmetry of the discriminating component, scaled by the difference ${P}_{n-1}-{P}_{1}$ between the maximum and minimum IPP of

**ρ**.

**U**unitary), are invariants under $Q\rho {Q}^{\mathrm{T}}$ (with

**Q**orthogonal). Therefore, the set of CP is complete, because parametrizes all the l indicated rotational invariants of

**ρ**. Furthermore, the CP (hence, the intrinsic Stokes parameters) are physically meaningful because they satisfy the following properties

- (a)
- As shown in Equation (20), the degree of purity ${P}_{nD}$ is given by a weighted square average of the CP. The degrees of population and correlation asymmetry constitute two complementary sources of purity. Full population asymmetry ${P}_{P}=1$ entails full purity $\left({P}_{nD}=1\right)$ and zero correlation asymmetry $\left({P}_{c}=0\right)$, while full correlation asymmetry ${P}_{c}=1$ implies full purity together with a certain amount of population asymmetry ${P}_{p}=\sqrt{\left(n/2-1\right)/\left(n-1\right)}$, in which case the state can be represented by a circularly polarized state embedded into an n-dimensional space.
- (b)
- The $n-1$ indices of population asymmetry ${M}_{k}$ are defined in a hierarchical manner $\left(0\le {M}_{1}\le \dots \le {M}_{n-1}\le 1\right)$, so that ${M}_{n-1}=0$ implies that the state is maximally mixed, while ${M}_{1}=1$ implies full population asymmetry (or population purity) ${P}_{p}=1$ and full overall purity ${P}_{nD}=1$, in which case the state can be represented by a linearly polarized state embedded into an n-dimensional space.
- (c)
- The $n\left(n-1\right)/2$ intrinsic coherences ${n}_{ij}$ hold all information on the correlations among the random variables that describe the system. Their values are constrained by those of ${M}_{k}$ because of the nonnegativity of the principal minors of
**ρ**. Moreover, the inherent chirality associated with the degree of correlation asymmetry, which has its origin in the handedness of the eigenstates of**ρ**, has been analyzed and characterized. - (d)
- All the information contained in
**ρ**(${n}^{2}-1$ free parameters) can be parametrized by an n-dimensional rotation (non-invariant $n\left(n-1\right)/2$ angular parameters) together with the $n-1$ indices of population asymmetry and the $n\left(n-1\right)/2$ intrinsic coherences of**ρ**.

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The polarization ellipse of a pure state

**ρ**represented with respect to its intrinsic reference frame ${X}_{1O}{X}_{2O}{X}_{3O}$.

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**MDPI and ACS Style**

Gil, J.J.
Sources of Asymmetry and the Concept of Nonregularity of *n*-Dimensional Density Matrices. *Symmetry* **2020**, *12*, 1002.
https://doi.org/10.3390/sym12061002

**AMA Style**

Gil JJ.
Sources of Asymmetry and the Concept of Nonregularity of *n*-Dimensional Density Matrices. *Symmetry*. 2020; 12(6):1002.
https://doi.org/10.3390/sym12061002

**Chicago/Turabian Style**

Gil, José J.
2020. "Sources of Asymmetry and the Concept of Nonregularity of *n*-Dimensional Density Matrices" *Symmetry* 12, no. 6: 1002.
https://doi.org/10.3390/sym12061002