1. Introduction
Uncertainty relations for position and momentum [
1] have always been deeply related to the foundations of Quantum Mechanics. For several decades, their axiomatization has been of ‘preparation’ type: an inviolable lower bound for the widths of the position and momentum distributions, holding in any quantum state. Such kinds of uncertainty relations, which are now known as
preparation uncertainty relations (PURs) have been later extended to arbitrary sets of
observables [
2,
3,
4,
5]. All PURs trace back to the celebrated Robertson’s formulation [
6] of Heisenberg’s uncertainty principle: for any two observables, represented by self-adjoint operators
A and
B, the product of the variances of
A and
B is bounded from below by the expectation value of their commutator; in formulae,
, where
is the variance of an observable measured in any system state
. In the case of position
Q and momentum
P, this inequality gives Heisenberg’s relation
. About 30 years after Heisenberg and Robertson’s formulation, Hirschman attempted a first statement of position and momentum uncertainties in terms of informational quantities. This led him to a formulation of PURs based on Shannon entropy [
7]; his bound was later refined [
8,
9], and extended to discrete observables [
10]. Also other entropic quantities have been used [
11]. We refer to [
12,
13] for an extensive review on entropic PURs.
However, Heisenberg’s original intent [
1] was more focused on the unavoidable disturbance that a measurement of position produces on a subsequent measurement of momentum [
14,
15,
16,
17,
18,
19,
20,
21]. Trying to give a better understanding of his idea, more recently new formulations were introduced, based on a ‘measurement’ interpretation of uncertainty, rather than giving bounds on the probability distributions of the target observables. Indeed, with the modern development of the quantum theory of measurement and the introduction of
positive operator valued measures and
instruments [
3,
22,
23,
24,
25,
26], it became possible to deal with approximate measurements of incompatible observables and to formulate
measurement uncertainty relations (MURs) for position and momentum, as well as for more general observables. The MURs quantify the degree of approximation (or inaccuracy and disturbance) made by replacing the original incompatible observables with a
joint approximate measurement of them. A very rich literature on this topic flourished in the last 20 years, and various kinds of MURs have been proposed, based on distances between probability distributions, noise quantifications, conditional entropy, etc. [
12,
14,
15,
16,
17,
18,
19,
20,
21,
27,
28,
29,
30,
31,
32].
In this paper, we develop a new information-theoretical formulation of MURs for position and momentum, using the notion of the
relative entropy (or
Kullback-Leibler divergence) of two probabilities. The relative entropy
is an informational quantity which is precisely tailored to quantify the amount of information that is lost by using an approximating probability
q in place of the target one
p. Although classical and quantum relative entropies have already been used in the evaluation of the performances of quantum measurements [
24,
27,
30,
33,
34,
35,
36,
37,
38,
39,
40], their first application to MURs is very recent [
41].
In [
41], only MURs for discrete observables were considered. The present work is a first attempt to extend that information-theoretical approach to the continuous setting. This extension is not trivial and reveals peculiar problems, that are not present in the discrete case. However, the nice properties of the relative entropy, such as its scale invariance, allow for a satisfactory formulation of the entropic MURs also for position and momentum.
We deal with position and momentum in two possible scenarios. Firstly, we consider the case of n-dimensional position and momentum, since it allows to treat either scalar particles, or vector ones, or even the case of multi-particle systems. This is the natural level of generality, and our treatment extends without difficulty to it. Then, we consider a couple made up of one position and one momentum component along two different directions of the n-space. In this case, we can see how our theory behaves when one moves with continuity from a highly incompatible case (parallel components) to a compatible case (orthogonal ones).
The continuous case needs much care when dealing with arbitrary quantum states and approximating observables. Indeed, it is difficult to evaluate or even bound the relative entropy if some assumption is not made on probability distributions. In order to overcome these technicalities and focus on the quantum content of MURs, in this paper we consider only the case of Gaussian preparation states and Gaussian measurement apparatuses [
2,
4,
5,
42,
43,
44,
45]. Moreover, we identify the class of the approximate joint measurements with the class of the joint POVMs satisfying the same symmetry properties of their target position and momentum observables [
3,
23]. We are supported in this assumption by the fact that, in the discrete case [
41], simmetry covariant measurements turn out to be the best approximations without any hypothesis (see also [
17,
19,
20,
29,
32] for a similar appearance of covariance within MURs for different uncertainty measures).
