# Quantum Wigner entropy

Zacharie Van Herstraeten<EMAIL_ADDRESS>Centre for Quantum Information
and Communication, École polytechnique de Bruxelles, CP 165/59, Université
libre de Bruxelles, 1050 Brussels, Belgium  Nicolas J. Cerf<EMAIL_ADDRESS>Centre for Quantum Information and Communication, École polytechnique de
Bruxelles, CP 165/59, Université libre de Bruxelles, 1050 Brussels, Belgium

###### Abstract

We define the Wigner entropy of a quantum state as the differential Shannon
entropy of the Wigner function of the state. This quantity is properly defined
only for states that possess a positive Wigner function, which we name Wigner-
positive states, but we argue that it is a proper measure of quantum
uncertainty in phase space. It is invariant under symplectic transformations
(displacements, rotations, and squeezing) and we conjecture that it is lower
bounded by $\ln\pi+1$ within the convex set of Wigner-positive states. It
reaches this lower bound for Gaussian pure states, which are natural minimum-
uncertainty states. This conjecture bears a resemblance with the Wehrl-Lieb
conjecture, and we prove it over the subset of passive states of the harmonic
oscillator which are of particular relevance in quantum thermodynamics. Along
the way, we present a simple technique to build a broad class of Wigner-
positive states exploiting an optical beam splitter and reveal an unexpectedly
simple convex decomposition of extremal passive states. The Wigner entropy is
anticipated to be a significant physical quantity, for example, in quantum
optics where it allows us to establish a Wigner entropy-power inequality. It
also opens a way towards stronger entropic uncertainty relations. Finally, we
define the Wigner-Rényi entropy of Wigner-positive states and conjecture an
extended lower bound that is reached for Gaussian pure states.

##  I Introduction

The phase-space formulation of quantum mechanics provides a complete framework
that echoes classical statistical mechanics. Quantum states and quantum
operators are described within this formulation by continuous functions of the
pair of canonical variables $x$ and $p$. These variables traditionally refer
to the position and momentum observables, but are also isomorphic to the
conjugate quadrature components of a mode of the electromagnetic field (we use
this quantum optics nomenclature in the present paper). The conversion from
quantum operators to quantum phase-space distributions is carried out via the
Wigner-Weyl transform Case (2008), which maps any linear operator $\hat{A}$
into a distribution $A(x,p)$ as

$A(x,p)=\dfrac{1}{\pi\hbar}\int\exp\left(2ipy/\hbar\right)\bra{x-y}\hat{A}\ket{x+y}\mathrm{d}y,$
(1)

where $\hbar$ denotes the Planck constant (we set $\hbar=1$ in the remainder
of this paper). Accordingly, the Wigner function of a quantum state is the
Wigner-Weyl transform of its density operator $\hat{\rho}$, written as
$W(x,p)$. The Wigner function comes as close to a probability distribution in
phase space as allowed by quantum mechanics. It indeed shares most properties
of a classical probability distribution. Notably, the marginal distributions
of $W(x,p)$ coincide with the probability distributions for $x$ and $p$,
respectively $\rho_{x}(x)=\bra{x}\hat{\rho}\ket{x}$ and
$\rho_{p}(p)=\bra{p}\hat{\rho}\ket{p}$, as it can easily be shown that $\int
W(x,p)\,\mathrm{d}p=\rho_{x}(x)$ and $\int W(x,p)\,\mathrm{d}x=\rho_{p}(p)$.
Also, the expectation value of any operator $\hat{A}$ in state $\hat{\rho}$ is
straightforwardly computed from its Wigner function through the overlap
formula Leonhardt (2010):

$\langle\hat{A}\rangle=\mathrm{Tr}\left[\hat{A}\,\hat{\rho}\right]=2\pi\iint
A(x,p)\,W(x,p)\,\mathrm{d}x\,\mathrm{d}p.$ (2)

However, it is well known that the Wigner function is not a true probability
distribution as it lacks positiveness Kenfack and Zyczkowski (2004). For
example, all pure non-Gaussian states have a Wigner function that admits
negative regions as a consequence of the Hudson theorem Hudson (1974). This is
the price to pay to the Heisenberg uncertainty principle, which forbids the
joint definition of noncommuting variables $x$ and $p$. Hence, several common
functionals of probability distributions, such as the Shannon differential
entropy, become in general ill defined if applied to Wigner functions.

