# Rapidly decaying Wigner functions are Schwartz functions

Felipe Hernández1 1Department of Mathematics, Stanford University, 450 Jane
Stanford Way, Building 380, Stanford, CA 94305 USA  and  C. Jess Riedel2
2Physics & Informatics Laboratories, NTT Research Inc., 940 Stewart Drive,
Sunnyvale, CA 94085, USA<EMAIL_ADDRESS>
###### Abstract.

We show that if the Wigner function of a (possibly mixed) quantum state decays
toward infinity faster than any polynomial in the phase space variables $x$
and $p$, then so do all of its derivatives, i.e., it is a Schwartz function on
phase space. This is equivalent to the condition that the Husimi function is a
Schwartz function, that the quantum state is a Schwartz operator in the sense
of Keyl et al., and, in the case of a pure state, that the wavefunction is a
Schwartz function on configuration space. We discuss the interpretation of
this constraint on Wigner functions and provide explicit bounds on Schwartz
seminorms.

##  1\. Introduction

In quantum mechanics, quantum states of $n$ degrees of freedom can be
represented by positive semidefinite trace-class operators on
$L^{2}({\mathbf{R}}^{n})$. Each quantum state $\rho$ is associated with a
kernel ${\mathcal{K}_{\rho}}$ through
$(\rho\phi)(x)=\int{\mathcal{K}_{\rho}}(x,y)\phi(x)\mathop{}\\!\mathrm{d}x$,
$\phi\in L^{2}({\mathbf{R}}^{n})$, and the corresponding _Wigner function_
${\mathcal{W}_{\rho}}$ is

${\mathcal{W}_{\rho}}(x,p):=\frac{1}{(2\pi)^{n}}\int e^{ip\cdot
y}{\mathcal{K}_{\rho}}(x-y/2,x+y/2)\mathop{}\\!\mathrm{d}y.$

We denote the set of all such Wigner function as
${\mathcal{V}}({\mathbf{R}}^{2n})$. Our main result is a relationship between
the decay of such Wigner functions and their smoothness.

To quantify this we use the Schwartz-type seminorms $|F|_{a,b}$ $:=$
$\sup_{x,p}|x^{a_{\mathrm{x}}}p^{a_{\mathrm{p}}}\partial_{x}^{b_{\mathrm{x}}}\partial_{p}^{b_{\mathrm{x}}}F(x,p)|$
of a function $F:{{\mathbf{R}}^{2n}}\to{\mathbf{C}}$ on phase space, with
multi-indices
$a=(a_{\mathrm{x}},a_{\mathrm{p}}),b=(b_{\mathrm{x}},b_{\mathrm{p}})\in(\mathbb{N}\cup\\{0\\})^{\times
2n}.$ With shorthand notation $|F|_{a}:=|F|_{a,0}$ for the seminorms that only
measure the decay of $F$, we say a function is _rapidly decaying_ when
$|F|_{a}<\infty$ and is a _Schwartz function_ when $|F|_{a,b}<\infty$ for all
$a,b$. We denote the sets of rapidly decaying and Schwartz function by
${\mathcal{D}}({\mathbf{R}}^{2n})$ and ${\mathcal{S}}({\mathbf{R}}^{2n})$,
respectively. Our main result:

######  Theorem 1.1.

If $\rho$ is a positive semidefinite operator whose Wigner function
${\mathcal{W}_{\rho}}$ exists and is rapidly decaying, then
${\mathcal{W}_{\rho}}$ is a Schwartz function.

The assumed rapid decay of ${\mathcal{W}_{\rho}}$ implies
$\infty>\int{\mathcal{W}_{\rho}}(\alpha)\mathop{}\\!\mathrm{d}\alpha=\operatorname{tr}[\rho]$
and hence that $\rho$ is trace-class and so a quantum state. Thus the theorem
can be rephrased as the set relation
${\mathcal{V}}({\mathbf{R}}^{2n})\cap{\mathcal{D}}({\mathbf{R}}^{2n})\subset{\mathcal{S}}({\mathbf{R}}^{2n})$.

In this paper we prove Theorem 1.1 in two different ways. The first proof is a
bit more abstract, making use of the twisted convolution. The second proof is
a bit more direct, using only basic objects, but requiring more computation.
The second proof also results in an explicit bound on the Schwartz seminorms
$|{\mathcal{W}_{\rho}}|_{a,b}$ of a Wigner function in terms of only its decay
seminorms $|{\mathcal{W}_{\rho}}|_{a}$ (Theorem 3.9).

In the rest of this introduction, we informally sketch the direct (second)
proof of Theorem 1.1 in order to give the reader intuition, but we stop short
of completing the computation. In the Sec. 2, we recall some notation and
basic properties around quantum mechanics in phase space, which can be skipped
by experienced readers. In Sec. 3 we present our two proofs of our main result
and exhibit explicit bounds on the Schwartz seminorms of a Wigner function in
terms of its decay seminorms. In Section 4 we connect our results to the
notion of Schwartz operators in the sense of Keyl et al. [keyl2016schwartz],
and in particular prove the equivalence of a large set of equivalent decay and
regularity conditions for various representations of the quantum state. In
Sec. 5, we make some concluding remarks about the “overparameterization” of a
quantum state by the Wigner function.

###  1.1. Motivation

Why might one think the decay of a Wigner function constrains its derivatives?
Consider a pure state
$\rho=\mathinner{|{\psi}\rangle}\\!\\!\mathinner{\langle{\psi}|}$ with
$\mathinner{|{\psi}\rangle}\in L^{2}({\mathbf{R}}^{n})$. We can see from the
identity

$|\widehat{\psi}(p)|^{2}=\int{\mathcal{W}_{\rho}}(x,p)\mathop{}\\!\mathrm{d}x$
(1.1)

that rapid decay (in both $x$ and $p$) of the Wigner function implies decay
(in $p$) of the Fourier transform $\widehat{\psi}$ of the wavefunction. This
implies that the wavefunction $\psi$ is smooth: $|\psi|_{0,b}<\infty$ for all
$b\in(\mathbb{N}\cup\\{0\\})^{\times n}$. Unfortunately, a bit of trial and
error suggests that it is not easy to generalize (1.1) and obtain a bound on
the mixed Schwartz seminorms $|\psi|_{a,b}$ (that is, to show that all the
derivatives of $\psi$ are not merely bounded but are also rapidly decaying).

A better way to approach Theorem 1.1 avoids privileging either the position or
momentum variables by performing a wavepacket decomposition of the quantum
state $\rho$. Using Gaussian wavepackets (coherent states), Zurek argued
[zurek2001sub-planck] that if the Wigner function ${\mathcal{W}_{\rho}}$ of
any quantum state is largely confined to a phase space region of volume
$S\sim\ell_{\mathrm{x}}\times\ell_{\mathrm{p}}$, then the smallest structure
it will develop is on scales of volume $\Delta
s\sim(\hbar/\ell_{\mathrm{x}})\times(\hbar/\ell_{\mathrm{p}})\sim\hbar^{2}/S$.
This argument was further supported by numerical studies of “typical” states
generated by chaotic quantum dynamics [zurek2001sub-planck].

###  1.2. Sketch of direct proof

Consider a family of wavepackets $\chi_{\alpha}$ of the form

$\chi_{(\alpha_{\mathrm{x}},\alpha_{\mathrm{p}})}(x)=e^{i(x-\alpha_{x}/2)\cdot\alpha_{p}}\chi(x-\alpha_{x}),$
(1.2)

for fixed smooth envelope function $\chi$ concentrated near the origin. (For
example, $\chi$ can be chosen to be a Gaussian.) Given the spectral
decomposition of a quantum state

$\rho=\sum_{j}\lambda_{j}\mathinner{|{\psi_{j}}\rangle}\\!\\!\mathinner{\langle{\psi_{j}}|},$
(1.3)

we can use the decomposition
$\psi_{j}=(2\pi)^{-n}\int\mathinner{\langle{\chi_{\alpha}|\psi_{j}}\rangle}\chi_{\alpha}\mathop{}\\!\mathrm{d}\alpha$
for each eigenfunction as an integral over phase space, which is a standard
calculation proven in Lemma 2.9. We can then express $\rho$ as

$\rho=\frac{1}{(2\pi)^{n}}\sum_{j}\lambda_{j}\int\mathinner{|{\chi_{\alpha}}\rangle}\\!\\!\mathinner{\langle{\chi_{\beta}}|}\mathinner{\langle{\chi_{\alpha}|\psi_{j}}\rangle}\mathinner{\langle{\psi_{j}|\chi_{\beta}}\rangle}\mathop{}\\!\mathrm{d}\alpha\mathop{}\\!\mathrm{d}\beta,$
(1.4)

Applying the Wigner transform to both sides, this yields a decomposition

${\mathcal{W}_{\rho}}=\frac{1}{(2\pi)^{n}}\int{\mathcal{W}_{\mathinner{|{\chi_{\alpha}}\rangle}\\!\mathinner{\langle{\chi_{\beta}}|}}}\mathinner{\langle{\chi_{\alpha}|\rho|\chi_{\beta}}\rangle}\mathop{}\\!\mathrm{d}\alpha\mathop{}\\!\mathrm{d}\beta.$
(1.5)

in terms of the Wigner transform
${\mathcal{W}_{\mathinner{|{\chi_{\alpha}}\rangle}\\!\mathinner{\langle{\chi_{\beta}}|}}}$
of the “off-diagonal” operator
$\mathinner{|{\chi_{\alpha}}\rangle}\\!\\!\mathinner{\langle{\chi_{\beta}}|}$.
Although
${\mathcal{W}_{\mathinner{|{\chi_{\alpha}}\rangle}\\!\mathinner{\langle{\chi_{\beta}}|}}}$
is not a Wigner function (because
$\mathinner{|{\chi_{\alpha}}\rangle}\\!\\!\mathinner{\langle{\chi_{\beta}}|}$
is not positive semidefinite for $\alpha\neq\beta$), it is known
[zurek2001sub-planck, toscano2006sub-planck] to be localized near the phase
space point $(\alpha+\beta)/2$ and has an oscillation with frequency roughly
$|\alpha-\beta|$.

Since $\mathinner{\langle{\chi_{\alpha}|\rho|\chi_{\alpha}}\rangle}$ is just a
convolution of the Wigner function ${\mathcal{W}_{\rho}}$, the rapid decay of
${\mathcal{W}_{\rho}}$ implies the rapid decay of
$\mathinner{\langle{\chi_{\alpha}|\rho|\chi_{\alpha}}\rangle}$, and then in
turn one can show the rapid decay of
$\mathinner{\langle{\chi_{\alpha}|\rho|\chi_{\beta}}\rangle}$ using the
Cauchy-Schwartz inequality:

$\mathinner{\langle{\chi_{\alpha}|\rho|\chi_{\beta}}\rangle}^{2}\leq\mathinner{\langle{\chi_{\alpha}|\rho|\chi_{\alpha}}\rangle}\mathinner{\langle{\chi_{\beta}|\rho|\chi_{\beta}}\rangle},$
(1.6)

which holds because $\rho$ is positive semidefinite. The assumed decay and
smoothness properties of $\chi$ additionally give an estimate of the form

$|{\mathcal{W}_{\mathinner{|{\chi_{\alpha}}\rangle}\\!\mathinner{\langle{\chi_{\beta}}|}}}|_{a,b}\leq
C(a,b)(1+|\alpha|+|\beta|)^{D(a,b)}.$ (1.7)

When combined with decomposition (1.5) of ${\mathcal{W}_{\rho}}$, this is
enough to show that all the Schwartz seminorms $|{\mathcal{W}_{\rho}}|_{a,b}$
are finite.

Our other proof requires additional machinery but still rests heavily on
wavepacket decompositions of the quantum state and on the Cauchy-Schwartz
inequality (1.6).