We now sketch the main results of the paper. In the vector case, we consider approximate joint measurements
of the position
and the momentum
. We find the following entropic MUR (Theorem 5, Remark 14): for every choice of two positive thresholds
, with
, there exists a Gaussian state
with position variance matrix
and momentum variance matrix
such that
for all Gaussian approximate joint measurements
of
and
. Here
and
are the distributions of position and momentum in the state
, and
is the distribution of
in the state
, with marginals
and
; the two marginals turn out to be noisy versions of
and
. The lower bound is strictly positive and it linearly increases with the dimension
n. The thresholds
and
are peculiar of the continuous case and they have a classical explanation: the relative entropy
if the variance of
p vanishes faster than the variance of
q, so that, given
, it is trivial to find a state
enjoying (
1) if arbtrarily small variances are allowed. What is relevant in our result is that the total loss of information
exceeds the lower bound even if we forbid target distributions with small variances.
The MUR (
1) shows that there is no Gaussian joint measurement which can approximate arbitrarily well both
and
. The lower bound (
1) is a consequence of the incompatibility between
and
and, indeed, it vanishes in the classical limit
. Both the relative entropies and the lower bound in (
1) are scale invariant. Moreover, for fixed
and
, we prove the existence and uniqueness of an optimal approximate joint measurement, and we fully characterize it.
In the scalar case, we consider approximate joint measurements
of the position
along the direction
and the momentum
along the direction
, where
. We find two different entropic MURs. The first entropic MUR in the scalar case is similar to the vector case (Theorem 3, Remark 11). The second one is (Theorem 1):
for all Gaussian states
and all Gaussian joint approximate measurements
of
and
. This lower bound holds for every Gaussian state
without constraints on the position and momentum variances
and
, it is strictly positive unless
and
are orthogonal, but it is state dependent. Again, the relative entropies and the lower bound are scale invariant.
The paper is organized as follows. In
Section 2, we introduce our target position and momentum observables, we discuss their general properties and define some related quantities (spectral measures, mean vectors and variance matrices, PURs for second order quantum moments, Weyl operators, Gaussian states).
Section 3 is devoted to the definitions and main properties of the relative and differential (Shannon) entropies.
Section 4 is a review on the entropic PURs in the continuous case [
7,
8,
9,
46], with a particular focus on their lack of scale invariance. This is a flaw due to the very definition of differential entropy, and one of the reasons that lead us to introduce relative entropy based MURs. In
Section 5 we construct the covariant observables which will be used as approximate joint measurements of the position and momentum target observables. Finally, in
Section 6 the main results on MURs that we sketched above are presented in detail. Some conclusions are discussed in
Section 7.
2. Target Observables and States
Let us start with the usual position and momentum operators, which satisfy the canonical commutation rules:
Each of the vector operators has
n components; it could be the case of a single particle in one or more dimensions (
), or several scalar or vector particles, or the quadratures of
n modes of the electromagnetic field. We assume the Hilbert space
to be irreducible for the algebra generated by the canonical operators
and
. An
observable of the quantum system
is identified with a
positive operator valued measure (POVM); in the paper, we shall consider observables with outcomes in
endowed with its Borel
-algebra
The use of POVMs to represent observables in quantum theory is standard and the definition can be found in many textbooks [
22,
23,
26,
47]; the alternative name “non-orthogonal resolutions of the identity” is also used [
3,
4,
5]. Following [
5,
23,
26,
31], a
sharp observable is an observable represented by a
projection valued measure (pvm); it is standard to identify a sharp observable on the outcome space
with the
k self-adjoint operators corresponding to it by spectral theorem. Two observables are
jointly measurable or
compatible if there exists a POVM having them as marginals. Because of the non-vanishing commutators, each couple
,
, as well as the vectors
,
, are not jointly measurable.
We denote by the trace class operators on , by the subset of the statistical operators (or states, preparations), and by the space of the linear bounded operators.
2.1. Position and Momentum
Our target observables will be either n-dimensional position and momentum (vector case) or position and momentum along two different directions of (scalar case). The second case allows to give an example ranging with continuity from maximally incompatible observables to compatible ones.
2.1.1. Vector Observables
As target observables we take
and
as in (
3) and we denote by
,
, their pvm’s, that is
Then, the distributions in the state
of a sharp position and a sharp momentum measurements (denoted by
and
) are absolutely continuous with respect to the Lebesgue measure; we denote by
and
their probability densities:
,
In the Dirac notation, if
and
are the improper position and momentum eigenvectors, these densities take the expressions
and
, respectively. The mean vectors and the variance matrices of these distributions will be given in (
7) and (
8).