In contrast, there exists a well-known distribution in quantum phase space
that behaves as a genuine probability distribution, namely the Husimi $Q$
function Leonhardt (2010), defined as
$Q(\alpha)=\bra{\alpha}\hat{\rho}\ket{\alpha}/\pi$. It corresponds to the
probability to measure state $\hat{\rho}$ in a coherent state $\ket{\alpha}$.
Remember that a coherent state $\ket{\alpha}$ is an eigenstate of the
annihilation operator $\hat{a}=\left(\hat{x}+i\hat{p}\right)/\sqrt{2}$ with
eigenvalue $\alpha$. Splitting the complex parameter $\alpha$ into two real
parameters $x$ and $p$ such that $\alpha=x+ip$ gives

$Q(x,p)=\dfrac{1}{\pi}\bra{x+ip}\hat{\rho}\ket{x+ip}\,.$ (3)

Despite lacking the nice properties of the Wigner function such as the overlap
formula (2), the Husimi function has the advantage of being positive, hence it
admits a properly defined entropy. The Shannon differential entropy of the
Husimi function is indeed known as the Wehrl entropy and is defined as
$h\left(Q\right)=-\iint Q(x,p)\ln Q(x,p)\,\mathrm{d}x\,\mathrm{d}p$. This
entropy is at the core of the Wehrl conjecture Wehrl (1979), later proven by
Lieb Lieb (1978, 2002), which states that the Wehrl entropy is lower-bounded
by $\ln\pi+1$ and that the only minimizers of $h(Q)$ are the coherent states
111The Wehrl conjecture is also often written as $h\left(Q\right)\geq 1$,
which simply originates from a different convention. If the Husimi Q-function
is defined without the $1/\pi$ prefactor but, instead, this additional
prefactor is inserted in the definition of $h\left(Q\right)$, we get an
additive constant $-\ln\pi$ which shifts the lower bound to 1..

Interestingly, there is a link between the Husimi function and Wigner function
of a state, as can simply be understood using the quantum optics language. To
this purpose, recall that the vacuum state $\ket{0}$ [or ground state of the
harmonic oscillator $\hat{H}=\left(\hat{p}^{2}+\hat{x}^{2}\right)/2$ in
natural units] admits the Wigner function
$W_{0}\left(x,p\right)=\exp\left(-x^{2}-p^{2}\right)/\pi$. Since a coherent
state $\ket{\alpha}$ is a displaced vacuum state, its Wigner function is then
$W_{\alpha}(x,p)=W_{0}\left(x^{\prime},p^{\prime}\right)$, where
$x^{\prime}=x-\sqrt{2}\,\mathrm{Re}(\alpha)$ and
$p^{\prime}=p-\sqrt{2}\,\mathrm{Im}(\alpha)$. Using this, the Husimi function
can be expressed from the overlap formula (2) as:

$\displaystyle Q(x,p)$ $\displaystyle=$
$\displaystyle\dfrac{1}{\pi}\mathrm{Tr}\left[\ket{x+ip}\\!\bra{x+ip}\,\hat{\rho}\right]$
(4) $\displaystyle=$ $\displaystyle 2\iint
W_{0}\left(\tilde{x}-\sqrt{2}x,\tilde{p}-\sqrt{2}p\right)W(\tilde{x},\tilde{p})\,\mathrm{d}\tilde{x}\,\mathrm{d}\tilde{p}.$

Thus, it appears that $Q$ is a convolution between $W$ and $W_{0}$, with a
rescaling factor of $\sqrt{2}$. In the language of random variables (and
provided $W$ is non-negative), we could say that if $(\tilde{x},\tilde{p})$ is
distributed according to $W$ and $(x_{0},p_{0})$ is distributed according to
$W_{0}$, then $(x,p)$ is distributed according to $Q$, with

$x=\left(\tilde{x}-x_{0}\right)/\sqrt{2}\quad\textrm{and}\quad
p=\left(\tilde{p}-p_{0}\right)/\sqrt{2}.$ (5)

This is a familiar relation in quantum optics, describing the action of a beam
splitter of transmittance $\eta=1/2$ onto the state $\hat{\rho}$ and the
vacuum state. Defining $\hat{\sigma}$ as the reduced state of the
corresponding output of the beam splitter, as shown in Fig. 1, we conclude
that the Wigner function of $\hat{\sigma}$ is precisely the Husimi function of
$\hat{\rho}$, namely,

