Generalized Anti-Wick Quantum States

TL;DR

This paper investigates a class of mixed quantum states called Toeplitz density operators, generalizing anti-Wick operators, and explores their properties using advanced functional analysis tools.

Contribution

It introduces and analyzes a new class of mixed states called Toeplitz density operators, extending the concept of anti-Wick operators with a rigorous mathematical framework.

Findings

Characterization of Toeplitz density operators as quantum states.

Connection between Toeplitz operators and anti-Wick operators.

Application of Feichtinger spaces in the analysis of these operators.

Abstract

The purpose of this Note is to study a simple class of mixed states and the corresponding density operators (matrices). These operators, which we call quite Toeplitz density operators correspond to states obtained from a fixed function ("window") by position-momentum translations, and reduce in the simplest case to the anti-Wick operators considered long ago by Berezin. The rigorous study of Toeplitz operators requires the use of classes of functional spaces defined by Feichtinger.

Equations

$$\rho (z)=\sum _j{\alpha }_{j}W{\psi }_j(z)$$

$${\varphi }_{z_0}(x)=(\pi \hslash {)}^{-n/4}{e}^{i{p}_0(x-x_0)}{e}^{-|x-x_0{|}^2/2\hslash }$$

$$\hat{\rho }=\sum _{z_{\lambda }\in \Lambda }{\mu }_{z_{\lambda }}{\hat{\Pi }}_{z_{\lambda }}$$

$$\rho (z)=\sum _{z_{\lambda }\in \Lambda }{\mu }_{z_{\lambda }}W{\varphi }_{z_{\lambda }}(z).$$

$$\rho (z)=\sum _{z_{\lambda }\in \Lambda }{\mu }_{z_{\lambda }}W{\varphi }_0(z-z_{\lambda }).$$

$$\rho (z)=\int \mu (z^{\prime })W{\varphi }_0(z-z^{{}^{\prime }})d^{2n}{z}^{\prime }=(\mu \ast W{\varphi }_0)(z).$$

$$\rho (z)=\int \mu (z^{\prime })W\varphi (z-z^{{}^{\prime }})d^{2n}{z}^{\prime }=(\mu \ast W\varphi )(z).$$

$$(\psi |\varphi {)}_{L^2}=\int \psi (x){\varphi }^{\ast }(x)d^{n}x$$

$$\hat{T}(z)=\hat{T}(x,p)=e^{-i(x\hat{p}-p\hat{x})/\hslash }$$

$${\hat{A}}_{\mathrm{W}}=\underset{\mathrm{W}}{\mathrm{Op}}(a)=\int a_{\sigma }(z)\hat{T}(z)d^{2n}z$$

$$a_{\sigma }(z)={\left(\frac{1}{2\pi \hslash }\right)}^n\int e^{-\frac{i}{\hslash }\sigma (z,z^{\prime })}a(z^{\prime })d^{2n}{z}^{\prime }.$$

$$W\psi (z)={\left(\frac{1}{2\pi \hslash }\right)}^n\int e^{-\frac{i}{\hslash }py}\psi (x+\frac{1}{2}y){\psi }^{\ast }(x-\frac{1}{2}y)d^{n}y.$$

$$W(\psi ,\varphi )(z)={\left(\frac{1}{2\pi \hslash }\right)}^n\int e^{-\frac{i}{\hslash }py}\psi (x+\frac{1}{2}y){\varphi }^{\ast }(x-\frac{1}{2}y)d^{n}y.$$

$$\int W(\psi ,\varphi )(z)d^{2n}z=(\psi |\varphi {)}_{L^2}.$$

$$\hat{\rho }\psi =\sum _j{\lambda }_j(\psi |{\varphi }_j{)}_{L^2}{\varphi }_j.$$

$$\rho =\sum _j{\lambda }_{j}W{\varphi }_j\in L^2({\mathbb{R}}^{2n})\cap L^{\infty }({\mathbb{R}}^{2n})$$

$$W\psi (z)={\left(\frac{1}{2\pi \hslash }\right)}^n\int e^{-\frac{i}{\hslash }py}\psi (x+\frac{1}{2}y){\psi }^{\ast }(x-\frac{1}{2}y)d^{n}y.$$

$$\hat{\rho }=(2\pi \hslash {)}^n\underset{\mathrm{W}}{\mathrm{Op}}(\rho ).$$

$$a(z)=\int e^{-\frac{i}{\hslash }py}K(x+\frac{1}{2}y,x-\frac{1}{2}y)d^{n}y$$

$$||\psi |{|}_{\varphi }=\int |W(\psi ,\varphi )(z)|d^{2n}z=||W(\psi ,\varphi )|{|}_{L^1}$$