2.1.2. Scalar Observables
As target observables we take the position along a given direction
and the momentum along another given direction
:
In this case we have , so that and are not jointly measurable, unless the directions and are orthogonal.
Their pvm’s are denoted by
and
, their distributions in a state
by
and
, and their corresponding probability densities by
and
:
,
Of course, the densities in the scalar case are marginals of the densities in the vector case. Means and variances will be given in (
11).
2.2. Quantum Moments
Let
be the set of states for which the second moments of position and momentum are finite:
Then, the mean vector and the variance matrix of the position
in the state
are
while for the momentum
we have
For
it is possible to introduce also the mixed ‘quantum covariances’
Since there is no joint measurement for the position and momentum , the quantum covariances are not covariances of a joint distribution, and thus they do not have a classical probabilistic interpretation.
By means of the moments above, we construct the three real
matrices
, the
-dimensional vector
and the symmetric
matrix
, with
We say
is the
quantum variance matrix of position and momentum in the state
. In [
2] dimensionless canonical operators are considered, but apart from this, our matrix
corresponds to their “noise matrix in real form”; the name “variance matrix” is also used [
44,
48].
In a similar way, we can introduce all the moments related to the position
and momentum
introduced in (
6). For
, the means and variances are respectively
Similarly to (
9), we have also the ‘quantum covariance’
. Then, we collect the two means in a single vector and we introduce the variance matrix:
Proposition 1. Let be a real symmetric block matrix with the same dimensions of a quantum variance matrix. Define In this case we have: , , , and The inequalities (
14) for
tell us exactly when a (positive semi-definite) real matrix
V is the quantum variance matrix of position and momentum in a state
. Moreover, they are the multidimensional version of the usual uncertainty principle expressed through the variances [
2,
3,
5], hence they represent a form of PURs. The block matrix
in the definition of
is useful to compress formulae involving position and momentum; moreover, it makes simpler to compare our equations with their frequent dimensionless versions (with
) in the literature [
43,
44].
Proof. Equivalences (
14) are well known, see e.g., [
3] (Section 1.1.5), [
5] (Equation (2.20)), and [
2] (Theorem 2). Then
.
By using the real block vector
, with arbitrary
and given
, the semi-positivity (
14) implies
which in turn implies
,
and (
15). Then, by choosing
, where
are the eigenvectors of
A (since
A is a real symmetric matrix,
for all
i), one gets the strict positivity of all the eigenvalues of
A; analogously, one gets
. ☐
Inequality (
15) for
and
becomes the uncertainty rule à la Robertson [
6] for the observables in (
6) (a position component and a momentum component spanning an arbitrary angle
):
Inequality (
16) is equivalent to
Since
are block matrices, their positive semi-definiteness can be studied by means of the Schur complements [
49,
50,
51]. However, as
are complex block matrices with a very peculiar structure, special results hold for them. Before summarizing the properties of
in the next proposition, we need a simple auxiliary algebraic lemma.
Lemma 1. Let A and B be complex self-adjoint matrices such that . Then , and the equality holds iff .
Proof. Let
and
be the ordered decreasing sequences of the eigenvalues of
A and
B, respectively. Then, by Weyl’s inequality,
implies
for every
i [
52] (Section III.2). This gives the first statement. Moreover, if
and
, we get
for every
i. Then
because
and
. ☐
Proposition 2. Let be a real symmetric matrix with the same dimensions of a quantum variance matrix. Then (or, equivalently, ) if and only if and Moreover, we have also the following properties for the various determinants: By interchanging
A with
B and
C with
in (
18)–(
22) equivalent results are obtained.
Proof. Since we already know that
implies the invertibility of
A, the equivalence between (
14) and (
18) with
follows from [
49] (Theorem 1.12 p. 34) (see also [
50] (Theorem 11.6) or [
51] (Lemma 3.2)).
In (
19), the first inequality follows by summing up the two inequalities in (
18). The last two ones are immediate by the positivity of
.