##  2\. Preliminaries

This section establishes our notation and reviews standard features of phase-
space representations of quantum mechanics. (To keep this paper self-
contained, we provide proofs of the lemmas in this section in the Appendix.)
Throughout this paper, we take $\chi\in{\mathcal{S}}({\mathbf{R}}^{n})$ to be
a fixed Schwartz function that is normalized,
$\|\chi\|_{L^{2}({\mathbf{R}}^{n})}=\int|\chi(y)|^{2}\mathop{}\\!\mathrm{d}y=1$,
but otherwise arbitrary.111A standard choice is to specialize to a Gaussian
coherent state $\chi(y)=\exp(-x^{2}/2)/\sqrt{(2\pi)^{n}}$ (especially when
used as in Subsection 2.2 as the reference wavefunction with respect to which
Husimi function is defined). However, this specialization is not necessary and
one could instead take $\chi$ to be, e.g., a smooth and compactly supported
wavefunction.

Experienced readers may prefer to skip directly to Sec. 3 for the proof of our
main result and only refer back to this section as necessary.

###  2.1. Notation

In what follows, a _wavefunction_ of $n$ continuous quantum degrees of freedom
is represented by a member of $L^{2}({\mathbf{R}}^{n})$ and denoted by $\psi$,
$\phi$, or $\chi$. A _quantum state_ is the possibly mixed generalization,
represented by a positive semidefinite (and hence self-adjoint) trace-class
operator on $L^{2}({\mathbf{R}}^{n})$ and denoted by $\rho$ or $\eta$. Vectors
on phase space are $\alpha,\beta,\gamma,\xi\in{{\mathbf{R}}^{2n}}$ with
position and momentum components denoted by (for example)
$\alpha_{\mathrm{x}},\xi_{\mathrm{p}}\in{\mathbf{R}}^{n}$. Multi-indices are
$a,b,c,d\in(\mathbb{N}\cup\\{0\\})^{\times 2n}$ (or
($\mathbb{N}\cup\\{0\\})^{\times n}$ in Sec. 4) with
$\alpha^{b}=\alpha_{\mathrm{x}}^{b_{\mathrm{x}}}\alpha_{\mathrm{p}}^{b_{\mathrm{p}}}=\prod_{i=1}^{2n}\alpha_{i}^{b_{i}}$,
$|b|=|b_{\mathrm{x}}|+|b_{\mathrm{p}}|=\sum_{i=1}^{2n}b_{i}$,
$b!=b_{\mathrm{x}}!b_{\mathrm{p}}!=\prod_{i=1}^{2n}b_{i}!$, and
$\binom{a}{b}=a!/((a-b)!b!)$. We use $b\leq c$ to mean $b_{i}\leq c_{i}$ for
all $i=1,2,\ldots 2n$.

The symplectic form is
$\alpha\wedge\beta=\alpha\cdot\Omega\cdot\beta=\alpha_{\mathrm{x}}\cdot\beta_{\mathrm{p}}-\alpha_{\mathrm{p}}\cdot\beta_{\mathrm{x}}$,
with $\Omega=\left(\begin{smallmatrix}0&I\\\ -I&0\end{smallmatrix}\right)$ an
antisymmetric matrix on ${\mathbf{R}}^{2n}$, $I$ the identity matrix on
${\mathbf{R}}^{n}$, and “$\cdot$” the dot product on ${\mathbf{R}}^{n}$ and
${\mathbf{R}}^{2n}$. The position and momentum operators are
$X=(X_{1},\ldots,X_{n})$ and $P=(P_{1},\ldots,P_{n})$, which are combined into
the phase-space operator $R=(X,P)$. For a given quantum state $\rho$ and
reference wavefunction $\chi\in{\mathcal{S}}({\mathbf{R}}^{n})$, some
associated functions over phase space, doubled phase space, and doubled
configuration space are ${\mathcal{W}_{\rho}}$, ${\mathcal{Q}^{\chi}_{\rho}}$,
${\mathcal{M}^{\chi}_{\rho}}$, ${\mathcal{F}_{\rho}}$, and
${\mathcal{K}_{\rho}}$ (defined below). We use “$*$” to denote the
convolution, $(f*g)(\alpha)=\int
f(\alpha-\beta)g(\beta)\mathop{}\\!\mathrm{d}\beta$. Given a matrix form
$\Omega^{\prime}$ we also define the twisted convolution

$(f\circledast_{\Omega^{\prime}}g)(\alpha)=\int
e^{i\alpha\cdot\Omega^{\prime}\cdot\beta/2}f(\alpha-\beta)g(\beta)\mathop{}\\!\mathrm{d}\beta.$
(2.1)

For any wavefunction $\phi\in L^{2}({\mathbf{R}}^{n})$, we denote the linear
functional associated with it using bra notation,
$\mathinner{\langle{\phi}|}=(\psi\mapsto\int\bar{\phi}(x)\psi(x)\mathop{}\\!\mathrm{d}x)\in{\mathcal{S}}^{\prime}({\mathbf{R}}^{n})$,
and denote the scalar result with a bra-ket,
$\mathinner{\langle{\phi|\psi}\rangle}=\mathinner{\langle{\phi}|}(\psi)=\int\bar{\phi}(x)\psi(x)\mathop{}\\!\mathrm{d}x$.
More generally, with an operator $E$ we write
$\mathinner{\langle{\phi|E|\psi}\rangle}=\mathinner{\langle{\phi}|}(E\psi)=\mathinner{\langle{E^{\dagger}\phi}|}(\psi)$.
For any two wavefunction $\phi_{1},\phi_{2}\in L^{2}({\mathbf{R}}^{n})$, we
use $\mathinner{|{\phi_{1}}\rangle}\\!\\!\mathinner{\langle{\phi_{2}}|}$ for
the rank-1 operator
$\psi\mapsto\mathinner{\langle{\phi_{2}|\psi}\rangle}\phi_{1}$.

###  2.2. The displacement operator and phase-space functions

In this subsection we recall standard results about quantum mechanics in phase
space (see, e.g., Chapter 1 of Ref. [folland1989harmonic]).

######  Definition 2.1.

For $\xi\in{{\mathbf{R}}^{2n}}$, define the _(Weyl generator) displacement
operator_

$D_{\xi}:=e^{i\xi\wedge R}=e^{i(\xi_{\mathrm{x}}\cdot P-\xi_{\mathrm{p}}\cdot
X)}.$ (2.2)

The following lemma describes the action of $D_{\xi}$ on an arbitrary
wavefunction.

######  Lemma 2.2.

For any $\phi\in L^{2}({\mathbf{R}}^{n})$,

$D_{\xi}\phi(y)=e^{i(y-\xi_{\mathrm{x}}/2)\cdot\xi_{\mathrm{p}}}\phi(y-\xi_{\mathrm{x}}).$
(2.3)

It’s easy to check these basic properties:
$D_{\alpha}D_{\beta}=e^{i\beta\wedge\alpha/2}D_{\alpha+\beta}$ and
$D_{\alpha}^{\dagger}=D_{-\alpha}$.

Now we introduce the quasicharacteristic function, the Wigner function, and
the Kernel.

######  Definition 2.3.

For a given quantum state $\rho$, the _quasicharacteristic function_ is

${\mathcal{F}_{\rho}}(\xi):=\operatorname{tr}[\rho D_{\xi}].$ (2.4)

where the trace is well defined because $\rho$ is trace-class and $D_{\xi}$ is
a bounded operator on $L^{2}({\mathbf{R}}^{n})$. Because a quantum state
$\rho$ is necessarily compact, it has a spectral decomposition
[gohberg2000trace]

$\begin{split}(\rho\phi)(x)&=\sum_{i=1}^{\infty}\psi_{i}(x)\mathinner{\langle{\psi_{i}|\phi}\rangle}\end{split}$
(2.5)

with unnormalized eigenvectors $\psi_{i}\in L^{2}({\mathbf{R}}^{n})$ and
associated kernel ${\mathcal{K}_{\rho}}$ satisfying
$(\rho\phi)(x)=\int{\mathcal{K}_{\rho}}(x,y)\phi(x)\mathop{}\\!\mathrm{d}x$
and

${\mathcal{K}_{\rho}}(x,y)=\sum_{i=1}^{\infty}\psi_{i}(x)\bar{\psi}_{i}(y)$
(2.6)

almost everywhere. Finally, we define the _Wigner function_ of $\rho$ as

${\mathcal{W}_{\rho}}(\alpha):=\frac{1}{(2\pi)^{n}}\int
e^{i\alpha_{\mathrm{p}}\cdot
y}{\mathcal{K}_{\rho}}(\alpha_{\mathrm{x}}-y/2,\alpha_{\mathrm{x}}+y/2)\mathop{}\\!\mathrm{d}y,$
(2.7)

where ${\mathcal{W}_{\rho}}\in L^{2}({\mathbf{R}}^{2n})$ because it is a
Fourier transform of ${\mathcal{K}_{\rho}}\in L^{2}({\mathbf{R}}^{2n})$ in one
variable.

More generally, we call ${\mathcal{W}_{E}}(\alpha):=(2\pi)^{-n}\int
e^{i\alpha_{\mathrm{p}}\cdot
y}{\mathcal{K}_{E}}(\alpha_{\mathrm{x}}-y/2,\alpha_{\mathrm{x}}+y/2)\mathop{}\\!\mathrm{d}y$
and ${\mathcal{F}_{E}}(\xi):=\operatorname{tr}[ED_{\xi}]$ the _Wigner
transform_ and _quasicharacteristic transform_ of any kernel operator $E$,
which in particular exists for any rank-1 operator
$E=\mathinner{|{\phi}\rangle}\\!\\!\mathinner{\langle{\psi}|}$ since
${\mathcal{K}_{\mathinner{|{\phi}\rangle}\\!\mathinner{\langle{\psi}|}}}\in
L^{2}({\mathbf{R}}^{2n})$.

######  Lemma 2.4.

For any trace-class kernel operator $E$, the corresponding Wigner transform
and quasicharacteristic transform are symplectic Fourier duals:

${\mathcal{W}_{E}}(\alpha)=\frac{1}{(2\pi)^{2n}}\int
e^{-i\alpha\wedge\xi}{\mathcal{F}_{E}}(\xi)\mathop{}\\!\mathrm{d}\xi.$ (2.8)

The preceding expression is sometimes used as the definition of the Wigner
transform, and it is notable for manifestly respecting the symplectic
structure of phase space. The perhaps more traditional definition (2.7) relies
on the kernel, and hence privileges position over momentum, but has the
advantage of being more obviously well-defined.

######  Lemma 2.5.

The Wigner function ${\mathcal{W}_{\rho}}$ is a Schwartz function if and only
if the kernel ${\mathcal{K}_{\rho}}$ is a Schwartz function.

Roughly speaking, this is because ${\mathcal{W}_{\rho}}$ and
${\mathcal{K}_{\rho}}$ are Fourier transforms of each other in _one_ of their
two variables (after the linear change of variables
$(x,y)\to(\bar{x}=(x+y)/2,\Delta x=x-y)$).

######  Lemma 2.6.

The twisted convolution of a rapidly decaying function with a Schwartz
function is itself a Schwartz function.

The proof is essentially the same as for the similar statement with the normal
convolution.

######  Lemma 2.7.

For any two quantum states $\rho$ and $\eta$,

$\displaystyle\operatorname{tr}[\rho\eta]=(2\pi)^{n}\int{\mathcal{W}_{\rho}}(\alpha){\mathcal{W}_{\eta}}(\alpha)\mathop{}\\!\mathrm{d}\alpha.$
(2.9)

Now we introduce the Husimi function and the so-called matrix element; these
are most often defined with respect to a preferred Gaussian reference
wavefunction, but we will allow more generality (see, e.g., Ref.
[klauder2007generalized]).

######  Definition 2.8.

Fixing a reference wavefunction $\chi\in{\mathcal{S}}({\mathbf{R}}^{n})$ that
is normalized
($\|\chi\|_{L^{2}({\mathbf{R}}^{n})}=\int|\chi(y)|^{2}\mathop{}\\!\mathrm{d}y=1$),
and Schwartz-class but otherwise arbitrary, we define the _matrix element_

${\mathcal{M}^{\chi}_{\rho}}(\alpha,\beta):=\mathinner{\langle{\chi_{\alpha}|\rho|\chi_{\beta}}\rangle},$
(2.10)

and the _Husimi function_

${\mathcal{Q}^{\chi}_{\rho}}(\alpha):=\mathinner{\langle{\chi_{\alpha}|\rho|\chi_{\alpha}}\rangle}={\mathcal{M}^{\chi}_{\rho}}(\alpha,\alpha).$
(2.11)

using the shorthand
$\mathinner{|{\chi_{\alpha}}\rangle}:=D_{\alpha}\mathinner{|{\chi}\rangle}$.