$$\mathcal{S}({\mathbb{R}}^n)\subset M^1({\mathbb{R}}^n)\subset C^0({\mathbb{R}}^n)\cap L^1({\mathbb{R}}^n)\cap F{L}^1({\mathbb{R}}^n)\cap L^2({\mathbb{R}}^n)$$

$$\psi \in M^1({\mathbb{R}}^n)⟺W\psi \in L^1({\mathbb{R}}^n).$$

$$M^1({\mathbb{R}}^n)\ast M^1({\mathbb{R}}^n)\subset M^1({\mathbb{R}}^n).$$

$${\hat{A}}_{\varphi }=\int a(z){\hat{\Pi }}_{\varphi }(z)d^{2n}z$$

$${\hat{\Pi }}_{\varphi }(z)\psi =(\psi |\hat{T}(z)\varphi {)}_{L^2}\hat{T}(z)\varphi .$$

$$(\psi |\hat{T}(z)\varphi {)}_{L^2}=(2\pi \hslash {)}^n\mathrm{Amb}(\psi ,\varphi )(z)$$

$$\mathrm{Amb}(\psi ,\varphi )(z)={\left(\frac{1}{2\pi \hslash }\right)}^n\int e^{-\frac{i}{\hslash }py}\psi (y+\frac{1}{2}x)\varphi (y-\frac{1}{2}x{)}^{\ast }{d}^{n}y.$$

$${\hat{A}}_{\varphi }\psi =(2\pi \hslash {)}^n\int a(z)\mathrm{Amb}(\psi ,\varphi )(z)\hat{T}(z)\varphi d^{2n}z$$

$${\hat{A}}_{\varphi }=(2\pi \hslash {)}^n\underset{\mathrm{W}}{\mathrm{Op}}(a\ast W\varphi ).$$

$$({\hat{\Pi }}_{\varphi }(z)\psi |\chi {)}_{L^2}=\int {\pi }_{\varphi }(z)W(\psi ,\chi )(z^{\prime })d^{2n}{z}^{\prime }$$

Full text

1 Introduction

Motivations

A quantum mixed state on $\mathbb{R}^{n}$ consists of a collection of pairs $\{(\psi_{j},\alpha_{j})\}$ where the $\psi_{j}\in L^{2}(\mathbb{R}^{n})$ are “pure states” and the $\alpha_{j}$ are probabilities summing up to one. Given a mixed state one defines its density operator as the bounded operator $\widehat{\rho}=\sum_{j}\alpha_{j}\widehat{\Pi}_{j}$ on $L^{2}(\mathbb{R}^{n})$ where $\widehat{\Pi}_{j}$ is the orthogonal projection in $L^{2}(\mathbb{R}^{n})$ onto the ray $\mathbb{C}\psi_{j}=\{\alpha\psi_{j}:\alpha\in\mathbb{C}\}$. The corresponding Wigner distribution is by definition the function $\rho\in L^{2}(\mathbb{R}^{2n})$ defined by

[TABLE]

here $W\psi_{j}$ is the usual Wigner transform of $\psi_{j}$. In this work we will deal with a particular class of mixed states, and their generalization to a continuous setting. Consider the ground state

[TABLE]

of a linear oscillator $H=\frac{1}{2}(|p-p_{0}|^{2}+|x-x_{0}|^{2})$ whose center $z_{0}=(x_{0},p_{0})$ is not know with precision. Suppose first we have some partial information telling us that the center $z_{0}$ is located somewhere on a lattice $\Lambda\subset\mathbb{R}^{2n}$ consisting of a discrete set of phase space points $z_{\lambda}=(x_{\lambda},p_{\lambda})$ and that there is a probability $\mu_{z_{\lambda}}$ that the system is precisely in the state $\phi_{z_{\lambda}}$. Denoting by $\widehat{\Pi}_{z_{\lambda}}$ the orthogonal projection on $\mathbb{C}\phi_{z_{\lambda}}$ the corresponding density operator

[TABLE]

has Wigner distribution

[TABLE]

Observing that $W\phi_{z_{\lambda}}(z)=W\phi_{0}(z-z_{\lambda})$ this reduces to the Gabor-type [12] expansion

[TABLE]

