On the Non-Uniqueness of Statistical Ensembles Defining a Density Operator and a Class of Mixed Quantum States with Integrable Wigner Distribution

It is standard to assume that the Wigner distribution of a mixed quantum state consisting of square-integrable functions is a quasi-probability distribution, that is that its integral is one and that the marginal properties are satisfied. However this is in general not true. We introduce a class of quantum states for which this property is satisfied, these states are dubbed"Feichtinger states"because they are defined in terms of a class of functional spaces (modulation spaces) introduced in the 1980's by H. Feichtinger. The properties of these states are studied, which gives us the opportunity to prove an extension to the general case of a result of Jaynes on the non-uniqueness of the statistical ensemble generating a density operator. As a bonus we obtain a result for convex sums of Wigner transforms.


Introduction
A useful device when dealing with density operators is the covariance matrix, whose existence is taken for granted in most elementary and advanced texts. However, a closer look shows that the latter only exists under rather stringent conditions on the involved quantum states. This question is related to another more general one. Assume that we are dealing with mixed quantum state {(ψ j , α j )} with ψ j ∈ L 2 (R n ), ||ψ j || = 1 and α j ≥ 0, j α j = 1 the index j belonging to some countable set. The Wigner distribution of that state, which is by definition the convex sum of Wigner transforms is usually referred to as a "quasi-probability". But such an interpretation makes sense only if R 2n ρ(z)dz = 1 (2) which is the case if ψ j ∈ L 2 (R n ), but it does not imply per se the absolute integrability of W ψ j : there are examples of square-integrable functions ψ such that W ψ ∈ L 2 (R n ) but W ψ / ∈ L 1 (R 2n ) [12]. In [2] Cordero and Tabacco give a simple example in dimension n = 1: the function ψ = χ [−/2,1/2] (the characteristic function of the interval [−/2, 1/2] belongs to both L 1 (R) and L 2 (R) but we have W ψ / ∈ L 1 (R 2 ). On the other hand, even when satisfied, condition (2) is not sufficient to ensure the existence of the covariances since these involve the calculation of second moments, and the integrability of ρ does not guarantee the convergence of such integrals. These difficulties are usually ignored in the physical literature, or dismissed by vague assumptions like "sufficiently fast" decrease of the Wigner distribution at infinity. The aim of this paper is to remedy this vagueness by proposing an adequate functional framework for a rigorous analysis of mixed states and of the associated density operator. For this purpose we will use the Feichtinger modulation spaces M 1 s (R n ), in instead of L 2 (R n ) as a "reservoir" for quantum states. In the simplest case, s = 0, we get the so-called Feichtinger algebra M 1 0 (R n ) = S 0 (R n ) which can be defined by We agree that it is not clear at all why the condition above should define a vector space -let alone an algebra! -since the Wigner transform is not additive; surprisingly enough this is however the case, and one sees that if we assume that the state {(ψ j , α j )} is such that ψ j ∈ S 0 (R n ) for every j then the normalization condition (2) is satisfied. We will also see that the existence of covariances like (3) (and hence the covariance matrix) is guaranteed if we make the sharper assumption that ψ j ∈ M 1 s (R n ) for some s ≥ 2. While the theory of modulation spaces has become a standard tool in time-frequency and harmonic analysis, it is somewhat less known in quantum physics. In [11] we have applied it to deformation quantization; in [4] spectral and regularity results for operators in modulation spaces are studied. One of the reasons why this theory is less popular in quantum mechanics might be that the usual treatments are given in terms of short-time Fourier transforms (also called Gabor transforms) instead of the Wigner transform as we do here, and this has the tendency to make the theory obscure for many physicists ignoring the simple relation between Wigner and Gabor transforms. We therefore speculate that this lack of communication between both communities is of a pedagogical nature. The use of Wigner transforms in the theory of modulation spaces actually has many advantages, for instance it makes the symplectic invariance of these spaces become obvious and thus links them directly to the Weyl-Wigner-Moyal formalism.
The notation used here is standard, the phase space variable is z = (x, p) with x = (x 1 , ..., x n ) ∈ R n and p = (p 1 , ..., p n ) ∈ (R n ) * ≡ R n . The scalar product on L 2 (R n ) is (φ(x) the complex conjugate of φ(x)) and the associated norm is denoted by ||ψ||.