######  Lemma 2.9.

For any trace-class operator $E$ and any $\chi\in L^{2}({\mathbf{R}}^{n})$
satisfying $\|\chi\|_{L^{2}({\mathbf{R}}^{n})}=1$,

$\displaystyle\operatorname{tr}[E]=\frac{1}{(2\pi)^{n}}\int\mathinner{\langle{\chi_{\alpha}|E|\chi_{\alpha}}\rangle}\mathop{}\\!\mathrm{d}\alpha$
(2.12)

In particular, for any $\phi,\psi\in L^{2}({\mathbf{R}}^{n})$

$\displaystyle\mathinner{\langle{\phi|\psi}\rangle}=\frac{1}{(2\pi)^{n}}\int\mathinner{\langle{\phi|\chi_{\alpha}}\rangle}\mathinner{\langle{\chi_{\alpha}|\psi}\rangle}\mathop{}\\!\mathrm{d}\alpha$
(2.13)

######  Lemma 2.10.

For any quantum state $\rho$ and reference wavefunction
$\chi\in{\mathcal{S}}({\mathbf{R}}^{n})$,

$\displaystyle{\mathcal{Q}^{\chi}_{\rho}}(\alpha)$
$\displaystyle=(2\pi)^{n}({\mathcal{W}_{\rho}}\ast{\mathcal{W}_{\chi}^{-}})(\alpha)=(2\pi)^{n}\int{\mathcal{W}_{\rho}}(\beta){\mathcal{W}_{\chi}}(\beta-\alpha)\mathop{}\\!\mathrm{d}\beta$
(2.14)

where ${{\mathcal{W}_{\chi}^{-}}}(\alpha):={\mathcal{W}_{\chi}}(-\alpha)$ is a
Schwartz function.

##  3\. Proof that rapidly decaying Wigner functions are Schwartz function

The first (more abstract) proof of our main result is given in subsection 3.1
below. The second (more direct) proof follows in subsection 3.2. These two
subsections are independent of each other and can be read in either order.

Both proofs will make crucial use of the Cauchy-Schwartz inequality in the
following form:

######  Lemma 3.1.

For any quantum state $\rho$, the Husimi function bounds the matrix element:

$|{\mathcal{M}^{\chi}_{\rho}}(\alpha,\beta)|^{2}\leq{\mathcal{Q}^{\chi}_{\rho}}(\alpha){\mathcal{Q}^{\chi}_{\rho}}(\beta)$
(3.1)

###### Proof.

We have

$|{\mathcal{M}^{\chi}_{\rho}}(\alpha,\beta)|^{2}=|\mathinner{\langle{\chi_{\alpha}|\rho|\chi_{\beta}}\rangle}|^{2}\leq\mathinner{\langle{\chi_{\alpha}|\rho|\chi_{\alpha}}\rangle}\mathinner{\langle{\chi_{\beta}|\rho|\chi_{\beta}}\rangle}={\mathcal{Q}^{\chi}_{\rho}}(\alpha){\mathcal{Q}^{\chi}_{\rho}}(\beta)$
(3.2)

where the inequality is just the Cauchy-Schwartz inequality applied to the
inner product
$\langle\phi_{1},\phi_{2}\rangle_{\rho}:=\mathinner{\langle{\phi_{1}|\rho|\phi_{2}}\rangle}$.
∎

######  Corollary 3.2.

If the Wigner function ${\mathcal{W}_{\rho}}$ of a quantum state $\rho$ is
rapidly decaying, then the Husimi function ${\mathcal{Q}^{\chi}_{\rho}}$ and
the matrix element ${\mathcal{M}^{\chi}_{\rho}}$ are also rapidly decaying.

###### Proof.

By Lemma 2.10, the Husimi function ${\mathcal{Q}^{\chi}_{\rho}}$ is a
convolution of the rapidly decaying ${\mathcal{W}_{\rho}}$ by the Schwartz
function ${\mathcal{W}_{\chi}^{-}}(\alpha)={\mathcal{W}_{\chi}}(-\alpha)$, so
${\mathcal{Q}^{\chi}_{\rho}}$ must also be rapidly decaying. We then get rapid
decay of ${\mathcal{M}^{\chi}_{\rho}}$ using Lemma 3.1. ∎

We now turn to the first strategy.

###  3.1. Abstract proof

Here is a sketch of our strategy: We obtain a reproducing formula expressing
${\mathcal{M}^{\chi}_{\rho}}$ as a twisted convolution of itself with a
Schwartz function constructed from $\chi$, showing that
${\mathcal{M}^{\chi}_{\rho}}$ must itself be a Schwartz function. Then we find
an integral expression for the Wigner function ${\mathcal{W}_{\rho}}$ in terms
of the matrix element ${\mathcal{M}^{\chi}_{\rho}}$, from which it follows
that ${\mathcal{W}_{\rho}}$ is a Schwartz function.

######  Lemma 3.3.

Let $\Omega^{\prime}=\left(\begin{smallmatrix}\Omega&0\\\
0&-\Omega\end{smallmatrix}\right)$ be a symplectic form on
${\mathbf{R}}^{4n}$. Then the matrix element ${\mathcal{M}^{\chi}_{\rho}}$
satisfies the following reproducing formula

${\mathcal{M}^{\chi}_{\rho}}=\frac{1}{(2\pi)^{2n}}({\mathcal{F}_{\chi}}\bar{\otimes}{\mathcal{F}_{\chi}})\circledast_{\Omega^{\prime}}{\mathcal{M}^{\chi}_{\rho}}$
(3.3)

where
${\mathcal{F}_{\chi}}=\operatorname{Tr}[\mathinner{|{\chi}\rangle}\\!\\!\mathinner{\langle{\chi}|}D_{\xi}]=\mathinner{\langle{\chi|D_{\xi}|\chi}\rangle}=\mathinner{\langle{\chi_{-\xi/2}|\chi_{\xi/2}}\rangle}$
is the quasicharacteristic function of the pure quantum state
$\mathinner{|{\chi}\rangle}\\!\\!\mathinner{\langle{\chi}|}$ and where
${\mathcal{F}_{\chi}}\bar{\otimes}{\mathcal{F}_{\chi}}:{\mathbf{R}}^{2n}\times{\mathbf{R}}^{2n}\to{\mathbf{C}}$
is defined by

$({\mathcal{F}_{\chi}}\bar{\otimes}{\mathcal{F}_{\chi}})(\xi,\omega):={\mathcal{F}_{\chi}}(\xi){\mathcal{F}_{\chi}}(-\omega).$
(3.4)

###### Proof.

We have:

$\begin{split}{\mathcal{M}^{\chi}_{\rho}}(\alpha,\beta)&=\mathinner{\langle{\chi_{\alpha}|\rho|\chi_{\beta}}\rangle}\\\
&=\frac{1}{(2\pi)^{n}}\int\mathinner{\langle{\chi_{\alpha}|\chi_{\alpha^{\prime}}}\rangle}\mathinner{\langle{\chi_{\alpha^{\prime}}|\rho|\chi_{\beta}}\rangle}\mathop{}\\!\mathrm{d}\alpha^{\prime}\\\
&=\frac{1}{(2\pi)^{2n}}\int\mathinner{\langle{\chi_{\alpha}|\chi_{\alpha^{\prime}}}\rangle}\mathinner{\langle{\chi_{\alpha^{\prime}}|\rho|\chi_{\beta^{\prime}}}\rangle}\mathinner{\langle{\chi_{\beta^{\prime}}|\chi_{\beta}}\rangle}\mathop{}\\!\mathrm{d}\alpha^{\prime}\mathop{}\\!\mathrm{d}\beta^{\prime}\\\
&=\frac{1}{(2\pi)^{2n}}\int
e^{i\alpha^{\prime}\wedge\alpha/2}{\mathcal{F}_{\chi}}(\alpha^{\prime}-\alpha){\mathcal{M}^{\chi}_{\rho}}(\alpha^{\prime},\beta^{\prime}){\mathcal{F}_{\chi}}(\beta-\beta^{\prime})e^{i\beta\wedge\beta^{\prime}/2}\mathop{}\\!\mathrm{d}\alpha^{\prime}\mathop{}\\!\mathrm{d}\beta^{\prime},\\\
&=\frac{1}{(2\pi)^{2n}}(({\mathcal{F}_{\chi}}\bar{\otimes}{\mathcal{F}_{\chi}})\circledast_{\Omega^{\prime}}{\mathcal{M}^{\chi}_{\rho}})(\alpha,\beta)\end{split}$
(3.5)

where to get the second line we use Eq. (2.13) of Lemma 2.9 for the inner
product of
$\mathinner{|{\chi_{\alpha}}\rangle},\rho\mathinner{|{\chi_{\beta}}\rangle}\in
L^{2}({\mathbf{R}}^{n})$ and to get the third line we use the lemma again for
the inner product of $\mathinner{|{\chi_{\beta}}\rangle}\in
L^{2}({\mathbf{R}}^{n}),\rho\mathinner{|{\chi_{\alpha}^{\prime}}\rangle}\in
L^{2}({\mathbf{R}}^{n})$. The final line is just the definition of the twisted
convolution with respect to the form $\Omega^{\prime}$. ∎

######  Corollary 3.4.

If the matrix element ${\mathcal{M}^{\chi}_{\rho}}$ is rapidly decaying, then
it is a Schwartz function.

###### Proof.

First note that $\chi$ and therefore
${\mathcal{K}_{\chi}}(x,y)=\chi(x)\bar{\chi}(y)$ are Schwartz functions. By
Lemma 2.5 this means ${\mathcal{W}_{\chi}}$ is a Schwartz function, so
therefore ${\mathcal{F}_{\chi}}$ is a Schwartz function by Lemma 2.4, meaning
that ${\mathcal{F}_{\chi}}\bar{\otimes}{\mathcal{F}_{\chi}}$ is a Schwartz
function. Then by Lemma 3.3, ${\mathcal{M}^{\chi}_{\rho}}$ is the twisted
convolution of a Schwartz function
(${\mathcal{F}_{\chi}}\bar{\otimes}{\mathcal{F}_{\chi}}$) against a rapidly
decaying function (${\mathcal{M}^{\chi}_{\rho}}$, by assumption), and is
therefore also Schwartz-class by Lemma 2.6. ∎

We now deploy Lemma 2.4 to recover the Wigner function ${\mathcal{W}_{\rho}}$
from the matrix element ${\mathcal{M}^{\chi}_{\rho}}$.