Let us now depart from the discrete case, and consider the somewhat more realistic situation where the center of the linear oscillator can be any point $z^{\prime}$ in phase space $\mathbb{R}^{2n}$; we assume the latter comes equipped with a certain Borel probability density $\mu$. Formula (5) suggests that we define in this case a generalized Wigner distribution by

[TABLE]

Notice that if one chooses for $\mu$ the atomic measure $\sum_{z_{\lambda}\in\Lambda}\mu_{z_{\lambda}}\delta_{z_{\lambda}}$ then (6) reduces to the discrete sum (4). It turns out that the operator $\widehat{\rho}$ obtained from $\rho$ using the Weyl correspondence is a well-known mathematical object: it is the anti-Wick operator with Weyl symbol $\rho$. Such operators were first considered by Berezin [2], and have been developed since by many independent authors. A further generalization of (6) now consists in replacing the standard Gaussian $\phi_{0}$ by an arbitrary square integrable function $\phi$ and to consider the Weyl transforms of functions of the type

[TABLE]

The operators thus obtained are called Toeplitz operators (or localization operators) in the mathematical literature; they are natural generalizations of anti-Wick operators [11, 27, 32]. So far, so good. The rub comes from the fact that it is not clear why the Weyl transform $\widehat{\rho}$ of (6) — let alone that of (7)! — should indeed be density operators. For this, $\widehat{\rho}$ has to satisfy three stringent conditions: (i) $\widehat{\rho}$ must be positive semi-definite: $\widehat{\rho}\geq 0$ and (ii) be self-adjoint: $\widehat{\rho}=\widehat{\rho}^{\ast}$; (iii) $\widehat{\rho}$ must be of trace class and have trace one: $\operatorname*{Tr}(\widehat{\rho})=1$. While it is easy to verify (i) (which implies (ii) since $L^{2}(\mathbb{R}^{n})$ is complex), it is the third condition which poses problem since it is certainly not trivially satisfied, as we will see in the course of this paper. Viewed in a broader perspective, quasi-distributions of the type $\rho=\mu\ast W\phi$ belong to the Cohen class [16, 25], which has a rich internal structure and is being currently very much investigated for its own sake both in time-frequency analysis [4, 5, 6, 8, 9, 31] and in quantum mechanics [19].

Main results

Notation and terminology

The scalar product on $L^{2}(R^{n})$ is defined by

[TABLE]

and we thus have $(\psi|\phi)_{L^{2}}=\langle\phi|\psi\rangle$ in Dirac bra-ket notation; in this notation $\widehat{\Pi}_{\phi}=|\phi\rangle\langle\phi|$. The phase space $T^{\ast}R^{n}\equiv R^{2n}$ will be equipped with the canonical symplectic structure $\sigma=\sum_{j=1}^{n}dp_{j}\wedge dx_{j}$, given in matrix notation by $\sigma(z,z^{\prime})=Jz\cdot z^{\prime}$ where $J=\begin{pmatrix}0&I\\ -I&0\end{pmatrix}$ is the standard symplectic matrix on $R^{2n}$. We denote by

[TABLE]

the Heisenberg–Weyl displacement operator. If $a\in\mathcal{S}^{\prime}(\mathbb{R}^{2n})$ is a symbol on $\mathbb{R}^{2n}$ the corresponding Weyl operator is

[TABLE]

where $a_{\sigma}$ is the symplectic Fourier transform of $a$, formally defined by

[TABLE]

The Wigner transform of $\psi\in L^{2}(\mathbb{R}^{n})$ is defined by

[TABLE]

Similarly, the cross-Wigner transform of two functions $\psi,\phi\in L^{2}(\mathbb{R}^{n})$ is defined by

[TABLE]

We have

[TABLE]

2 Density Operators: Summary

A density operator on $L^{2}(\mathbb{R}^{n})$ is by definition a bounded operator $\widehat{\rho}$ which is semidefinite positive: $\widehat{\rho}\geq 0$ (and hence self-adjoint), and has trace $\operatorname*{Tr}(\widehat{\rho})=1$. The set $\mathcal{D}=\mathcal{D}(L^{2}(\mathbb{R}^{n}))$ of density operators is a convex subset of the space $\mathcal{L}_{1}=\mathcal{L}_{1}(L^{2}(\mathbb{R}^{n}))$ of trace class operators (recall that the latter is a two-sided ideal of the algebra $\mathcal{B}=\mathcal{B}(L^{2}(\mathbb{R}^{n}))$ of bounded operators on $L^{2}(\mathbb{R}^{n})$). In particular density operators are compact operators, and hence, by the spectral theorem, there exists an orthonormal basis $(\phi_{j})_{j}$ of $L^{2}(\mathbb{R}^{n})$ and coefficients satisfying $\lambda_{j}\geq 0$ and $\sum_{j}\lambda_{j}=1$ such that $\widehat{\rho}$ can be written as a convex sum $\sum_{j}\lambda_{j}\widehat{\Pi}_{\phi_{j}}$ of orthogonal projections $\widehat{\Pi}_{\phi_{j}}$ converging in the strong operator topology, that is, counting the multiplicities,