$W_{\hat{\sigma}}(x,p)=Q_{\hat{\rho}}(x,p),$ (6)

where

$\hat{\sigma}=\mathrm{Tr}_{2}\left[\hat{U}_{\frac{1}{2}}\left(\hat{\rho}\otimes\ket{0}\bra{0}\right)\hat{U}_{\frac{1}{2}}^{\dagger}\right].$
(7)

Here $\hat{U}_{\frac{1}{2}}$ denotes the beam-splitter unitary of
transmittance $\eta=1/2$, while $\mathrm{Tr}_{2}$ denotes a reduced trace over
one of the modes, say the second mode. From Eq. (6), it appears that the
entropy of the Wigner function of $\hat{\sigma}$ is nothing else but the Wehrl
entropy of $\hat{\rho}$ in this particular setup. A natural question then
arises : can we give an intrinsic meaning to the entropy of a Wigner function
independently of this particular setup?

Figure 1:  Reduced output state $\hat{\sigma}$ of a balanced beam splitter (of
transmittance $\eta=1/2$) when the input state is $\hat{\rho}$, as described
in Eq. (7). The Wigner function of $\hat{\sigma}$ coincides with the Husimi
$Q$ function of $\hat{\rho}$; hence it is positive. Consequently, the Wigner
entropy of $\hat{\sigma}$ is equal to the Wehrl entropy of $\hat{\rho}$.

In this paper, we will answer by the affirmative. First, let us notice that
the setup of Fig. 1 ensures that the output state $\hat{\sigma}$ always has a
positive Wigner function (see Appendix B). In general, we will denote the
quantum states admitting a positive Wigner function [i.e., states such that
$W(x,p)\geq 0$, $\forall x,p$] as Wigner-positive states. For such states, it
is possible to compute the Shannon differential entropy of their Wigner
function. We make the leap and define the Wigner entropy of any Wigner-
positive state $\hat{\rho}$ as

$h\left(W\right)=-\iint W(x,p)\,\ln W(x,p)\,\mathrm{d}x\,\mathrm{d}p$ (8)

where

$W(x,p)=\frac{1}{\pi}\int\exp\left(2ipy\right)\bra{x-y}\hat{\rho}\ket{x+y}\mathrm{d}y$
(9)

is the Wigner function of $\hat{\rho}$. We argue that, although it is limited
to Wigner-positive states, the Wigner entropy is a natural measure in order to
characterize quantum uncertainty in phase space: it bears information about
the uncertainty of the marginal distributions of the $x$ and $p$ variables as
well as their correlations in phase space. In contrast with the Wehrl entropy,
it is not the classical entropy of the outcome of a specific measurement,
namely, a joint $(x,p)$ measurement (called heterodyne detection or eight-port
homodyne detection in quantum optics). Of course, in the special case where a
Wigner-positive state can be prepared using the setup of Fig. 1, its Wigner
entropy can be viewed simply as the Wehrl entropy of the corresponding input
state, but the definition goes further and the Wigner entropy remains relevant
for Wigner-positive states that cannot be built in this way.

The Wigner entropy $h(W)$ enjoys interesting properties. First, unlike the
Wehrl entropy $h(Q)$, it is invariant under symplectic transformations
(displacement, rotation, and squeezing) in phase space. Such transformations,
which are ubiquitous in quantum optics, correspond to the set of all Gaussian
unitaries in state space. We stress that a sensible measure of phase-space
uncertainty must remain invariant under symplectic transformations since these
are also area-preserving transformations in phase space. In contrast, $h(Q)$
is greater for squeezed states than for coherent states. As it can be
understood from Fig. 1, this preference simply originates from the fact that
one input of the balanced beam splitter is itself a coherent state. Second,
the Wigner entropy $h(W)$ can be related to the entropy of the marginal
distributions $h\left(\rho_{x}\right)$ and $h\left(\rho_{p}\right)$, but also
encompasses the $x$-$p$ correlations. Shannon information theory establishes a
relation between the entropy of a joint distribution and its marginal
entropies, namely, $h(x,p)=h(x)+h(p)-I$, where $I\geq 0$ is the mutual
information Cover and Thomas (1991). Applied to the Wigner entropy, this gives
the inequality $h(W)\leq h\left(\rho_{x}\right)+h\left(\rho_{p}\right)$. This
means that a lower bound on the Wigner entropy implies in turn a lower bound
on the sum of the marginal entropies.