######  Lemma 3.5.

For any quantum state $\rho$,

${\mathcal{W}_{\rho}}(\alpha)=\frac{1}{(2\pi)^{3n}}\int
e^{-i(\alpha-\beta/2)\wedge\xi}{\mathcal{M}^{\chi}_{\rho}}(\beta-\xi/2,\beta+\xi/2)\mathop{}\\!\mathrm{d}\xi\mathop{}\\!\mathrm{d}\beta.$
(3.6)

###### Proof.

We have:

$\begin{split}{\mathcal{W}_{\rho}}(\alpha)&=\frac{1}{(2\pi)^{2n}}\int
e^{-i\alpha\wedge\xi}\operatorname{tr}[D_{\xi/2}\rho
D_{\xi/2}]\mathop{}\\!\mathrm{d}\xi\\\ &=\frac{1}{(2\pi)^{3n}}\int
e^{-i\alpha\wedge\xi}\mathinner{\langle{\chi_{\beta}|D_{\xi/2}\rho
D_{\xi/2}|\chi_{\beta}}\rangle}\mathop{}\\!\mathrm{d}\xi\mathop{}\\!\mathrm{d}\beta\\\
&=\frac{1}{(2\pi)^{3n}}\int
e^{-i\alpha\wedge\xi}\mathinner{\langle{\chi|D_{-\beta}D_{\xi/2}\rho
D_{\xi/2}D_{\beta}|\chi}\rangle}\mathop{}\\!\mathrm{d}\xi\mathop{}\\!\mathrm{d}\beta\\\
&=\frac{1}{(2\pi)^{3n}}\int
e^{-i(\alpha-\beta/2)\wedge\xi}\mathinner{\langle{\chi|D_{\xi/2-\beta}\rho
D_{\xi/2+\beta}|\chi}\rangle}\mathop{}\\!\mathrm{d}\xi\mathop{}\\!\mathrm{d}\beta\\\
&=\frac{1}{(2\pi)^{3n}}\int
e^{-i(\alpha-\beta/2)\wedge\xi}{\mathcal{M}^{\chi}_{\rho}}(\beta-\xi/2,\beta+\xi/2)\mathop{}\\!\mathrm{d}\xi\mathop{}\\!\mathrm{d}\beta,\end{split}$
(3.7)

where for the first line we use Lemma 2.4, for the second line we use the
trace formula from Lemma 2.9, and for the fourth line we use the composition
identity $D_{\alpha}D_{\beta}=e^{i\beta\wedge\alpha/2}D_{\alpha+\beta}$ for
the displacement operator. ∎

######  Corollary 3.6.

For any quantum state $\rho$, if the matrix element
${\mathcal{M}^{\chi}_{\rho}}$ is a Schwartz function, then
${\mathcal{W}_{\rho}}$ is a Schwartz function.

###### Proof.

Lemma 3.5 shows that ${\mathcal{W}_{\rho}}$ can be obtained from
${\mathcal{M}^{\chi}_{\rho}}$ by (a) multiplying by the phase function
$e^{i\beta\wedge\xi/2}$ (quadratic in the variables $\beta$ and $\xi$), (b)
applying a symplectic Fourier transform (exchanging the variable $\xi$ for the
variable $\alpha$), and then (c) integrating over the variable $\beta$. All
three of these operations preserve Schwartz-class functions, so
${\mathcal{W}_{\rho}}$ is a Schwartz function. ∎

With all the hard work done, our main result follows easily.

######  Theorem 1.1.

If $\rho$ is a positive semidefinite operator whose Wigner function
${\mathcal{W}_{\rho}}$ exists and is rapidly decaying, then
${\mathcal{W}_{\rho}}$ is a Schwartz function.

###### Proof.

The rapid decay of ${\mathcal{W}_{\rho}}$ means
$\infty>\int{\mathcal{W}_{\rho}}(\alpha)\mathop{}\\!\mathrm{d}\alpha=\operatorname{tr}[\rho]$,
so $\rho$ is trace-class and hence a quantum state. We then conclude that
${\mathcal{M}^{\chi}_{\rho}}$ is rapidly decaying by Corollary 3.2, so
${\mathcal{M}^{\chi}_{\rho}}$ is a Schwartz function by Corollary 3.4.
Therefore, ${\mathcal{W}_{\rho}}$ is a Schwartz function by Corollary 3.6. ∎

This proof is constructive and so can in principle be used to derive effective
bounds for the Schwartz seminorms for ${\mathcal{W}_{\rho}}$ in terms of only
the decay seminorms. However, computing the bounds through this method would
be very laborious, so instead we use a more direct method in the next
subsection.

###  3.2. Direct proof

The strategy rests on showing that the Schwartz seminorms of
${\mathcal{W}_{\mathinner{|{\chi_{\alpha}}\rangle}\\!\mathinner{\langle{\chi_{\beta}}|}}}$
depend only polynomially on $\alpha$ and $\beta$. In this section, we will use
for convenience the Schwartz type norms

$\|F\|_{a,b}:=\sum_{a^{\prime}\leq a}\sum_{b^{\prime}\leq
b}|F|_{a^{\prime},b^{\prime}}$ (3.8)

and the corresponding shorthand $\|F\|_{a}=\|F\|_{a,0}=\sum_{a^{\prime}\leq
a}|F|_{a^{\prime},0}$.

######  Lemma 3.7.

For any quantum state $\rho$ and any reference wavefunction
$\chi\in{\mathcal{S}}({\mathbf{R}}^{n})$,

${\mathcal{W}_{\rho}}(\gamma)=\frac{1}{(2\pi)^{2n}}\int{\mathcal{W}_{\mathinner{|{\chi_{\alpha}}\rangle}\\!\mathinner{\langle{\chi_{\beta}}|}}}(\gamma){\mathcal{M}^{\chi}_{\rho}}(\alpha,\beta)\mathop{}\\!\mathrm{d}\alpha\mathop{}\\!\mathrm{d}\beta.$
(3.9)

###### Proof.

Given the spectral decomposition
$\rho=\sum_{j}\lambda_{j}\mathinner{|{\psi_{j}}\rangle}\\!\\!\mathinner{\langle{\psi_{j}}|}$
we use Lemma 2.4 to get

$\begin{split}{\mathcal{W}_{\rho}}(\gamma)&=\frac{1}{(2\pi)^{2n}}\int
e^{-i\gamma\wedge\xi}\operatorname{tr}[\rho
D_{\xi}]\mathop{}\\!\mathrm{d}\xi\\\ &=\frac{1}{(2\pi)^{4n}}\int
e^{-i\gamma\wedge\xi}\mathinner{\langle{\chi_{\beta}|D_{\xi}|\chi_{\alpha}}\rangle}\mathinner{\langle{\chi_{\alpha}|\rho|\chi_{\beta}}\rangle}\mathop{}\\!\mathrm{d}\alpha\mathop{}\\!\mathrm{d}\beta\mathop{}\\!\mathrm{d}\xi\\\
&=\frac{1}{(2\pi)^{4n}}\int
e^{-i\gamma\wedge\xi}{\mathcal{F}_{\mathinner{|{\chi_{\alpha}}\rangle}\\!\mathinner{\langle{\chi_{\beta}}|}}}(\xi){\mathcal{M}^{\chi}_{\rho}}(\alpha,\beta)\mathop{}\\!\mathrm{d}\alpha\mathop{}\\!\mathrm{d}\beta\mathop{}\\!\mathrm{d}\xi\end{split}$
(3.10)

where to get the second line we apply Lemma 2.9 to the trace-class operator
$\rho D_{\xi}$ twice. Using Lemma 2.4 yields (3.9). ∎

######  Lemma 3.8.

The Schwartz seminorms of the Wigner transform of the off-diagonal operator
$\mathinner{|{\chi_{\alpha}}\rangle}\\!\\!\mathinner{\langle{\chi_{\beta}}|}$
obey

$\displaystyle|{\mathcal{W}_{\mathinner{|{\chi_{\alpha}}\rangle}\\!\mathinner{\langle{\chi_{\beta}}|}}}|_{a,b}$
$\displaystyle\leq\sum_{c\leq b}\sum_{d\leq a}\sum_{e\leq c}\sum_{f\leq
d}\binom{b}{c}\binom{a}{d}\binom{c}{e}\binom{d}{f}2^{-|d|}|{\mathcal{W}_{\chi}}|_{a-d,b-c}|\alpha^{\hat{e}+f}\beta^{\hat{c}-\hat{e}+d-f}|$
(3.11) $\displaystyle\leq
4^{|a|+|b|}(1+|\alpha|+|\beta|)^{|a|+|b|}\|{\mathcal{W}_{\chi}}\|_{a,b},$
(3.12)

where we use a hat to swap the position and momentum components of a multi-
index:
$\hat{a}=\widehat{(a_{\mathrm{x}},a_{\mathrm{p}})}:=(a_{\mathrm{p}},a_{\mathrm{x}})$.

###### Proof.

First note that

$\begin{split}{\mathcal{F}_{\mathinner{|{\chi_{\alpha}}\rangle}\\!\mathinner{\langle{\chi_{\beta}}|}}}(\xi)&=\operatorname{tr}[\mathinner{|{\chi_{\alpha}}\rangle}\\!\\!\mathinner{\langle{\chi_{\beta}}|}D_{\xi}]=\mathinner{\langle{\chi|D_{-\beta}D_{\xi}D_{\alpha}|\chi}\rangle}\\\
&=e^{i(\alpha+\beta)\wedge\xi/2+i\beta\wedge\alpha/2}\mathinner{\langle{\chi|D_{\xi+\alpha-\beta}|\chi}\rangle}=e^{i\bar{\alpha}\wedge\xi+i\bar{\alpha}\wedge\Delta\alpha/2}{\mathcal{F}_{\chi}}(\xi+\Delta\alpha)\end{split}$
(3.13)

where in the last line we introduced the shorthand
$\bar{\alpha}=(\alpha+\beta)/2$ and $\Delta\alpha=\alpha-\beta$. Then
${\mathcal{W}_{\mathinner{|{\chi_{\alpha}}\rangle}\\!\mathinner{\langle{\chi_{\beta}}|}}}$
is related to ${\mathcal{W}_{\chi}}$ by

$\begin{split}{\mathcal{W}_{\mathinner{|{\chi_{\alpha}}\rangle}\\!\mathinner{\langle{\chi_{\beta}}|}}}(\gamma)&=\int
e^{-i\gamma\wedge\xi}{\mathcal{F}_{\mathinner{|{\chi_{\alpha}}\rangle}\\!\mathinner{\langle{\chi_{\beta}}|}}}(\xi)\mathop{}\\!\mathrm{d}\xi\\\
&=\int
e^{i(\bar{\alpha}-\gamma)\wedge\xi+i\bar{\alpha}\wedge\Delta\alpha/2}{\mathcal{F}_{\chi}}(\xi+\Delta\alpha)\mathop{}\\!\mathrm{d}\xi\\\
&=\int
e^{i(\bar{\alpha}-\gamma)\wedge(\xi-\Delta\alpha)+i\bar{\alpha}\wedge\Delta\alpha/2}{\mathcal{F}_{\chi}}(\xi)\mathop{}\\!\mathrm{d}\xi\\\
&=e^{i(\gamma-\bar{\alpha}/2)\wedge\Delta\alpha}{\mathcal{W}_{\chi}}(\gamma-\bar{\alpha}),\end{split}$
(3.14)

so

$\begin{split}\gamma^{a}\partial_{\gamma}^{b}{\mathcal{W}_{\mathinner{|{\chi_{\alpha}}\rangle}\\!\mathinner{\langle{\chi_{\beta}}|}}}(\gamma)&=\gamma^{a}\partial_{\gamma}^{b}e^{i(\gamma-\bar{\alpha}/2)\wedge\Delta\alpha}{\mathcal{W}_{\chi}}(\gamma-\bar{\alpha})\\\
&=\sum_{c\leq
b}\binom{b}{c}e^{i(\gamma-\bar{\alpha}/2)\wedge\Delta\alpha}(i\Omega\cdot\Delta\alpha)^{c}((\gamma-\bar{\alpha})+\bar{\alpha})^{a}\partial_{\gamma}^{b-c}{\mathcal{W}_{\chi}}(\gamma-\bar{\alpha})\end{split}$
(3.15)

where in the second line we used

$\begin{split}\partial_{\gamma}^{b}e^{i\gamma\wedge\xi}=\partial_{\gamma_{\mathrm{x}}}^{b_{\mathrm{x}}}\partial_{\gamma_{\mathrm{p}}}^{b_{\mathrm{p}}}e^{i\gamma_{\mathrm{x}}\cdot\xi_{\mathrm{p}}-i\gamma_{\mathrm{p}}\cdot\xi_{\mathrm{x}}}=(i\xi_{\mathrm{p}})^{b_{\mathrm{x}}}(-i\xi_{\mathrm{x}})^{b_{\mathrm{p}}}e^{i\gamma_{\mathrm{x}}\cdot\xi_{\mathrm{p}}-i\gamma_{\mathrm{p}}\cdot\xi_{\mathrm{x}}}=(i\Omega\cdot\xi)^{b}e^{i\gamma\wedge\xi}\end{split}$
(3.16)

Then, using
$|(i\Omega\cdot\xi)^{b}|=|\xi_{\mathrm{p}}^{b_{\mathrm{x}}}\xi_{\mathrm{x}}^{b_{\mathrm{p}}}|=|\xi^{\hat{b}}|$,
it follows that