[TABLE]

By definition, the “Wigner distribution” of $\widehat{\rho}$ is the function

[TABLE]

where $W\psi_{j}$ is the usual Wigner transform of $\phi_{j}$, defined for $\psi\in L^{2}(\mathbb{R}^{n})$ by

[TABLE]

Proposition 1

The Wigner distribution (16) is $(2\pi\hbar)^{-n}$ times the Weyl symbol of $\widehat{\rho}$:

[TABLE]

The proof of this result immediately follows from:

Lemma 2

The orthogonal projection $\widehat{\Pi}_{\phi}$ of $L^{2}(\mathbb{R}^{n})$ on $\mathbb{C}\phi$ ($\phi\in L^{2}(\mathbb{R}^{n})$) has Weyl symbol $(2\pi\hbar)^{n}W\phi$.

Proof. By definition $\widehat{\Pi}_{\phi}\psi=(\psi|\phi)_{L^{2}}\phi$ so the distributional kernel of $\widehat{\Pi}_{\phi}$ is the function $K_{\phi}(x,y)=\phi(x)\phi^{\ast}(y)$; the Weyl symbol of an operator $\widehat{A}=\operatorname*{Op}\nolimits_{\mathrm{W}}(a)$ being related to the kernel $K$ of $\widehat{A}$ by the formula [16, 32]

[TABLE]

the lemma follows.

A class of windows $\phi$ playing a privileged role in the study of density operators is the Feichtinger algebra [13, 15], which is the simplest modulation space (a non-exhaustive list of references on the topic of modulation spaces is [14, 25, 28, 34]. In [16], Chapters 16 and 17 we have given a succinct account of the theory using the Wigner transform instead of the traditional short-time Fourier transforms approach). By definition Feichtinger’s algebra $M^{1}(\mathbb{R}^{n})=S_{0}(\mathbb{R}^{n})$ consists of all distributions $\psi\in\mathcal{S}^{\prime}(\mathbb{R}^{n})$ such that $W(\psi,\phi)\in L^{1}(\mathbb{R}^{2n})$ for some window $\phi\in\mathcal{S}(\mathbb{R}^{n})$; when this condition holds, we have $W(\psi,\phi)\in L^{1}(\mathbb{R}^{2n})$ for all windows $\phi\in\mathcal{S}(\mathbb{R}^{n})$ and the formula

[TABLE]

defines a norm on the vector space $M^{1}(\mathbb{R}^{n})$; another choice of window $\phi^{\prime}$ the leads to an equivalent norm and one shows that $M^{1}(\mathbb{R}^{n})$ is a Banach space for the topology thus defined. We have the following continuous inclusions:

[TABLE]

and $\mathcal{S}(\mathbb{R}^{n})$ is dense in $M^{1}(\mathbb{R}^{n})$. Moreover, for every $\psi\in L^{2}(\mathbb{R}^{n})$ we have the equivalence

[TABLE]

A particularly important property of $M^{1}(\mathbb{R}^{n})$ is its metaplectic invariance. Let $\widehat{S}\in\operatorname*{Mp}(n)$ (the metaplectic group) cover $S\in\operatorname*{Sp}(n)$; then $W(\widehat{S}\psi,\widehat{S}\phi)=W(\psi,\phi)\circ S^{-1}$ [16]; it follows from this covariance formula and the fact that the choice of $\phi$ is irrelevant, that $\widehat{S}\psi\in M^{1}(\mathbb{R}^{n})$ if and only if $\psi\in M^{1}(\mathbb{R}^{n})$. Also, $M^{1}(\mathbb{R}^{n})$ is invariant under the action of the translations $\widehat{T}(z)$. One can show [25] that $M^{1}(\mathbb{R}^{n})$ is the smallest Banach algebra containing $\mathcal{S}(\mathbb{R}^{n})$ and having metaplectic and translational invariance.