The Modulation Spaces
We give here a brief review of the main definitions and properties of the class of modulations spaces we will need. Modulation spaces were introduced by Feichtinger in the early 1980's [6,7,8]. The most complete treatment can be found in the book [12] by Gröchenig; also see the recent review paper [14]. In [9] (Chapters 16 and 17) modulation spaces are studied from the point of view of the Wigner transform which we use here. We will denote by W (ψ, φ) the cross-Wigner function of it is defined by When ψ = φ one obtains the usual Wigner transform Recall [9,10] that W (ψ, φ) is a continuous function belonging to L 2 (R 2n ) and that as well as Taking ψ = φ it follows that, in particular, hence the integral condition (2) holds as soon as In what follows s is a non-negative real number: s ≥ 0. We set z = (x, p) and The function z −→ z is the Weyl symbol of the elliptic pseudodifferential operator (1 − ∆) 1/2 where ∆ is the Laplacian in the z variables. We denote by L 1 s (R 2n ) the weighted L 1 -space is defined by that is, ρ ∈ L 1 s (R 2n ) if and only if we have Due to the submultiplicativity of the weights these space are in fact Banach algebras with respect to convolution (see [18]). The same is true for the spaces M 1 s (R n ).
for every φ ∈ S(R n ) (the Schwartz space of test functions deceasing rapidly at infinity).
It turns out that it suffices to check that condition (13) holds for one function φ = 0 for it then holds for all; moreover the mappings ψ −→ ||ψ|| φ,M 1 s defined by form a family of equivalent norms, and the topology on M 1 s (R n ) thus defined makes it into a Banach space. We have the chain of inclusions where S(R n ) is the Schwartz space of tests functions and F(L 1 (R n )) is the space of Fourier transforms Fψ of the elements ψ of L 1 (R n ). Observe that and one proves [12] that Recall that the metaplectic group Mp(n) is the unitary representation on L 2 (R n ) of the symplectic group. Sp(n). The modulation spaces M 1 s (R n ) are invariant under the action of metaplectic group: if S ∈ Mp(n) and ψ ∈ M 1 s (R n ) then Sψ ∈ M 1 s (R n ). This property actually follows from the symplectic covariance property of the Wigner transform, where S ∈ Sp(n) is the projection of S ∈ Mp(n) (see [9] for a detailed study of the metaplectic representation and symplectic covariance). When s = 0 we write M 1 0 (R n ) = S 0 (R n ) (the Feichtinger algebra); clearly M 1 s (R n ) ⊂ S 0 (R n ) for all s ≥ 0. In addition to being a vector space, S 0 (R n ) is a Banach algebra for both pointwise multiplication and convolution. It is actually the smallest Banach algebra invariant under the action of the metaplectic group Mp(n) and phase space translations. As mentioned in the introduction, the Feichtinger algebra can be characterized by the condition (4): S 0 (R n ) is the vector space of all ψ ∈ L 2 (R n ) such that W ψ ∈ L 1 (R 2n ).

Feichtinger States
Consider, as in the introduction, a mixed quantum state {(ψ j , α j )} and denote by ρ = j α j ρ j , ρ = j α j W ψ j the corresponding density operator and its Wigner distribution; ρ j is the orthogonal projection on the ray Cψ j , that is Observe that (2π ) n ρ is the Weyl symbol of the operator ρ [9].  Proof. It is an immediate consequence of metaplectic invariance of the modulation spaces M 1 s (R n ). The Wigner distribution ρ of a Feichtinger state is a bona-fide quasidistribution: Proposition 4 Let {(ψ j , α j )} be a Feichtinger state. Then ρ ∈ L 1 (R 2n ) and the marginal properties hold, and we have Proof. Since M 1 s (R n ) ⊂ M 1 0 (R n ) = S 0 (R n ) we automatically have W ψ j ∈ L 1 (R 2n ) for every j hence ρ ∈ L 1 (R 2n ). On the other hand we know that the marginal conditions hold if both ψ j and F ψ j are integrable [10], and this is precisely the case here in view of the inclusion (14). It follows that since the α j sum up to one and the ψ j are unit vectors in.L 2 (R n ).

Remark 5
The assumption that {(ψ j , α j )} is a Feichtinger state is crucial since it ensures us that the marginal properties (19) and (20) hold.
Modulation spaces also allow to define rigorously the covariance matrix Σ of a state. It is the symmetric 2n × 2 × n matrix defined by where z, the expectation vector, is given by (all vectors z are viewed as column matrices in these definitions). Assuming z = 0 the covariance matrix explicitly given by where Σ XP = (σ 2 x j p k ) 1≤j,k,≤n with σ 2 x j p k given by (3), and so on.
(i) Then the covariance matrix Σ is well-defined; (ii) the Fourier transform F ρ of the Wigner distribution of ρ is twice continuously differentiable: F ρ ∈ C 2 (R 2n ).
Proof. It suffices to assume that s = 2; we then have Setting z α = x α if 1 ≤ α ≤ n and z α = p α if n + 1 ≤ α ≤ 2n we have z = ( z 1 , ..., z n ) where z α is given by the absolutely convergent integral similarly, the integral is also absolutely convergent in view of the trivial inequalities |z α z β | ≤ 1 + |z| 2 . We have differentiating twice under the integration sign we get hence the estimates