$\begin{split}|{\mathcal{W}_{\mathinner{|{\chi_{\alpha}}\rangle}\\!\mathinner{\langle{\chi_{\beta}}|}}}|_{a,b}&\leq\sum_{c\leq
b}\sum_{d\leq
a}\binom{b}{c}\binom{a}{d}|{\mathcal{W}_{\chi}}|_{a-d,b-c}|\Delta\alpha^{\hat{c}}\bar{\alpha}^{d}|\\\
&\leq\sum_{c\leq b}\sum_{d\leq a}\sum_{e\leq c}\sum_{f\leq
d}\binom{b}{c}\binom{a}{d}\binom{c}{e}\binom{d}{f}2^{-|d|}|{\mathcal{W}_{\chi}}|_{a-d,b-c}|\alpha^{\hat{e}+f}\beta^{\hat{c}-\hat{e}+d-f}|\\\
&\leq
4^{|a|+|b|}(1+|\alpha|+|\beta|)^{|a|+|b|}\|{\mathcal{W}_{\chi}}\|_{a,b}.\end{split}$
(3.17)

To get the third line, we bound the terms in the sum on the second line using
$2^{-|d|}\leq 1$,
$|{\mathcal{W}_{\chi}}|_{a-d,b-c}\leq\|{\mathcal{W}_{\chi}}\|_{a,b}$, and
$|\alpha^{\hat{e}+f}\beta^{\hat{c}-\hat{e}+d-f}|\leq(|1+|\alpha|+|\beta|)^{|a|+|b|}$
and then sum the binomial coefficients. ∎

######  Theorem 3.9.

For any quantum state $\rho$ and reference state
$\chi\in{\mathcal{S}}({\mathbf{R}}^{n})$, the following inequality holds:

$\displaystyle|{\mathcal{W}_{\rho}}|_{a,b}$
$\displaystyle\leq(2\pi)^{5n}2^{4(|a|+|b|+n)}\|{\mathcal{W}_{\chi}}\|_{a,b}\|{\mathcal{W}_{\chi}}\|_{2(a+\hat{b})+6}\|{\mathcal{W}_{\rho}}\|_{2(a+\hat{b})+4}.$
(3.18)

Note that the right-hand side contains only the decay norms of
${\mathcal{W}_{\rho}}$, and the left-hand side contains an arbitrary Schwartz
seminorm, so this implies Theorem 1.1. We also observe that, on the right-hand
side, the position and momentum indices are flipped in the derivative multi-
index
$\hat{b}=\widehat{(b_{\mathrm{x}},b_{\mathrm{p}})}=(b_{\mathrm{p}},b_{\mathrm{x}})$
when it contributes to a coordinate power (rather than a derivative power).

###### Proof.

We start with Eq. (3.9) of Lemma 3.7 and apply Lemma 3.1 followed by Eq.
(3.11) from Lemma 3.8 to get

$\begin{split}|\gamma^{a}\partial_{\gamma}^{a}{\mathcal{W}_{\rho}}(\gamma)|&\leq\frac{1}{(2\pi)^{2n}}\int|\gamma^{a}\partial_{\gamma}^{a}{\mathcal{W}_{\mathinner{|{\chi_{\alpha}}\rangle}\\!\mathinner{\langle{\chi_{\beta}}|}}}(\gamma)||{\mathcal{Q}^{\chi}_{\rho}}(\alpha)|^{1/2}|{\mathcal{Q}^{\chi}_{\rho}}(\beta)|^{1/2}\mathop{}\\!\mathrm{d}\alpha\mathop{}\\!\mathrm{d}\beta\\\
&\leq\frac{1}{(2\pi)^{2n}}\sum_{c\leq b}\sum_{d\leq a}\sum_{e\leq
c}\sum_{f\leq
d}\binom{b}{c}\binom{a}{d}\binom{c}{e}\binom{d}{f}2^{-|d|}|{\mathcal{W}_{\chi}}|_{d-a,b-c}\\\
&\qquad\times\int\frac{1}{\prod_{j=1}^{2n}(1+|\alpha_{j}|^{2})}|\alpha^{\hat{e}+f}|\prod_{j=1}^{2n}(1+|\alpha_{j}|^{2})|{\mathcal{Q}^{\chi}_{\rho}}(\alpha)|^{1/2}\mathop{}\\!\mathrm{d}\alpha\\\
&\qquad\times\int\frac{1}{\prod_{j=1}^{2n}(1+|\beta_{j}|^{2})}|\beta^{\hat{c}-\hat{e}+d-f}|\prod_{j=1}^{2n}(1+|\beta_{j}|^{2})|{\mathcal{Q}^{\chi}_{\rho}}(\beta)|^{1/2}\mathop{}\\!\mathrm{d}\beta.\end{split}$
(3.19)

To compute the integral over $\alpha$, we use

$\begin{split}|\alpha^{\hat{e}+f}|\prod_{j=1}^{2n}(1+|\alpha_{j}^{2}|)|{\mathcal{Q}^{\chi}_{\rho}}(\alpha)|^{1/2}&\leq
2^{2n}\|{\mathcal{Q}^{\chi}_{\rho}}\|_{2\hat{e}+2f+4}^{1/2}\leq
2^{2n}\|{\mathcal{Q}^{\chi}_{\rho}}\|_{2(a+\hat{b})+4}^{1/2},\end{split}$
(3.20)

and likewise for the integral over $\beta$. Integrating over $\alpha$ with
$\int(1+|\alpha_{j}|^{2})^{-1}\mathop{}\\!\mathrm{d}\alpha_{j}=\pi$ and
likewise for $\beta$, we obtain

$\displaystyle|{\mathcal{W}_{\rho}}|_{a,b}$
$\displaystyle\leq\left(\frac{\pi}{2}\right)^{2n}\|{\mathcal{Q}^{\chi}_{\rho}}\|_{2(a+\hat{b})+4}\sum_{c\leq
b}\sum_{d\leq a}\sum_{e\leq c}\sum_{f\leq
d}\binom{b}{c}\binom{a}{d}\binom{c}{e}\binom{d}{f}2^{-|d|}|{\mathcal{W}_{\chi}}|_{d-a,b-c}$
$\displaystyle\leq\left(\frac{\pi}{2}\right)^{2n}2^{2(|a|+|b|)}\|{\mathcal{W}_{\chi}}\|_{a,b}\|{\mathcal{Q}^{\chi}_{\rho}}\|_{2(a+\hat{b})+4}$
(3.21)

using $2^{-|d|}\leq 1$ and
$|{\mathcal{W}_{\chi}}|_{d-a,b-c}\leq\|{\mathcal{W}_{\chi}}\|_{a,b}$. We then
bound the decay seminorms of ${\mathcal{Q}^{\chi}_{\rho}}$ using Lemma 2.10:

$\begin{split}\alpha^{a}{\mathcal{Q}^{\chi}_{\rho}}(\alpha)=(2\pi)^{n}\int{\mathcal{W}_{\rho}}(\alpha-\beta)((\alpha-\beta)+\beta)^{a}{\mathcal{W}_{\chi}^{-}}(\beta)\mathop{}\\!\mathrm{d}\beta\end{split}$
(3.22)

so

$\begin{split}|\alpha^{a}{\mathcal{Q}^{\chi}_{\rho}}(\alpha)|&\leq(2\pi)^{n}\sum_{b\leq
a}\binom{a}{b}\int\frac{1}{\prod_{j=1}^{2n}(1+|\beta_{j}|^{2})}|(\alpha-\beta)^{b}{\mathcal{W}_{\rho}}(\alpha-\beta)|\\\
&\qquad\qquad\qquad\qquad\qquad\times|\beta^{a-b}|\prod_{j=1}^{2m}(1+|\beta_{j}|^{2})|{\mathcal{W}_{\chi}^{-}}(\beta)|\mathop{}\\!\mathrm{d}\beta\\\
&\leq(2\pi)^{n}2^{|a|}\pi^{2n}\|{\mathcal{W}_{\rho}}\|_{a}\|{\mathcal{W}_{\chi}}\|_{a+2}\end{split}$
(3.23)

using $\|{\mathcal{W}_{\chi}}\|_{a+2}=\|{\mathcal{W}_{\chi}^{-}}\|_{a+2}$.
Summing over the seminorms in the norm,

$\displaystyle\|{\mathcal{Q}^{\chi}_{\rho}}\|_{a}=\sum_{b\leq
a}|{\mathcal{Q}^{\chi}_{\rho}}|_{b}$ $\displaystyle\leq
2^{n}\pi^{3n}\sum_{b\leq
a}2^{|b|}\|{\mathcal{W}_{\rho}}\|_{b}\|{\mathcal{W}_{\chi}}\|_{b+2}$ (3.24)
$\displaystyle\leq
2^{3n+|a|}\pi^{3n}\|{\mathcal{W}_{\rho}}\|_{a}\|{\mathcal{W}_{\chi}}\|_{a+2},$
(3.25)

and then inserting into (3.21) yields (3.18). ∎

##  4\. Schwartz states

In this section, we extend our main result by establishing an equivalence
between the Schwartz-class and rapid-decay properties of many different
representations of the quantum state. First, we will give a notion of Schwartz
class to a set of orthogonal wavefunction $\\{\psi_{i}\\}$ appearing in a
spectral decomposition
${\mathcal{K}_{\rho}}(x,y)=\sum_{i}\psi_{i}(x)\bar{\psi}_{i}(y)$. Then, we
recall the definition of a Schwartz operator as identified by Keyl et al.
[keyl2016schwartz]. Finally, we will prove our large equivalence theorem.

To guarantee that ${\mathcal{K}_{\rho}}$ is a Schwartz function, it is, of
course, not sufficient for each $\psi_{i}$ to be a Schwartz function. For
example, if each $\psi_{i}$ is a Gaussian wavepacket with increasing variance
$\sigma^{2}_{i}\propto i$ centered on the origin, then the overall variance
$\langle X^{2}\rangle=\operatorname{Tr}[X^{2}\rho]=\int
x^{2}{\mathcal{K}_{\rho}}(x,x)\mathop{}\\!\mathrm{d}x=\sum_{i}\int
x^{2}|\psi_{i}(x)|^{2}\mathop{}\\!\mathrm{d}x=\sum_{i}p_{i}\sigma_{i}^{2}$ can
diverge if the norms $p_{i}=\mathinner{\langle{\psi_{i}|\psi_{i}}\rangle}$ are
decreasing slowly, so that ${\mathcal{K}_{\rho}}$ is not a Schwartz function
even though each $\psi_{i}$ is. Instead, we consider the following
definition.222Note that the multi-indices $a,b,c,d$ in this section are
$n$-dimensional rather than $2n$-dimensional because the wavefunction $\psi$
and the kernel ${\mathcal{K}_{\rho}}$ take arguments in position space rather
than phase space.

######  Definition 4.1.

A set $\\{\psi_{i}\\}$ of unnormalized wavefunctions ($\psi_{i}\in
L^{2}({\mathbf{R}}^{n})$ for all $i$) is _jointly Schwartz_ when
$|\\{\psi_{i}\\}|_{a,b}<\infty$ for all
$a,b\in({\mathbb{N}}\cup\\{0\\})^{\times n}$, where the Schwartz seminorms of
such a set are defined by

$|\\{\psi_{i}\\}|_{a,b}^{2}:=\sup_{x}\sum_{i}\left|x^{a}\partial_{x}^{b}\psi_{i}(x)\right|^{2}.$
(4.1)

This is denoted
$\\{\psi_{i}\\}\in{\mathcal{S}}_{\mathrm{j}}({\mathbf{R}}^{n})$.

Note that this seminorm (4.1) is not simply a function of the seminorm
$|{\mathcal{K}_{\rho}}|_{(a,c),(b,d)}$ of the kernel, nor is it simply a
function of the individual seminorms
$|\psi_{i}|_{a,b}:=\sup_{x}|x^{a}\partial_{x}^{b}\psi_{i}(x)|$ of the
wavefunctions $\psi_{i}$. However, one can check that when the set
$\\{\psi_{i}\\}$ is finite, the jointly Schwartz property is equivalent to the
condition that all the wavefunctions are Schwartz functions individually,
$\\{\psi_{i}\\}\subset{\mathcal{S}}({\mathbf{R}}^{n})$.

