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

Abstract

It is standard to assume that the Wigner distribution of a mixed quantum state consisting of square-integrable functions is a quasi-probability distribution, i.e., that its integral is one and that the marginal properties are satisfied. However, this is generally not true. We introduced 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 1980s by H. Feichtinger. The properties of these states were studied, giving us the opportunity to prove an extension to the general case of a result due to Jaynes on the non-uniqueness of the statistical ensemble, generating a density operator.

$τ$-quantization and $τ$-Cohen classes distributions of Feichtinger operators

“…Hence, Feichtinger operators are trace class operators and we can compute their trace as follows tr(S) = R d K S (x, x) dx. In [7] operators having such a decomposition have been studied and called Feichtinger states in case tr(S) = 1, but there the link between these operators and the work [10] was not established, which is one of our main observations. Then the τ -Wigner distribution of S is defined in the following way…”

“…). We recall another frequently used time-frequency representation, the so-called cross-τ -Wigner distribution of f and g in L 2 (R d ) defined by (7) W…”

“…We now interpret (7) as the cross-τ -Wigner distribution of the rank-one operator f ⊗ g. Hence, it is natural to define the τ -Wigner distribution of an operator S with kernel K S in the following way:…”

Toeplitz density operators and their separability properties

“…The importance of density operators in quantum mechanics comes from the fact that they represent (and are identified with) "mixed quantum states"; these are mixtures of L 2 -normalized "pure states" (ψ j ) j in L 2 (R n ) weighted by probabilities μ j ≥ 0 summing up to one; the corresponding mixed state is then by definition the operator ρ = j μ j ψ j and represents the maximal knowledge one has about the system under consideration. It is not difficult [14,18] to check that the operator ρ thus defined indeed is a density operator; note that the decomposition j μ j ψ j of ρ has no reason to be unique (Jayne's theorem, see however [19] where we compare different expansions of pure states). A density operator is de facto a Weyl operator in view of Schwartz's kernel theorem; its Weyl symbol is (2π h) n ρ where ρ is the "Wigner distribution of ρ" defined by…”

Toeplitz Density Operators and their Separability Properties

“…The importance of density operators in quantum mechanics comes from the fact that they represent (and are identified with) "mixed quantum states"; these are mixtures of L 2 -normalized "pure states" (ψ j ) j∈F in L 2 (R n ) (F some discrete set) weighted by probabilities µ j ≥ 0 summing up to one; the corresponding mixed state is then by definition ρ = j µ j Π ψ j and represents the maximal knowledge one has about the system under consideration. It is not difficult [15,17] to check that the operator ρ thus defined indeed is a density operator; note that the decomposition j µ j Π ψ j of ρ has no reason to be unique (Jayne's theorem, see however [18]). A density operator is de facto a Weyl operator in view of Schwartz's kernel theorem; its Weyl symbol is (2π ) n ρ where ρ is the "Wigner distribution of ρ" defined by…”