The equality in (
20) is Schur’s formula for the determinant of block matrices ([
49], Theorem 1.1 p. 19). Then, the first inequality is immediate by the lemma above and the trivial relation
; the second one follows from (
19):
The equality
is equivalent to
; since the latter two determinants are evaluated on ordered positive matrices by (
19), they coincide if and only if the respective arguments are equal (Lemma 1); this shows the equivalence in (
21). Then, by (
18), the self-adjoint matrix
is both positive semi-definite and negative semi-definite; hence it is null, that is,
.
Finally,
gives
trivially. Conversely,
implies
by (
20); since
by (
19), Lemma 1 then implies
and so
. ☐
By (
18) and (
19), every time three matrices
define the quantum variance matrix of a state
, the same holds for
. This fact can be used to characterize when two positive matrices
A and
B are the diagonal blocks of some quantum variance matrix, or two positive numbers
and
are the position and momentum variances of a quantum state along the two directions
and
.
Proposition 3. Two real matrices and , having the dimension of the square of a length and momentum, respectively, are the diagonal blocks of a quantum variance matrix if and only if Two real numbers and , having the dimension of the square of a length and momentum, respectively, are such that and for some state ρ if and only if Proof. For
A and
B, the necessity follows from (
19). The sufficiency comes from (
18) by choosing
.
For
and
, the necessity follows from (
15). The sufficiency comes from (
18) with
and for example the following choices of
A and
B:
In the last two cases, we chose A and B in such a way that when restricted to the linear span of . ☐
2.3. Weyl Operators and Gaussian States
In the following, we shall introduce Gaussian states, Gaussian observables and covariant observables on the phase-space. In all these instances, the Weyl operators are involved; here we recall their definition and some properties (see e.g., [
4] (Section 5.2) or [
5] (Section 12.2), where, however, the definition differs from ours in that the Weyl operators are composed with the map
of (
13)).
Definition 1. The Weyl operators
are the unitary operators defined by The Weyl operators (
23) satisfy the composition rule
in particular, this implies the commutation relation
These commutation relations imply the translation property
due to this property, the Weyl operators are also known as
displacement operators.
With a slight abuse of notation, we shall sometimes use the identification
where
is a block column vector belonging to the phase-space
; here, the first block
is a position and the second block
is a momentum.
By means of the Weyl operators, it is possible to define the characteristic function of any trace-class operator.
Definition 2. For any operator , its characteristic function is the complex valued function defined by Note that is the inverse of a length and is the inverse of a momentum, so that w is a block vector living in the space regarded as the dual of the phase-space.
Instead of the characteristic function, sometimes the so called Weyl transform
is introduced [
4,
44].
By [
4] (Proposition 5.3.2, Theorem 5.3.3), we have
and the following trace formula holds:
,
As a corollary [
4] (Corollary 5.3.4), we have that a state
is pure if and only if
By [
53] (Lemma 3.1) or [
26] (Proposition 8.5.(e)), the trace formula also implies
Moreover, the following inversion formula ensures that the characteristic function
completely characterizes the state
[
4] (Corollary 5.3.5):
The last two integrals are defined in the weak operator topology.
Finally, for
, the moments (
7)–(
10) can be expressed as in [
4] (Section 5.4):
Definition 3 ([
2,
3,
4,
5,
44,
48])
. A state ρ is Gaussian iffor a vector and a real matrix such that . The condition
is necessary and sufficient in order that the function (
31) defines the characteristic function of a quantum state [
4] (Theorem 5.5.1), [
5] (Theorem 12.17). Therefore, Gaussian states are exactly the states whose characteristic function is the exponential of a second order polynomial [
4] (Equation (5.5.49)), [
5] (Equation (12.80)).
We shall denote by
the set of the Gaussian states; we have
. By (
30), the vectors
,
and the matrices
,
,
characterizing a Gaussian state
are just its first and second order quantum moments introduced in (
7)–(
9). By (
31), the corresponding distributions of position and momentum are Gaussian, namely
Proposition 4 (Pure Gaussian states)
. For , we have if and only if ρ is pure.
Proof. The trace formula (
28) and (
31) give
, and this implies the statement. ☐
Proposition 5 (Minimum uncertainty states).
For , we have if and only if ρ is a pure Gaussian state and it factorizes into the product of minimum uncertainty states up to a rotation of .
Proof. If
, then the equivalence (
22) gives
, so that the variance matrices
and
have a common eigenbasis
. Thus, all the corresponding couples of position
and momentum
have minimum uncertainties:
. Therefore, if we consider the factorization of the Hilbert space
corresponding to the basis
, all the partial traces of the state
on each factor
are minimum uncertainty states. Since for
the minimum uncertainty states are pure and Gaussian, the state
is a pure product Gaussian state.