In the light of these considerations, we introduce a conjecture on the Wigner
entropy, which resembles the Wehrl conjecture. As anticipated in Hertz  _et
al._ (2017), we conjecture that the Wigner entropy of any Wigner-positive
state $\hat{\rho}$ satisfies

$h\left(W\right)\geq\ln\pi+1.$ (10)

As we will show, this bound is reached by all Gaussian pure states, which
appears consistent with the Hudson theorem Hudson (1974). It implies (but is
stronger than) the entropic uncertainty relation of Białynicki-Birula and
Mycielski Białynicki-Birula and Mycielski (1975), namely,
$h\left(\rho_{x}\right)+h\left(\rho_{p}\right)\geq\ln\pi+1$. Importantly,
conjecture (10) also implies the Wehrl conjecture since we have shown that the
Husimi function of any state $\hat{\rho}$ is the Wigner function of some
Wigner-positive state $\hat{\sigma}$ in a particular setup (see Fig. 1).
However, the converse is not true as there exist Wigner-positive states whose
Wigner function cannot be written as the Husimi function of a physical state
(an example will be shown in Sec. IV).

The paper is organized as follows. In Sec. II, we start by recalling some
basics of the symplectic formalism and then define the Wigner entropy of a
Wigner-positive state as a distinctive information-theoretical measure of its
uncertainty in phase space. In Sec. III, we discuss the characterization of
the set of Wigner-positive states and focus on the particular subset of phase-
invariant Wigner-positive states. Then, in Sec. IV, we turn to the main
conjecture and provide a proof for some special case of phase-invariant
Wigner-positive states, namely the passive states. Finally, we conclude in
Sec. V and provide an example application of the Wigner entropy, namely the
Wigner entropy-power inequality. Further, in Appendix A, we extend the Wigner
entropy and define the Wigner-Rényi entropy of Wigner-positive states. We also
discuss a natural extension of the conjectured lower bound. In Appendix B, we
present a quantum-optics-inspired method for generating a large variety of
Wigner-positive states with a balanced beam splitter, extending on Fig. 1.
Appendix C is devoted to the detailed analysis of the set of Wigner-positive
states when considering the Fock space restricted to two photons as this
provides a helpful illustration of our results. Finally, Appendix D provides
more details on the derivation of the formula [Eq. (43)] at the heart of our
proof.

##  II Wigner entropy of a state

In this paper, we restrict our considerations to a single bosonic mode (one
harmonic oscillator) for simplicity, although the definition of the Wigner
entropy and the corresponding conjecture should extend to the multidimensional
case. Let us briefly review the symplectic formalism for one bosonic mode. Let
$\mathbf{\hat{x}}=(\hat{x},\hat{p})^{\intercal}$ be the vector of quadrature
operators (or position and momentum canonical operators) satisfying
$[\hat{x}_{j},\hat{x}_{k}]=i\,\Omega_{jk}$, with the matrix

$\mathbf{\Omega}=\begin{pmatrix}0&1\\\ -1&0\end{pmatrix}$ (11)

being the symplectic form. The coherence vector (also called the displacement
vector) of a state $\hat{\rho}$ is defined as

$\mathbf{c}=\langle\mathbf{\hat{x}}\rangle\coloneq\mathrm{Tr}(\mathbf{\hat{x}}\,\hat{\rho}),$
(12)

where $\langle\cdot\rangle$ stands for the expectation value in state
$\hat{\rho}$, while the covariance matrix $\mathbf{\Gamma}$ of state
$\hat{\rho}$ is defined as

$\Gamma_{jk}=\langle\\{\hat{x}_{j}-\langle\hat{x}_{j}\rangle,\hat{x}_{k}-\langle\hat{x}_{k}\rangle\\}\rangle$
(13)

where $\\{\cdot,\cdot\\}$ stands for the anticommutator. The set of Gaussian
states contains those for which the Wigner function $W(x,p)$ is Gaussian;
hence these states are completely characterized by their first- and second-
order moments $\mathbf{c}$ and $\mathbf{\Gamma}$. The set of Gaussian
unitaries in state space is isomorphic to the set of symplectic
transformations in phase space. Formally, a symplectic transformation is an
affine map on the space of quadrature operators which is defined by a
symplectic matrix $\mathbf{S}$ and a displacement vector $\mathbf{d}$, namely,