It turns out that $M^{1}(\mathbb{R}^{n})$ is in addition an algebra for both pointwise product and convolution; in fact if $\psi,\psi^{\prime}\in M^{1}(\mathbb{R}^{n})$ then $||\psi\ast\psi^{\prime}||_{\phi}\leq||\psi||_{L^{1}}||\psi^{\prime}||_{\phi}$ so we have $L^{1}(\mathbb{R}^{n})\ast M^{1}(\mathbb{R}^{n})\subset M^{1}(\mathbb{R}^{n})$ hence in particular

[TABLE]

Taking Fourier transforms we conclude that $M^{1}(\mathbb{R}^{n})$ is also closed under pointwise product.

3 Toeplitz Operators: Definitions and Properties

Let $\phi$ (hereafter called window) be in $M^{1}(\mathbb{R}^{n})$. By definition, the Toeplitz operator $\widehat{A}_{\phi}=\operatorname*{Op}\nolimits_{\phi}(a)$ with window $\phi$ and symbol $a$ is

[TABLE]

where $\widehat{\Pi}_{\phi}:L^{2}(\mathbb{R}^{n})\longrightarrow L^{2}(\mathbb{R}^{n})$ is the orthogonal projection onto $\widehat{T}(z)\phi$, that is

[TABLE]

We observe that $(\psi|\widehat{T}(z)\phi)_{L^{2}}$ is, up to a factor, the cross-ambiguity transform of the pair $(\psi,\phi)$; in fact ([16], §11.4.1)

[TABLE]

where

[TABLE]

We can therefore rewrite the definition (19) of $\widehat{A}_{\phi}$ as

[TABLE]

which is essentially the definition of single-windowed Toeplitz operators given in the time-frequency analysis literature (see e.g. [8, 9, 7, 34]). Toeplitz operators are linear continuous operators $\mathcal{S}(\mathbb{R}^{n})\longrightarrow\mathcal{S}^{\prime}(\mathbb{R}^{n}$); in view of Schwartz’s kernel theorem they are thus automatically Weyl operators. In fact:

Proposition 3

The Toeplitz operator $\widehat{A}_{\phi}=\operatorname*{Op}\nolimits_{\phi}(a)$ has Weyl symbol $(2\pi\hbar)^{n}(a\ast W\phi)$, that is

[TABLE]

Proof. It is sufficient to assume that $a\in\mathcal{S}(\mathbb{R}^{2n})$. Let $\pi_{\phi}(z)$ be the Weyl symbol of the orthogonal projection $\widehat{\Pi}_{\phi}(z)$ on $\widehat{T}(z)\phi$; we thus have

[TABLE]

for all $\psi,\chi\in\mathcal{S}(\mathbb{R}^{n})$ (see e.g. [16], §10.1.2). Lemma 2 and the translational covariance of the Wigner transform ([16], §9.2.2) imply that we have

[TABLE]

and hence

[TABLE]

Using definition (19) we thus have, by the Fubini–Tonnelli theorem,

[TABLE]

hence the Weyl symbol of $\widehat{A}_{\phi}$ is $a=(2\pi\hbar)^{n}W\phi\ast\mu$ as claimed.

We next prove an extension of the usual symplectic covariance result [17, 16, 30]

[TABLE]

for Weyl operators to the Toeplitz case. We recall the associated symplectic covariance formulas ([16], §8.1.3 and 10.3.1)

[TABLE]

Corollary 4

Let $\widehat{A}_{\phi}=\operatorname*{Op}\nolimits_{\phi}(a)$ and $\widehat{S}\in\operatorname*{Mp}(n)$ have projection $S\in\operatorname*{Sp}(n)$. We have

[TABLE]

Proof. We begin by noticing that we have $\widehat{S}^{-1}\phi\in M^{1}(\mathbb{R}^{n})$ since $\phi\in M^{1}(\mathbb{R}^{n})$ hence $\widehat{S}^{-1}\phi$ is a bona fide window. Since $W\phi(z-z^{\prime})=W(\widehat{T}(z^{\prime})\phi)(z)$ we have, setting $z^{\prime\prime}=S^{-1}z^{\prime}$ and using the relations (28),

[TABLE]

It follows, using the covariance formula (27) for Weyl operators that

[TABLE]

which is precisely (29) in view of formula (24) in Proposition 3.