The converse is immediate. ☐
5. Approximate Joint Measurements of Position and Momentum
In order to deal with MURs for position and momentum observables, we have to introduce the class of approximate joint measurements of position and momentum, whose marginals we will compare with the respective sharp observables. As done in [
3,
4,
18,
57], it is natural to characterize such a class by requiring suitable properties of covariance under the group of space translations and velocity boosts: namely, by
approximate joint measurement of position and momentum we will mean any POVM on the product space of the position and momentum outcomes sharing the same covariance properties of the two target sharp observables. As we have already discussed, two approximation problems will be of our concern: the approximation of the position and momentum vectors (vector case, with outcomes in the phase-space
), and the approximation of one position and one momentum component along two arbitrary directions (scalar case, with oucomes in
). In order to treat the two cases altogether, we consider POVMs with outcomes in
, which we call
bi-observables; they correspond to a measurement of
m position components and
m momentum components. The specific covariance requirements will be given in the Definitions 5–7.
In studying the properties of probability measures on
, a very useful notion is that of the characteristic function, that is, the Fourier cotransform of the measure at hand; the analogous quantity for POVMs turns out to have the same relevance. Different names have been used in the literature to refer to the characteristic function of POVMs, or, more generally, quantum instruments, such as characteristic operator or operator characteristic function [
3,
24,
34,
44,
58,
59,
60,
61,
62]. As a variant, also the symplectic Fourier transform quite often appears [
5] (Section 12.4.3). The characteristic function has been used, for instance, to study the quantum analogues of the infinite-divisible distributions [
3,
34,
58,
59,
60,
62] and measurements of Gaussian type [
5,
44,
61]. Here, we are interested only in the latter application, as our approximating bi-observables will typically be Gaussian. Since we deal with bi-observables, we limit our definition of the characteristic function only to POVMs on
, which have the same number of variables of position and momentum type.
Being measures, POVMs can be used to construct integrals, whose theory is presented e.g., in [
26] (Section 4.8) and [
4] (Section 2.9, Proposition 2.9.1).
Definition 4. Given a bi-observable ,
the characteristic function
of is the operator valued function , with In this definition the dimensions of the vector variables
and
are the inverses of a length and momentum, respectively, as in the definition of the characteristic function of a state (
27). This definition is given so that
is the usual characteristic function of the probability distribution
on
.
5.1. Covariant Vector Observables
In terms of the pvm’s (
4), the translation property (
25) is equivalent to the symmetry properties
and they are taken as the transformation property defining the following class of POVMs on
[
23,
26,
44,
53,
57].
Definition 5. A covariant phase-space observable
is a bi-observable satisfying the covariance relation We denote by the set of all the covariant phase-space observables.
The interpretation of covariant phase-space observables as approximate joint measurements of position and momentum is based on the fact that their marginal POVMs
have the same symmetry properties of
and
, respectively. Although
and
are not jointly measurable, the following well-known result says that there are plenty of covariant phase-space observables [
4] (Theorem 4.8.3), [
63,
64]. In (
43) below, we use the parity operator
on
, which is such that
Proposition 8. The covariant phase-space observables are in one-to-one correspondence with the states on , so that we have the identification ; such a correspondence is given by The characteristic function (
41) of a measurement
has a very simple structure in terms of the characteristic function (
27) of the corresponding state
.
Proposition 9. The characteristic function of is given byand the characteristic function of the probability is In (
44) we have used the identification (
26). The characteristic function of a state is introduced in (
27).
Proof. By the commutation relations (
24) we have
Then, we get
where we used the formula (
29). By (
42) and the definition (
27), we get (
44). Again by (
27), we get (
45). ☐
In terms of probability densities, measuring
on the state
yields the density function
. Then, by (
45), the densities of the marginals
and
are the convolutions
where
f and
g are the sharp densities introduced in (
5). By the arbitrariness of the state
, the marginal POVMs of
turn out to be the convolutions (or ‘smearings’)
(see e.g., [
23] (Section III, Equations (2.48) and (2.49))).