$\mathbf{\hat{x}}\to\mathbf{S}\mathbf{\hat{x}}+\mathbf{d}.$ (14)

The symplectic matrix $\mathbf{S}$ is a real matrix that must preserve the
symplectic form, that is,
$\mathbf{S}\mathbf{\Omega}\mathbf{S}^{\intercal}=\mathbf{\Omega}$, which
implies in particular that $\det\mathbf{S}=1$. The displacement vector
$\mathbf{d}$ is an arbitrary real vector. The first- and second-order moments
of a state $\rho$ evolve under such a symplectic transformation as

$\mathbf{c}\to\mathbf{S}\mathbf{c}+\mathbf{d}\,,\qquad\mathbf{\Gamma}=\mathbf{S}\mathbf{\Gamma}\mathbf{S}^{\intercal}\,.$
(15)

In the special case of Gaussian states, this completely characterizes the
evolution of the state under the Gaussian unitary.

The core of this paper is the definition of an information-theoretical measure
of uncertainty in phase space, which we call the Wigner entropy $h(W)$, where
$h(\cdot)$ denotes the Shannon differential entropy functional and $W(x,p)$ is
the Wigner function of $\hat{\rho}$ [see Eqs. (8) and (9)]. As already
mentioned, it only applies to Wigner-positive states since, otherwise, the
definition of the entropy entails the logarithm of a negative number. We note
it as a functional of $W$ but, of course, it is eventually a functional of the
state $\hat{\rho}$ since $W$ itself depends on $\hat{\rho}$.

In contrast with the Shannon entropy of a discrete variable, the Shannon
differential entropy of a continuous variable does not have an absolute
meaning (it depends on the scale of the variable) and it becomes negative if
the probability distribution is highly peaked Cover and Thomas (1991).
However, when applied to a Wigner function, a natural scale is provided here
by the area $\hbar$ of a unit cell in phase space. Hence, the Wigner entropy
has a meaning per se and it is legitimate to conjecture a lower bound, namely,
Eq. (10), when setting $\hbar=1$. Further, it is natural to extend on this and
consider a lower bound on the differential Rényi entropy of the Wigner
function of any Wigner-positive state, a quantity that we define as the
Wigner-Rényi entropy (see Appendix A).

The Wigner entropy $h(W)$ has the nice property to be invariant under
symplectic transformations. Consider the symplectic transformation
$\mathbf{\hat{x}}\to\mathbf{\hat{x}^{\prime}}=\mathbf{S}\mathbf{\hat{x}}+\mathbf{d}$
and let us denote as $W$ and $W^{\prime}$ the Wigner function of the input and
output states, respectively. The change of variables corresponding to this
transformation gives

$W^{\prime}(x^{\prime},p^{\prime})=\frac{W(x,p)}{|\det\mathbf{S}|},$ (16)

which indeed implies that

$\displaystyle h(W^{\prime})$ $\displaystyle=$ $\displaystyle-\iint
W^{\prime}(x^{\prime},p^{\prime})\,\ln
W^{\prime}(x^{\prime},p^{\prime})\,\mathrm{d}x^{\prime}\,\mathrm{d}p^{\prime}$
(17) $\displaystyle=$ $\displaystyle-\iint
W(x,p)\ln\left(\frac{W(x,p)}{|\det\mathbf{S}|}\right)\,\mathrm{d}x\,\mathrm{d}p$
$\displaystyle=$ $\displaystyle h(W)+\ln|\det\mathbf{S}|$ $\displaystyle=$
$\displaystyle h(W),$

where we have used the fact that $W$ is normalized and the fact that
$\mathbf{S}$ is a symplectic matrix ($\det\mathbf{S}=1$).