4 Toeplitz Quantum States

The following boundedness result was proven by Gröchenig in [24], Thm. 3; it is a particular case of Thm. 3.1 in Cordero and Gröchenig [7]:

Proposition 5

Let $a\in M^{1}(\mathbb{R}^{2n})$, then $\widehat{A}=\operatorname*{Op}\nolimits_{\mathrm{W}}(a)$ is of trace class: $\widehat{A}\in\mathcal{L}_{1}(L^{2}(\mathbb{R}^{n}))$ and $||\widehat{A}||_{\mathcal{L}_{1}}\leq C||a||_{M^{1}}$ for some $C>0$ ($||\cdot||_{\mathcal{L}_{1}}$ the trace norm and $||\cdot||_{M^{1}}$ a norm on $M^{1}(\mathbb{R}^{2n})$).

Our main result makes use of the class of symbols $M^{1,\infty}(\mathbb{R}^{2n})$, which is a particular modulation space containing $L^{1}(\mathbb{R}^{2n})$. It is defined as follows [25]: $a\in M^{1,\infty}(\mathbb{R}^{2n})$ if and only if for some (and hence every) $b\in\mathcal{S}(\mathbb{R}^{2n})$

[TABLE]

When $b$ describes $\mathcal{S}(\mathbb{R}^{2n})$ the mappings $a\longmapsto||a||_{b}$ defined by

[TABLE]

We have the following important convolution property between the symbol class $M^{1,\infty}(\mathbb{R}^{2n})$ and the Feichtinger algebra $M^{1}(\mathbb{R}^{2n})$:

[TABLE]

(Prop. 2.4 in [7]).

Let us state and prove our main result. We assume that $\mu$ is a Borel probability density function on $\mathbb{R}^{2n}$ with respect to the Lebesgue measure: $\mu\in L^{1}(\mathbb{R}^{2n})$ and

[TABLE]

Theorem 6

Let $\mu\in M^{1,\infty}(\mathbb{R}^{2n})$ and $\phi\in M^{1}(\mathbb{R}^{n})$. Then

[TABLE]

is a density operator, and there exists $C>0$ such that

[TABLE]

Proof. Let us prove that $\widehat{\rho}\in\mathcal{L}_{1}(L^{2}(\mathbb{R}^{n}))$. In view of Proposition 5 it is sufficient to show that the Weyl symbol $a=(2\pi\hbar)^{n}(\mu\ast W\phi)$ is in $M^{1}(\mathbb{R}^{2n})$. We begin by observing that the condition $\phi\in M^{1}(\mathbb{R}^{n})$ implies that $W\phi\in M^{1}(\mathbb{R}^{2n})$; this property is in fact a consequence of the more general result Prop.2.5 in [7] but we give here a direct independent proof. Let $b\in\mathcal{S}(\mathbb{R}^{2n})$; denoting by $W^{2n}$ the cross-Wigner transform on $\mathbb{R}^{2n}$ we have (property (14))

[TABLE]

and hence $W\phi\in M^{1}(\mathbb{R}^{2n})$ as claimed. In view of the convolution property (31) we have $\mu\ast W\phi\in M^{1}(\mathbb{R}^{2n})$ as desired. Let us show that $\widehat{\rho}\geq 0$ (and hence $\widehat{\rho}^{\ast}=\widehat{\rho}$) for all $\psi\in L^{2}(\mathbb{R}^{n})$. We have

[TABLE]

since by definition

[TABLE]

we get

[TABLE]

Let us finally prove that $\operatorname*{Tr}(\widehat{\rho})=1$. We have $a\in M^{1}(\mathbb{R}^{2n})\subset L^{1}(\mathbb{R}^{2n})$ hence ([10]; also [16] §12.3.2):

[TABLE]

But we have

[TABLE]

and hence, since $||\phi||_{L^{2}}=1$,

[TABLE]

that is $\operatorname*{Tr}(\widehat{\rho})=1$.

A typical example is provided by anti–Wick quantization. Choose for window $\phi$ the standard coherent state $\phi_{0}$:

[TABLE]

Its Wigner transform is given by [1, 16, 21, 30]

[TABLE]

and is thus a classical probability density

[TABLE]

It follows that $\mu\ast\phi$ is itself a probability density and that $\operatorname*{Tr}(\widehat{\rho})=1$ (for an alternative proof see Boggiatto and Cordero [4], Thm. 2.4).

Acknowledgement 7

This work was written while the author was holding the Giovanni Prodi visiting professorship at the Julius-Maximilians-Universität Würzburg during the summer semester 2019.