Let us remark that the distribution of the approximate position observable
in a state
is the distribution of the sum of two independent random vectors: the first one is distributed as the sharp position
in the state
, the second one is distributed as the sharp position
in the state
. In this sense, the approximate position
looks like a sharp position plus an independent noise given by
. Of course, a similar fact holds for the momentum. However, this statement about the distributions can not be extended to a statement involving the observables. Indeed, since
and
are incompatible, nobody can jointly observe
,
and
, so that the convolutions (
46) do not correspond to sums of random vectors that actually exist when measuring
.
5.2. Covariant Scalar Observables
Now we focus on the class of approximate joint measurements of the observables
and
representing position and momentum along two possibly different directions
and
(see
Section 2.1.2). As in the case of covariant phase-space observables, this class is defined in terms of the symmetries of its elements: we require them to transform as if they were joint measurements of
and
. Recall that
and
denote the spectral measures of
,
.
Due to the commutation relation (
24), the following covariance relations hold
for all
and
. We employ covariance to define our class of approximate joint measurements of
and
.
Definition 6. A-covariant bi-observable
is a POVM such that We denote by the class of such bi-observables.
So, our approximate joint measurements of and will be all the bi-observables in the class .
Example 1. The marginal of a covariant phase-space observable along the directions and is a -covariant bi-observable. Actually, it can be proved that, if , all -covariant bi-observables can be obtained in this way.
It is useful to work with a little more generality, and merge Definitions 5 and 6 into a single notion of covariance.
Definition 7. Suppose J is a real matrix. A POVM is a J -covariant observable on if Thus, approximate joint observables of
and
are just
J-covariant observables on
for the choice of the
matrix
On the other hand, covariant phase-space observables constitute the class of -covariant observables on , where is the identity map of .
5.3. Gaussian Measurements
When dealing with Gaussian states, the following class of bi-observables quite naturally arises.
Definition 8. A POVM is a Gaussian bi-observable
iffor two vectors , a real matrix and a real symmetric matrix satisfying the condition We set . The triple is the set of the parameters of the Gaussian observable .
In this definition, the vector
has the dimension of a length, and
of a momentum; similarly, the matrices
,
decompose into blocks of different dimensions. The condition (
49) is necessary and sufficient in order that the function (
48) defines the characteristic function of a POVM.
For unbiased Gaussian measurements, i.e., Gaussian bi-observables with
, the previous definition coincides with the one of [
5] (Section 12.4.3). It is also a particular case of the more general definition of Gaussian observables on arbitrary (not necessarily symplectic) linear spaces that is given in [
43,
44]. We refer to [
5,
44] for the proof that Equation (
48) is actually the characteristic function of a POVM.
Measuring the Gaussian observable
on the Gaussian state
yields the probability distribution
whose characteristic function is
hence the output distribution is Gaussian,
5.3.1. Covariant Gaussian Observables
For Gaussian bi-observables, J-covariance has a very easy characterization.
Proposition 10. Suppose is a Gaussian bi-observable on with parameters . Let J be any real matrix. Then, the POVM is a J-covariant observable if and only if .
Proof. For
, we let
and
be the two POVMs on
given by
By the commutation relations (
24) for the Weyl operators, we immediately get
we have also
Since
for all
, by comparing the last two expressions we see that
if and only if
which in turn is equivalent to
. ☐
Vector Observables
Let us point out the structure of the Gaussian approximate joint measurements of and .
Proposition 11. A bi-observable is Gaussian if and only if the state σ is Gaussian. In this case, the covariant bi-observable is Gaussian with parameters Proof. By comparing (
31), (
44) and (
48), and using the fact that
if and only if
and
, we have the first statement. Then, for
, we see immediately that
is a Gaussian observable with the above parameters. ☐
We call
the class of the Gaussian covariant phase-space observables. By (
50), observing
on a Gaussian state
yields the normal probability distribution
, with marginals
When and , we have an unbiased measurement.
Scalar Observables
We now study the Gaussian approximate joint measurements of the target observables
and
defined in (
6).