Independence of the Statistical Ensemble
Several distinct statistical ensembles can give rise to the same density matrix. For instance, given an arbitrary mixed state {(ψ j , α j )} as above, the density matrix ρ = j α j ρ j where the ρ j are the orthogonal projection on the rays Cψ j can be written, using the spectral decomposition theorem, as ρ = j λ j ρ ′ j where the λ j are the eigenvalues of ρ and the ρ ′ j is the orthogonal projections on the rays Cφ j the φ j being the eigenvectors corresponding to the λ j .
The main result of this section is a generalization to the infinite-dimensional case of a result due to Jaynes [15]. Recall that a partial isometry is an operator whose restriction to the orthogonal complement of its null-space is an isometry [13].
Proposition 7 Let {(ψ j , λ j )} be a mixed state and (φ j ) and orthonormal basis of L 2 (R n ) and write ψ j = k a jk φ k . (i) The operator A defined by is a Hilbert-Schmidt operator and ρ = A A * is the density matrix of the state {(ψ j , λ j )}. (ii) Two mixed states {(ψ j , λ j )} and {(ψ ′ j , λ ′ j )} generate the same density matrix ρ = A A * if and only if there exists a partial isometry U of hence A is a Hilbert-Schmidt operator. It follows that A * is also Hilbert-Schmidt, hence A A * is a trace class operator with unit trace: Tr( A A * ) = Tr( A * A) = 1 the second equality in view of (24). Let us show that in fact ρ = A A * , that is ρψ = k λ k (ψ|ψ k )ψ k for every ψ ∈ L 2 (R n ). It is sufficient to show that this identity holds for the basis vectors φ j , that is for every j. Using the expansions we can rewrite this identity as On the other hand, by definition of A we have which is the same thing as ρφ j . (ii) Let U be a partial isometry of L 2 (R n ); then A ′ A ′ * = A A * = ρ. Suppose conversely that A ′ A ′ * = A A * . A classical result from the theory of Hilbert spaces (Douglas' lemma [5]) tells us that there exists a partial isometry U such that A ′ = A U so we are done..

Remark 8
The result above has been considered and proven by [1,16,17] in the case of quantum states having a finite numbers of elements. Their proofs do not immediately extend to the infinite dimensional case.
An immediate consequence is that the property of being a Feichtinger state is invariant under transformations preserving the density matrix.
Corollary 9 Let {(ψ j , α j ) : j ∈ J}be a Feichtinger state where J is a finite set of indices. Then every state {(φ j , β j ) generating the same density matrix ρ is also a Feichtinger state. In particular the spectral decomposition ρ = j∈J λ j ρ j consists of orthogonal projections ρ j on rays Cψ j where ψ j ∈ M 1 s (R n ).
Proof. Assume that ψ j ∈ M 1 s (R n ) for every j. In view of Proposition 7 there exist finite linear relations φ k = j a jk ψ j for each index k hence φ k ∈ M 1 s (R n ) for every k since M 1 s (R n ) is a vector space.

Remark 10
The proof does not trivially extend to the general case where the index set J is infinite because of convergence problems. The question whether the Corollary extends to the general case is open.
As a bonus we obtain the following new result about convex sums of Wigner distributions.
Corollary 11 Let (ψ j ) j be a sequence in M 1 s (R n ). Let (φ j ) j be a sequence of functions in L 2 (R n ) and sequences (α j ) and (β j ) of positive numbers such that j α j = j β j = 1. We assume that ||ψ j || = ||φ j || = 1 for all j. If we have Proof. Both series are absolutely convergent in view of (7); for instance The function ρ = (2π ) n j a j W ψ j = (2π ) n j β j W φ j is the Wigner distribution of a density matrix generated by the Feichtinger state {(ψ j , α j )}. In view of Corollary 9 we must then have φ j ∈ M 1 s (R n ) for every j.

Discussion
We have seen that the class of modulation spaces M 1 s (R n ) provide us with a very convenient framework for the study of the Wigner distribution of density matrices; it is far less restrictive than the conventional use of the Schwartz space S(R n ) which requires that the functions and all their derivatives be zero at infinity. In addition, the topology of modulation spaces is simpler since they are defined by a norm making them to Banach spaces, while that of S(R n ) is defined by a family of semi-norms making it to a Fréchet space. Another particularly attractive feature of modulation spaces is that they allow to introduce an useful class of Banach Gelfand triples (see [3]). For instance, (S 0 (R n ), L 2 (R n ), S ′ 0 (R n )) where S ′ 0 (R n ) is the dual space of the Feichtinger algebra S 0 (R n ) is such a triple. S ′ 0 (R n ) consists of all ψ ∈ S ′ (R n ) such that W (ψ, φ) ∈ L ∞ (R 2n ) for one (and hence all) φ ∈ S 0 (R n ); the duality bracket is simply given by the pairing Since S 0 (R n ) is the smallest Banach space isometrically invariant under the action of the metaplectic group its dual is essentially the largest space of distributions with this property. The use of such triples makes the use of the Dirac bra-ket notation much more natural and rigorous. For instance, objects like ψ|φ automatically have a meaning for all φ ∈ S 0 (R n ) and all ψ ∈ S ′ 0 (R n ).