######  Lemma 4.2.

For any quantum state $\rho$, the set of unnormalized wavefunctions
$\\{\psi_{i}\\}$ of the spectral decomposition is jointly Schwartz if and only
if the kernel ${\mathcal{K}_{\rho}}$ is a Schwartz function.

###### Proof.

First assume that
$\\{\psi_{i}\\}\in{\mathcal{S}}_{\mathrm{j}}({\mathbf{R}}^{n})$. Then

$\begin{split}|{\mathcal{K}_{\rho}}|_{(a,c),(b,d)}&=\sup_{x,y}\left|x^{a}y^{c}\partial_{x}^{b}\partial_{y}^{d}K(x,y)\right|\\\
&=\sup_{x,y}\left|\sum_{i}\left(x^{a}\partial_{x}^{b}\psi_{i}(x)\right)\left(y^{c}\partial_{y}^{d}\bar{\psi}_{i}(y)\right)\right|\\\
&\leq\sup_{x,y}\left(\sum_{i}\left|x^{a}\partial_{x}^{b}\psi_{i}(x)\right|^{2}\right)^{1/2}\left(\sum_{i}\left|y^{c}\partial_{y}^{d}\psi_{i}(y)\right|^{2}\right)^{1/2}\\\
&=\left(\sup_{x}\sum_{i}\left|x^{a}\partial_{x}^{b}\psi_{i}(x)\right|^{2}\right)^{1/2}\left(\sup_{y}\sum_{i}\left|y^{c}\partial_{y}^{d}\psi_{i}(y)\right|^{2}\right)^{1/2}\\\
&=|\\{\psi_{i}\\}|_{a,b}|\\{\psi_{i}\\}|_{c,d},\end{split}$ (4.2)

where the third line is the Cauchy-Schwartz inequality. Therefore,
$\\{\psi_{i}\\}\in{\mathcal{S}}_{\mathrm{j}}({\mathbf{R}}^{n})\Rightarrow{\mathcal{K}_{\rho}}\in{\mathcal{S}}({\mathbf{R}}^{2n})$.
To see the inverse, note that

$\begin{split}|{\mathcal{K}_{\rho}}|_{(a,a),(b,b)}&=\sup_{x,y}\left|x^{a}y^{a}\partial_{x}^{b}\partial_{y}^{b}K(x,y)\right|\\\
&\geq\sup_{x}\left|\left(x^{a}y^{a}\partial_{x}^{b}\partial_{y}^{b}K\right)(x,x)\right|\\\
&=\sup_{x}\sum_{i}|x^{a}\partial_{x}^{b}\psi_{i}|^{2}\\\ \end{split}$ (4.3)

where the inequality holds because ${\mathcal{K}_{\rho}}$ is a Schwartz
function. This quantity diverges by definition if
$\\{\psi_{i}\\}\notin{\mathcal{S}}_{\mathrm{j}}({\mathbf{R}}^{n})$, implying
${\mathcal{K}_{\rho}}\notin{\mathcal{S}}({\mathbf{R}}^{2n})$. ∎

The natural way to characterize the Schwartz class of quantum states was
suggested by Keyl et al. [keyl2016schwartz] as those quantum states $\rho$
with bounded expectation value for all symmetric polynomials in $X$ and $P$
(i.e., $\operatorname{tr}[X^{a}P^{b}\rho P^{b}X^{a}]<\infty$ for all
$a,b\in(\mathbb{N}\cup\\{0\\})^{\times n}$). More generally, for arbitrary
operators (i.e., not necessarily positive semidefinite, self-adjoint, or
trace-class), they define:

######  Definition 4.3.

An operator $E$ is a Schwartz operator, denoted
$E\in{\mathcal{S}}(L^{2}({\mathbf{R}}^{n}))$, when $|E|_{a,b,c,d}<\infty$ for
all $a,b,c,d\in({\mathbb{N}}\cup\\{0\\})^{\times n}$, where the Schwartz
operator seminorms are

$|E|_{a,b,c,d}:=\sup_{|\psi|,|\phi|=1}\left|\left\langle\psi\left|X^{a}P^{b}EP^{c}X^{d}\right|\phi\right\rangle\right|.$
(4.4)

Here, the supremum is taken over all normalized wavefunctions $\psi,\phi\in
L^{2}({\mathbf{R}}^{n})$.

Equipped with Definitions 4.1 and 4.3, we can state a theorem that subsumes
the results of Sec. 3.

######  Theorem 4.4.

For any Schwartz-class reference wavefunction
$\chi\in{\mathcal{S}}({\mathbf{R}}^{n})$ and for any quantum state (i.e.,
positive semidefinite trace-class operator on $L^{2}({\mathbf{R}}^{n})$)
$\rho$, with spectral decomposition $\\{\psi_{i}\\}$, quasicharacteristic
function ${\mathcal{F}_{\rho}}$, Wigner function ${\mathcal{W}_{\rho}}$,
kernel ${\mathcal{K}_{\rho}}$, Husimi function ${\mathcal{Q}^{\chi}_{\rho}}$,
and matrix element ${\mathcal{M}^{\chi}_{\rho}}$, the following conditions are
equivalent:

  * •

${\mathcal{W}_{\rho}}\in{\mathcal{S}}({\mathbf{R}}^{2n})$

  * •

${\mathcal{W}_{\rho}}\in{\mathcal{D}}({\mathbf{R}}^{2n})$

  * •

${\mathcal{Q}^{\chi}_{\rho}}\in{\mathcal{S}}({\mathbf{R}}^{2n})$

  * •

${\mathcal{Q}^{\chi}_{\rho}}\in{\mathcal{D}}({\mathbf{R}}^{2n})$

  * •

${\mathcal{M}^{\chi}_{\rho}}\in{\mathcal{S}}({\mathbf{R}}^{4d})$

  * •

${\mathcal{M}^{\chi}_{\rho}}\in{\mathcal{D}}({\mathbf{R}}^{4d})$

  * •

${\mathcal{F}_{\rho}}\in{\mathcal{S}}({\mathbf{R}}^{2n})$

  * •

${\mathcal{K}_{\rho}}\in{\mathcal{S}}({\mathbf{R}}^{2n})$

  * •

$\\{\psi_{i}\\}\in{\mathcal{S}}_{\mathrm{j}}({\mathbf{R}}^{n})$

  * •

$\rho\in{\mathcal{S}}(L^{2}({\mathbf{R}}^{n}))$

Furthermore, if the set $\\{\psi_{i}\\}$ is finite (e.g., if the state is
pure, $\rho=|\psi\rangle\\!\langle\psi|$), then the condition
$\\{\psi_{i}\\}\subset{\mathcal{S}}({\mathbf{R}}^{n})$ is also equivalent to
the above.

###### Proof.

We have:

${\mathcal{W}_{\rho}}\in{\mathcal{S}}({\mathbf{R}}^{2n})\Rightarrow{\mathcal{W}_{\rho}}\in{\mathcal{D}}({\mathbf{R}}^{2n})$
because
${\mathcal{S}}({\mathbf{R}}^{2n})\subset{\mathcal{D}}({\mathbf{R}}^{2n})$.

${\mathcal{W}_{\rho}}\in{\mathcal{D}}({\mathbf{R}}^{2n})\Rightarrow{\mathcal{Q}^{\chi}_{\rho}},{\mathcal{M}^{\chi}_{\rho}}\in{\mathcal{D}}({\mathbf{R}}^{2n})$
by Corollary 3.2.

${\mathcal{M}^{\chi}_{\rho}}\in{\mathcal{D}}({\mathbf{R}}^{2n})\Rightarrow{\mathcal{M}^{\chi}_{\rho}}\in{\mathcal{S}}({\mathbf{R}}^{2n})$
by Corollary 3.4.

${\mathcal{M}^{\chi}_{\rho}}\in{\mathcal{S}}({\mathbf{R}}^{2n})\Rightarrow{\mathcal{W}_{\rho}}\in{\mathcal{S}}({\mathbf{R}}^{2n})$
by Corollary 3.6.

${\mathcal{Q}^{\chi}_{\rho}}\in{\mathcal{S}}({\mathbf{R}}^{2n})\Rightarrow{\mathcal{Q}^{\chi}_{\rho}}\in{\mathcal{D}}({\mathbf{R}}^{2n})$
because
${\mathcal{S}}({\mathbf{R}}^{2n})\subset{\mathcal{D}}({\mathbf{R}}^{2n})$.

${\mathcal{M}^{\chi}_{\rho}}\in{\mathcal{S}}({\mathbf{R}}^{2n})\Rightarrow{\mathcal{M}^{\chi}_{\rho}}\in{\mathcal{D}}({\mathbf{R}}^{2n})$
because
${\mathcal{S}}({\mathbf{R}}^{2n})\subset{\mathcal{D}}({\mathbf{R}}^{2n})$.

${\mathcal{W}_{\rho}}\in{\mathcal{S}}({\mathbf{R}}^{2n})\Leftrightarrow{\mathcal{F}_{\rho}}\in{\mathcal{S}}({\mathbf{R}}^{2n})$
by Lemma 2.4.

${\mathcal{W}_{\rho}}\in{\mathcal{S}}({\mathbf{R}}^{2n})\Leftrightarrow{\mathcal{K}_{\rho}}\in{\mathcal{S}}({\mathbf{R}}^{2n})$
by Lemma 2.5.

${\mathcal{K}_{\rho}}\in{\mathcal{S}}({\mathbf{R}}^{2n})\Leftrightarrow\\{\psi_{i}\\}\in{\mathcal{S}}_{\mathrm{j}}({\mathbf{R}}^{2n})$
by Lemma 4.2.

$\rho\in{\mathcal{S}}(L^{2}({\mathbf{R}}^{n}))\Leftrightarrow{\mathcal{W}_{\rho}}\in{\mathcal{S}}({\mathbf{R}}^{2n})$
by Proposition 3.18 in Ref. [keyl2016schwartz]. ∎

We say a quantum state satisfying the above equivalent conditions is a
_Schwartz state_.

Note that ${\mathcal{K}_{\rho}}\in{\mathcal{D}}({\mathbf{R}}^{2n})$ is _not_
an equivalent condition, being strictly weaker than the other conditions
above.333For instance, the plateau wavefunction
$\psi(y)=\\{1\,\mathrm{if}\,0\leq y\leq 1,0\,\mathrm{otherwise}\\}$ is
compactly supported in position space but in momentum space decays to infinity
only as a polynomial. This is essentially because ${\mathcal{K}_{\rho}}$ is a
spatial representation, so momentum information is encoded only in its
derivatives, whereas ${\mathcal{W}_{\rho}}$, ${\mathcal{Q}^{\chi}_{\rho}}$,
and ${\mathcal{M}^{\chi}_{\rho}}$ are phase-space representations whose decay
constrains both space and momentum features. Similarly,
${\mathcal{F}_{\rho}}\in{\mathcal{D}}({\mathbf{R}}^{2n})$ is not an equivalent
condition because rapid decay of the derivatives of ${\mathcal{W}_{\rho}}$
does not assure that ${\mathcal{W}_{\rho}}$ has rapid decay.444Consider the
$n=1$ quantum state
$\rho=\sum_{k=0}^{\infty}|\psi_{k}\rangle\\!\langle\psi_{k}|$ with
$\psi_{k}(y)=\psi_{0}(y-z_{k})=(6/\pi^{2})k^{-2}\exp[-(y-z_{k})^{2}/2]/\sqrt{2\pi}$
with $z_{k}=k^{3}$. This is a mixture of Gaussians of equal variance, so the
derivatives are all rapidly decreasing and
${\mathcal{F}_{\rho}}\in{\mathcal{D}}({\mathbf{R}}^{2n})$, but the mean
$\operatorname{tr}[\rho X]$ diverges so ${\mathcal{W}_{\rho}}$ is not a
Schwartz function.