Proposition 12. A Gaussian bi-observable with parameters is in if and only if , where J is given by (47). In this case, the condition (49) is equivalent to Proof. The first statement follows from Proposition 10. Then, the matrix inequality (
49) reads
which is equivalent to (
52). ☐
We write
for the class of the Gaussian
-covariant phase-space observables. An observable
is thus characterized by the couple
. From (
50) with
given by (
47), we get that measuring
on a Gaussian state
yields the probability distribution
with
and
given by (
12). Its marginals with respect to the first and second entry are, respectively,
Example 2. Let us construct an example of an approximate joint measurement of and , by using a noisy measurement of position along followed by a sharp measurement of momentum along . Let Δ
be a positive real number yielding the precision of the position measurement, and consider the POVM on given by The characteristic function of is Therefore, is a Gaussian bi-observable with parameters , and , where J is given by (47) and , and . This implies ; in particular, the set is non-empty. Moreover, the lower bound is attained, cf. (52). Example 3. Let us consider the case ; now the target observables and are compatible and we can define a pvm on by setting for all . Its characteristic function isThen, with parameters , , and given by (47). Note that can be regarded as the limit case of the observables of the previous example when and . 7. Conclusions
We have extended the relative entropy formulation of MURs given in [
41] from the case of discrete incompatible observables to a particular instance of continuous target observables, namely the position and momentum vectors, or two components of them along two possibly non parallel directions. The entropic MURs we found share the nice property of being scale invariant and well-behaved in the classical and macroscopic limits. Moreover, in the scalar case, when the angle spanned by the position and momentum components goes to
, the entropic bound correctly reflects their increasing compatibility by approaching zero with continuity.
Although our results are limited to the case of Gaussian preparation states and covariant Gaussian approximate joint measurements, we conjecture that the bounds we found still hold for arbitrary states and general (not necessarily covariant or Gaussian) bi-observables. Let us see with some more detail how this should work in the case when the target observables are the vectors and .
The most general procedure should be to consider the error function
for an arbitrary POVM
on
and any state
. First of all, we need states for which neither the position nor the momentum dispersion are too small; the obvious generalization of the test states (
81) is
Then, the most general definitions of the entropic divergence and incompatibility degree are:
It may happen that
is not absolutely continuous with respect to
, or
with respect to
; in this case, the error function and the entropic divergence take the value
by definition. So, we can restrict to bi-observables that are (weakly) absolutely continuous with respect to the Lebesgue measure. However, the true difficulty is that, even with this assumption, here we are not able to estimate (
94), hence (
95). It could be that the symmetrization techniques used in [
17,
19] can be extended to the present setting, and one can reduce the evaluation of the entropic incompatibility index to optimizing over all covariant bi-observables. Indeed, in the present paper we a priori selected only covariant approximating measurements; we would like to understand if, among all approximating measurements, the relative entropy approach selects covariant bi-observables by itself. However, even if
is covariant, there remains the problem that we do not know how to evaluate (
94) if
and
are not Gaussian. It is reasonable to expect that some continuity and convexity arguments should apply, and the bounds in Theorem 5 might be extended to the general case by taking dense convex combinations. Also the techniques used for the PURs in [
8,
9] could be of help in order to extend what we did with Gaussian states to arbitrary states. This leads us to conjecture:
Conjecture (
96) is also supported since the uniqueness of the optimal approximating bi-observable in Theorem 5(i) is reminiscent of what happens in the discrete case of two Fourier conjugated mutually unbiased bases (MUBs); indeed, in the latter case, the optimal bi-observable is actually unique among all the bi-observables, not only the covariant ones (see [
41] (Theorem 5)).
Similar considerations obviously apply also to the case of scalar target observables. We leave a more deep investigation of equality (
96) to future work.
As a final consideration, one could be interested in finding error/disturbance bounds involving sequential measurements of position and momentum, rather than considering all their possible approximate joint measurements. As sequential measurements are a proper subset of the set of all the bi-observables, optimizing only over them should lead to bounds that are greater than
. This is the reason for which in [
41] an error/disturbance entropic bound, denoted by
and dinstinct from
, was introduced. However, it was also proved that the equality
holds when one of the target observables is discrete and sharp. Now, in the present paper, only sharp target observables are involved; although the argument of [
41] can not be extended to the continuous setting, the optimal approximating joint observables we found in Theorems 3(i) and 5(i)
actually are sequential measurements. Indeed, the optimal bi-observable in Theorem 3(i) is one of the POVMs described in Examples 2 and 3 (see (
74)); all these bi-observables have a (trivial) sequential implementation in terms of an unsharp measurement of
followed by sharp
. On the other hand, in the vector case, it was shown in ([
67], Corollary 1) that all covariant phase-space observables can be obtained as a sequential measurement of an unsharp version of the position
followed by the sharp measurement of the momentum
. Therefore,
also for target position and momentum observables, in both the scalar and vector case.