##  5\. Discussion

Although the Wigner formalism provides a complete representation of quantum
states and dynamics, it is often regarded as less fundamental. (It only really
becomes uniquely preferred in the classical limit, and under, e.g., certain
symmetry demands to distinguish it from other deformations of classical
mechanics; see for instance the introduction of Ref.
[degosson2016bornjordan].) One practical reason is that computations are often
more difficult using the Moyal product555Of course, the peculiar features of
the Moyal product are not just a matter of practicalities: Because the Moyal
bracket has phase-space derivatives of arbitrarily high order, the dynamics of
the Wigner function are non-local., the so-called “$\star$-genvalue
equations”, and so on [curtright2014concise]. Another reason is that the state
space is awkward to define.

Formally, the state space of valid Wigner functions can be delineated with the
quantum generalization [srinivas1975nonclassical, broecker1995mixed] of
Bochner’s theorem [bochner1933monotone]. (See also illuminating discussion and
further generalizations to some discrete spaces in Ref.
[dangniam2015quantum].) This definition is sufficiently opaque that most
physicists simply think of the allowed pure-state Wigner functions as the
image of the Wigner transform of the space of allowed quantum states,
$L^{2}({\mathbf{R}}^{n})$, if they think of it at all. In particular, many
simple (even positive-valued) functions on $L^{1}({\mathbf{R}}^{2n})$ are not
the Wigner functions of any quantum states.

In contrast, the $L^{2}({\mathbf{R}}^{n})$ (pure) state space of the
Schrödinger representation is relatively simple to understand, and the
“parameterization” of that space is natural in the sense that all possible
functions are allowed modulo only the single, easy-to-interpret constraint of
normalization. This becomes even clearer in the case of a finite-dimensional
quantum system, where there are no complications related to the continuum and
where any complex-valued function over configuration space suffices as a (not
necessarily normalized) state. Delineating the corresponding set of Wigner
functions for finite-dimensional systems is much more subtle
[dangniam2015quantum].

In this sense, the Wigner representation is “overparameterized”. One can think
of our Theorem 1.1 as better characterizing this overparameterization: the
regularity of the _interior_ of the Wigner function in terms of its
derivatives of any order is tightly controlled by the Wigner function’s decay
toward infinity, a feature that is obviously not shared by all normalized
functions over ${\mathbf{R}}^{2n}$. With the seminorm bound of Theorem 3.9,
one can also recover a version of the uncertainty principle. For example,
because the supremum of the gradient of ${\mathcal{W}_{\rho}}$ is bounded by
the decay seminorms of ${\mathcal{W}_{\rho}}$, it is impossible for
${\mathcal{W}_{\rho}}$ to be supported in a ball of too small a radius.

##  6\. Acknowledgements

CJR thanks Jukka Kiukas and Reinhard Werner for helpful discussion. FH was
supported by the Fannie and John Hertz Foundation Fellowship.

## Appendix A

Here we here recall the proofs of some standard results referenced in the main
body of this paper.

######  Lemma 2.2.

For any $\phi\in L^{2}({\mathbf{R}}^{n})$,

$D_{\xi}\phi(y)=e^{i(y-\xi_{\mathrm{x}}/2)\cdot\xi_{\mathrm{p}}}\phi(y-\xi_{\mathrm{x}}).$
(A.1)

###### Proof.

First, let us prove the statement for some Schwartz function
$\chi\in{\mathcal{S}}({\mathbf{R}}^{n})$. Consider the following differential
equation:

$\partial_{t}f_{t}=iR\wedge\xi f_{t}$ (A.2)

Solutions $f_{t}$ preserve the $L^{2}$ norm (because the operator on the right
is anti-Hermitian), so solutions are unique. Moreover, since $D_{t\xi}f_{0}$
is a solution to (A.2), any solution to (A.2) with initial condition
$f_{0}=\chi$ satisfies $f_{1}=D_{\xi}\chi$.

It remains to check that the function

$f_{t}(y)=e^{i(y-t\xi_{\mathrm{x}}/2)\cdot(t\xi_{\mathrm{p}})}\chi(y-t\xi_{\mathrm{x}})$
(A.3)

solves (A.2). We check this by a direct computation,

$\begin{split}\partial_{t}f_{t}(y)&=i((y-t\xi_{\mathrm{x}})\cdot\xi_{\mathrm{p}})f_{t}(y)-e^{i(y-t\xi_{\mathrm{x}}/2)\cdot(t\xi_{\mathrm{p}})}\xi_{\mathrm{x}}\cdot(\nabla\chi)(y-t\xi_{\mathrm{x}})\\\
&=i\xi_{\mathrm{p}}\cdot
yf_{t}(y)-it\xi_{\mathrm{x}}\cdot\xi_{\mathrm{p}}f_{t}(y)-e^{i(y-t\xi_{\mathrm{x}}/2)\cdot(t\xi_{\mathrm{p}})}\xi_{\mathrm{x}}\cdot(\nabla\chi)(y-t\xi_{\mathrm{x}})\\\
&=i\xi_{\mathrm{p}}\cdot Xf_{t}(y)-i\xi_{\mathrm{x}}\cdot
Pf_{t}(y).\end{split}$ (A.4)

Having proven the claim for $\chi\in{\mathcal{S}}({\mathbf{R}}^{n})$, we can
extend it to any $\phi\in L^{2}({\mathbf{R}}^{n})$ by using the density of the
Schwartz function in $L^{2}({\mathbf{R}}^{n})$, i.e., by approximating
$D_{\xi}\phi$ to accuracy $\epsilon$ with some choice of
$D_{\xi}\chi^{(\epsilon)}\in{\mathcal{S}}({\mathbf{R}}^{n})$ and taking
$\epsilon\to 0$. ∎

######  Lemma 2.4.

For any trace-class kernel operator $E$, the corresponding Wigner transform
and quasicharacteristic transform are symplectic Fourier duals:

${\mathcal{W}_{E}}(\alpha)=\frac{1}{(2\pi)^{2n}}\int
e^{-i\alpha\wedge\xi}{\mathcal{F}_{E}}(\xi)\mathop{}\\!\mathrm{d}\xi.$ (A.5)

###### Proof.

By linearity it suffices to check the case that $E$ is a rank-1 state of the
form $E=\mathinner{|{\psi}\rangle}\\!\\!\mathinner{\langle{\phi}|}$. Moreover,
by the continuity of the Wigner transform in $L^{2}({\mathbf{R}}^{n})\times
L^{2}({\mathbf{R}}^{n})$ and the density of Schwartz functions in
$L^{2}({\mathbf{R}}^{n})$, we may furthermore assume that
$\psi,\phi\in{\mathcal{S}}({\mathbf{R}}^{n})$. In this case,

$\begin{split}\operatorname{tr}[ED_{\xi}]&=\mathinner{\langle{\phi|D_{\xi}|\psi}\rangle}=\mathinner{\langle{\phi|D_{\xi/2}D_{\xi/2}|\psi}\rangle}=\langle
D_{-\xi/2}\phi,D_{\xi/2}\psi\rangle\\\ &=\int
e^{i(z+\xi_{\mathrm{x}}/4)\cdot\xi_{\mathrm{p}}/2}e^{i(z-\xi_{\mathrm{x}}/4)\cdot\xi_{\mathrm{p}}/2}\bar{\phi}(z+\xi_{\mathrm{x}}/2)\psi(z-\xi_{\mathrm{x}}/2)\mathop{}\\!\mathrm{d}z\\\
&=\int
e^{iz\cdot\xi_{\mathrm{p}}}\bar{\phi}(z+\xi_{\mathrm{x}}/2)\psi(z-\xi_{\mathrm{x}}/2)\mathop{}\\!\mathrm{d}z\end{split}$
(A.6)

where to get the second line we use Lemma 2.2. Applying this identity into the
right-hand side of (A.5), we get

$\begin{split}\frac{1}{(2\pi)^{2n}}\int
e^{-i\alpha\wedge\xi}{\mathcal{F}_{E}}(\xi)\mathop{}\\!\mathrm{d}\xi&=\frac{1}{(2\pi)^{2n}}\int
e^{-i\alpha_{\mathrm{x}}\cdot\xi_{\mathrm{p}}+i\alpha_{\mathrm{p}}\cdot\xi_{\mathrm{x}}}e^{iz\cdot\xi_{\mathrm{p}}}\bar{\phi}(z+\xi_{\mathrm{x}}/2)\psi(z-\xi_{\mathrm{x}}/2)\mathop{}\\!\mathrm{d}z\mathop{}\\!\mathrm{d}\xi_{\mathrm{x}}\mathop{}\\!\mathrm{d}\xi_{\mathrm{p}}\\\
&=\frac{1}{(2\pi)^{n}}\int
e^{i\alpha_{\mathrm{p}}\cdot\xi_{\mathrm{x}}}\bar{\phi}(\alpha_{\mathrm{x}}+\xi_{\mathrm{x}}/2)\psi(\alpha_{\mathrm{x}}-\xi_{\mathrm{x}}/2)\mathop{}\\!\mathrm{d}\xi_{\mathrm{x}},\end{split}$
(A.7)

where the last line follows from the Fourier inversion formula $\int
e^{i(z-\alpha_{\mathrm{x}})\cdot\xi_{\mathrm{p}}}f(\alpha_{\mathrm{x}})\mathop{}\\!\mathrm{d}z\mathop{}\\!\mathrm{d}\xi_{\mathrm{p}}=(2\pi)^{n}f(z)$.
This is the definition of the Wigner transform ${\mathcal{W}_{E}}(\alpha)$ of
$E=\mathinner{|{\psi}\rangle}\\!\\!\mathinner{\langle{\phi}|}$. ∎

######  Lemma 2.5.

The Wigner function ${\mathcal{W}_{\rho}}$ is a Schwartz function if and only
if the kernel ${\mathcal{K}_{\rho}}$ is a Schwartz function.

###### Proof.

First assume the kernel ${\mathcal{K}_{\rho}}(x,y)$ is a Schwartz function.
Then the function $g(z,\Delta z)={\mathcal{K}_{\rho}}(z-\Delta z/2,z+\Delta
z/2)$ is a Schwartz function since it is related to ${\mathcal{K}_{\rho}}$
merely by a linear change in variables (a $45^{\circ}$ rotation). And of
course we have ${\mathcal{W}_{\rho}}(x,p)=(2\pi)^{-n}\int e^{ip\cdot\Delta
z}g(x,\Delta z)\mathop{}\\!\mathrm{d}\Delta z$, so ${\mathcal{W}_{\rho}}$ must
also be a Schwartz function since it is just the $n$-dimensional Fourier
transform of $g$ (exchanging the variable $\Delta z$ for $p$ but leaving the
variable $z$). The argument works the same in the opposite direction, so we
conclude that ${\mathcal{K}_{\rho}}$ is a Schwartz function if and only if
${\mathcal{W}_{\rho}}$ is a Schwartz function. ∎

######  Lemma 2.6.

The twisted convolution of a rapidly decaying function with a Schwartz
function is itself a Schwartz function.

###### Proof.

Let $F\in{\mathcal{S}}({\mathbf{R}}^{2n})$ and
$G\in{\mathcal{D}}({\mathbf{R}}^{2n})$. Then recall the definition of the
twisted convolution,

$F\circledast_{\Omega^{\prime}}G(\alpha)=\int
e^{i\alpha\cdot\Omega^{\prime}\cdot\alpha^{\prime}/2}F(\alpha-\alpha^{\prime})G(\alpha^{\prime})\mathop{}\\!\mathrm{d}\alpha^{\prime}.$
(A.8)

We claim that for any multi-index
$a=(a_{1},\dots,a_{2n})\in(\mathbb{N}\cup\\{0\\})^{\times 2n}$, there exist
constants $C(a,a^{\prime})$ such that

$\partial^{a}(F\circledast_{\Omega^{\prime}}G)(\alpha)=\sum_{a^{\prime}\leq
a}C(a,a^{\prime})\int
e^{i\alpha\cdot\Omega^{\prime}\cdot\alpha^{\prime}/2}(\partial^{a^{\prime}}F)(\alpha-\alpha^{\prime})(i\Omega^{\prime}\cdot\alpha^{\prime})^{a-a^{\prime}}G(\alpha^{\prime})\mathop{}\\!\mathrm{d}\alpha^{\prime}.$
(A.9)

Equation (A.9) is easily checked by induction on $|a|$. Now applying the
triangle inequality we can estimate

$|\partial^{a}(F\circledast_{\Omega^{\prime}}G)(\alpha)|\leq\sum_{a^{\prime}\leq
a}C(\alpha,\alpha^{\prime})\int|\partial^{a^{\prime}}F(\alpha-\alpha^{\prime})||(\Omega^{\prime}\cdot\alpha^{\prime})^{a-a^{\prime}}G(\alpha^{\prime})|\mathop{}\\!\mathrm{d}\alpha^{\prime},$
(A.10)

which is the finite sum of convolutions of the rapidly decaying functions
$|\partial_{\alpha}^{a^{\prime}}F(\alpha)|$ and
$|(\Omega^{\prime}\cdot\alpha)^{a-a^{\prime}}G(\alpha)|$. Therefore
$\partial^{a}(F\circledast G)$ is rapidly decaying. Since every partial
derivative of $F\circledast G$ is rapidly decaying, it follows that
$F\circledast G\in{\mathcal{S}}({\mathbf{R}}^{2n})$. ∎

(Note that a similar statement for normal convolutions can be proven in almost
exactly the same way.)

######  Lemma 2.7.

For any two quantum states $\rho$ and $\eta$,

$\displaystyle\operatorname{tr}[\rho\eta]=(2\pi)^{n}\int{\mathcal{W}_{\rho}}(\alpha){\mathcal{W}_{\eta}}(\alpha)\mathop{}\\!\mathrm{d}\alpha.$
(A.11)

###### Proof.

We first decompose $\rho$ and $\eta$ using the spectral theorem,

$\rho=\sum_{j}\lambda_{j}\mathinner{|{\psi_{j}}\rangle}\\!\\!\mathinner{\langle{\psi_{j}}|},\qquad\qquad\eta=\sum_{j}\mu_{j}\mathinner{|{\phi_{j}}\rangle}\\!\\!\mathinner{\langle{\phi_{j}}|},$
(A.12)

with $\lambda_{j},\mu_{j}\geq 0$ and $\sum_{j}\lambda_{j}=\sum_{j}\mu_{j}=1$.
Then
$\operatorname{tr}[\rho\eta]=\sum_{j,k}\lambda_{j}\mu_{k}|\\!\mathinner{\langle{\psi_{j}|\phi_{k}}\rangle}\\!|^{2}$.
The result then follows from

$\begin{split}\int{\mathcal{W}_{\psi}}(\alpha){\mathcal{W}_{\phi}}(\alpha)\mathop{}\\!\mathrm{d}\alpha&=\frac{1}{(2\pi)^{2n}}\int
e^{ip\cdot(y+z)}{\mathcal{K}_{\rho}}(x-y/2,x+y/2)\\\
&\qquad\qquad\qquad\times{\mathcal{K}_{\eta}}(x-z/2,x+z/2)\mathop{}\\!\mathrm{d}y\mathop{}\\!\mathrm{d}z\mathop{}\\!\mathrm{d}x\mathop{}\\!\mathrm{d}p\\\
&=\frac{1}{(2\pi)^{n}}\int\psi(x-y/2)\bar{\psi}(x+y/2)\phi(x+y/2)\bar{\phi}(x-y/2)\mathop{}\\!\mathrm{d}y\mathop{}\\!\mathrm{d}x\\\
&=\frac{1}{(2\pi)^{n}}\int\psi(x_{-})\bar{\psi}(x_{+})\phi(x_{+})\bar{\phi}(x_{-})\mathop{}\\!\mathrm{d}x_{+}\mathop{}\\!\mathrm{d}x_{-}\\\
&=\frac{1}{(2\pi)^{n}}|\mathinner{\langle{\psi|\phi}\rangle}|^{2},\end{split}$
(A.13)

along with ${\mathcal{W}_{\rho}}=\sum_{j}\lambda_{j}{\mathcal{W}_{\psi_{j}}}$
and ${\mathcal{W}_{\mu}}=\sum_{j}\mu_{j}{\mathcal{W}_{\phi_{j}}}$. The sums
can be interchanged because everything converges absolutely. ∎

######  Lemma 2.9.

For any trace-class operator $E$ and any $\chi\in L^{2}({\mathbf{R}}^{n})$
satisfying $\|\chi\|_{L^{2}({\mathbf{R}}^{n})}=1$,

$\displaystyle\operatorname{tr}[E]=\frac{1}{(2\pi)^{n}}\int\mathinner{\langle{\chi_{\alpha}|E|\chi_{\alpha}}\rangle}\mathop{}\\!\mathrm{d}\alpha.$
(A.14)

In particular, for any $\phi,\psi\in L^{2}({\mathbf{R}}^{n})$

$\displaystyle\mathinner{\langle{\phi|\psi}\rangle}=\frac{1}{(2\pi)^{n}}\int\mathinner{\langle{\phi|\chi_{\alpha}}\rangle}\mathinner{\langle{\chi_{\alpha}|\psi}\rangle}\mathop{}\\!\mathrm{d}\alpha.$
(A.15)

###### Proof.

We start by proving (A.15). For $\psi,\phi\in L^{2}({\mathbf{R}}^{n})$,

$\begin{split}\int\mathinner{\langle{\phi|\chi_{\alpha}}\rangle}\mathinner{\langle{\chi_{\alpha}|g}\rangle}\mathop{}\\!\mathrm{d}\alpha&=\int\Bigg{[}\left(\int\bar{\phi}(z)e^{i(z-\alpha_{\mathrm{x}}/2)\cdot\alpha_{\mathrm{p}}}\chi(z-\alpha_{\mathrm{x}})\mathop{}\\!\mathrm{d}z\right)\\\
&\qquad\quad\times\left(\int\psi(z)\bar{\chi}(z^{\prime}-\alpha_{\mathrm{x}})e^{-i(z^{\prime}-\alpha_{\mathrm{x}}/2)\cdot\alpha_{\mathrm{p}}}\mathop{}\\!\mathrm{d}z^{\prime}\right)\Bigg{]}\mathop{}\\!\mathrm{d}\alpha\\\
&=\int\bar{\phi}(z)\psi(z^{\prime})e^{i(z-z^{\prime})\cdot\alpha_{\mathrm{p}}}\chi(z-\alpha_{\mathrm{x}})\bar{\chi}(z^{\prime}-\alpha_{\mathrm{x}})\mathop{}\\!\mathrm{d}z\mathop{}\\!\mathrm{d}z^{\prime}\mathop{}\\!\mathrm{d}\alpha_{\mathrm{x}}\mathop{}\\!\mathrm{d}\alpha_{\mathrm{p}}\\\
&=(2\pi)^{n}\int\bar{\phi}(z)\psi(z)\left(\int|\chi(z-\alpha_{\mathrm{x}})|^{2}\mathop{}\\!\mathrm{d}\alpha_{\mathrm{x}}\right)\mathop{}\\!\mathrm{d}z\\\
&=(2\pi)^{n}\mathinner{\langle{\phi|\psi}\rangle}\end{split}$ (A.16)

where to get from the second to the third line we use the Fourier inversion
formula.

Then, if $E$ is any trace-class operator, we can write using the singular
value decomposition (which one can obtain from the spectral theorem applied to
the polar decomposition $E=U\sqrt{EE^{\dagger}}$)

$E=\sum_{j}\sigma_{j}\mathinner{|{\phi_{j}}\rangle}\\!\\!\mathinner{\langle{\psi_{j}}|}$
(A.17)

for some orthonormal bases $\phi_{j}$, $\psi_{j}$ on
$L^{2}({\mathbf{R}}^{n})$. Here the singular values $\sigma_{j}$ are
nonnegative and satisfy $\sum_{j}\sigma_{j}=\|E\|_{1}<\infty$. In this case we
can expand the trace using this sum and apply (A.16)

$\begin{split}\operatorname{tr}[E]&=\sum_{j}\sigma_{j}\mathinner{\langle{\psi_{j}|\phi_{j}}\rangle}\\\
&=\sum_{j}\sigma_{j}\frac{1}{(2\pi)^{n}}\int\mathinner{\langle{\chi_{\alpha}|\phi_{j}}\rangle}\mathinner{\langle{\psi_{j}|\chi_{\alpha}}\rangle}\mathop{}\\!\mathrm{d}\alpha\\\
&=\frac{1}{(2\pi)^{n}}\int\mathinner{\langle{\chi_{\alpha}|E|\chi_{\alpha}}\rangle}\mathop{}\\!\mathrm{d}\alpha.\end{split}$
(A.18)

The last line is obtained by swapping the integral and the sum, which can be
done because the sum is absolutely convergent. ∎

######  Lemma 2.10.

For any quantum state $\rho$ and reference wavefunction
$\chi\in{\mathcal{S}}({\mathbf{R}}^{n})$,

$\displaystyle{\mathcal{Q}^{\chi}_{\rho}}(\alpha)$
$\displaystyle=(2\pi)^{n}({\mathcal{W}_{\rho}}\ast{\mathcal{W}_{\chi}^{-}})(\alpha)=(2\pi)^{n}\int{\mathcal{W}_{\rho}}(\beta){\mathcal{W}_{\chi}}(\beta-\alpha)\mathop{}\\!\mathrm{d}\beta$
(A.19)

where ${{\mathcal{W}_{\chi}^{-}}}(\alpha):={\mathcal{W}_{\chi}}(-\alpha)$ is a
Schwartz function.

###### Proof.

We have

$\displaystyle{\mathcal{Q}^{\chi}_{\rho}}(\alpha)$
$\displaystyle=\mathinner{\langle{\chi_{\alpha}|\rho|\chi_{\alpha}}\rangle}=\operatorname{tr}[\rho(\mathinner{|{\chi_{\alpha}}\rangle}\\!\\!\mathinner{\langle{\chi_{\alpha}}|})]$
(A.20)
$\displaystyle=(2\pi)^{n}\int{\mathcal{W}_{\rho}}(\beta){\mathcal{W}_{D_{\alpha}\mathinner{|{\chi}\rangle}\\!\mathinner{\langle{\chi}|}D_{\alpha}^{\dagger}}}(\beta)\mathop{}\\!\mathrm{d}\beta$
(A.21)
$\displaystyle=(2\pi)^{n}\int{\mathcal{W}_{\rho}}(\beta){\mathcal{W}_{\chi}}(\beta-\alpha)\mathop{}\\!\mathrm{d}\beta$
(A.22)

where we get the second line from Lemma 2.7 and the third line from the fact
that the map $\eta\mapsto D_{\alpha}\eta D_{\alpha}^{\dagger}$ on quantum
states corresponds to a displacement of the Wigner function by $\alpha$:

$\displaystyle{\mathcal{W}_{D_{\alpha}\eta D_{\alpha}^{\dagger}}}(\beta)$
$\displaystyle=\frac{1}{(2\pi)^{2n}}\int
e^{-i\beta\wedge\xi}{\mathcal{F}_{D_{\alpha}\eta
D_{\alpha}^{\dagger}}}(\xi)\mathop{}\\!\mathrm{d}\xi$ (A.23)
$\displaystyle=\frac{1}{(2\pi)^{2n}}\int
e^{-i\beta\wedge\xi}\operatorname{tr}[D_{\alpha}\eta
D_{-\alpha}D_{\xi}]\mathop{}\\!\mathrm{d}\xi$ (A.24)
$\displaystyle=\frac{1}{(2\pi)^{2n}}\int
e^{-i(\beta-\alpha)\wedge\xi}\operatorname{tr}[\eta
D_{\xi}]\mathop{}\\!\mathrm{d}\xi$ (A.25)
$\displaystyle={\mathcal{W}_{\eta}}(\beta-\alpha)$ (A.26)

Furthermore, since $\chi$ is a Schwartz function, so is
${\mathcal{K}_{\chi}}(x,y)=\bar{\chi}(x)\chi(y)$, and hence by Lemma 2.5 we
have that ${\mathcal{W}_{\chi}^{-}}(\alpha)={\mathcal{W}_{\chi}}(-\alpha)$ is
a Schwartz function. ∎

## References