Note that this invariance can also be understood as a sole consequence of the
fact that symplectic transformations conserve areas in phase space since
$\det\mathbf{S}=1$. Indeed, for any functional $F$, we have

$\displaystyle\iint
F\big{(}W^{\prime}(x^{\prime},p^{\prime})\big{)}\,\mathrm{d}x^{\prime}\,\mathrm{d}p^{\prime}$
(18) $\displaystyle=\iint
F\left(\frac{W(x,p)}{|\det\mathbf{S}|}\right)\,|\det\mathbf{S}|\,\mathrm{d}x\,\mathrm{d}p$
$\displaystyle=\iint F\big{(}W(x,p)\big{)}\,\mathrm{d}x\,\mathrm{d}p.$

The special case of Gaussian states is very easy to deal with. A
straightforward calculation shows that the Wigner entropy of a Gaussian state
$\hat{\rho}$ is given by

$h(W)=\ln\left(2\pi\sqrt{\det\mathbf{\Gamma}}\right)+1=\ln(\pi/\mu)+1,$ (19)

where $\mu=\mathrm{Tr}\hat{\rho}^{2}=1/(2\sqrt{\det\mathbf{\Gamma}})\leq 1$
stands for the purity of the state. All Gaussian states that are connected
with a symplectic transformation obviously conserve their purity since
$\det\mathbf{\Gamma^{\prime}}=\det(\mathbf{S}\mathbf{\Gamma}\mathbf{S}^{\intercal})=\det\mathbf{\Gamma}$,
which confirms that their Wigner entropy is invariant. The lowest value of
$h(W)$ among Gaussian states is then reached for pure states ($\mu=1$) and is
given by $\ln\pi+1$, as expected. This is the value of the Wigner entropy of
all coherent states and squeezed states (regardless the squeezing parameter,
squeezing orientation, and coherence vector). Accordingly, the Gaussian pure
states would be the minimum-Wigner-uncertainty states. The difficult task
remains, however, to prove that non-Gaussian Wigner-positive states cannot
violate this lower bound (see Sec. IV).

Provided this conjecture is valid, the Wigner function of any Wigner-positive
state can be classically simulated from the Wigner function of the vacuum
state (or any other Gaussian pure state). More precisely, information theory
tells us that the difference $\Delta=h(W)-\ln\pi-1$ can be viewed as the
number of independent equiprobable random bits that are needed, on average, to
generate deterministically one random $(x,p)$ instance drawn from the Wigner
function of state $\rho$ from one random $(x,p)$ instance drawn from the
Wigner function of the vacuum state (or any Gaussian pure state). Of course,
this results holds at the asymptotic limit only, that is, around
$N\times\Delta$ bits of extra randomness are needed for converting $N$ random
instances of $(x,p)\sim W_{0}$ into $N$ random instances of $(x,p)\sim W$ by
deterministic means when $N\to\infty$.

##  III Wigner-positive states

As explained in Sec. II, the Wigner entropy naturally appears as an
information-theoretic measure of uncertainty in phase space, but is only
properly defined for positive Wigner functions. For this reason, we devote
this section to the quantum states with positive Wigner functions, which we
call Wigner-positive states. Note that Wigner positivity is a particular case
of $\eta$-positivity for $\eta=0$ Narcowich (1989); Bröcker and Werner (1995).
Quantum Wigner-positive states of a single mode are described by a Wigner
function $W(x,p)$ that respects the condition

$W(x,p)\geq 0\qquad\forall x,p.$ (20)

Restricting to pure states, the set of Wigner-positive states is well known:
the Hudson theorem establishes that Gaussian pure states are the only pure
quantum states with a positive Wigner function Hudson (1974). When it comes to
mixed states, however, the situation becomes more difficult since the mixing
of states enables one to build non-Gaussian Wigner-positive states. The
characterization of the set of Wigner-positive mixed states has been attempted
Bröcker and Werner (1995); Mandilara  _et al._ (2009), but the resulting
picture is somehow complex. Just like writing a necessary and sufficient
condition for a Wigner function to correspond to a positive-semidefinite
density operator is a hard task, it appears cumbersome to express a necessary
and sufficient condition for a density operator to be associated with a
positive Wigner function.

On a more positive note, the set of Wigner-positive states is convex since a
mixture of Wigner-positive states is itself Wigner positive. Taking advantage
of this property, we may focus on the extremal states of the convex set, as
pictured in Fig. 2. These are the states that cannot be obtained as a mixture
of other states of the set. Conversely, any state of the set can be generated
as a mixture of these extremal states. This brings a simplification in the
proof of the main conjecture, namely, expressing a lower bound on the Wigner
entropy of an arbitrary Wigner-positive state (see Sec. IV). Indeed, the
Shannon entropy being concave, the entropy of a mixture is lower-bounded by
the entropy of its components, that is,

$h(p_{1}W_{1}+p_{2}W_{2})\geq p_{1}\,h(W_{1})+p_{2}\,h(W_{2}),$ (21)

where $p_{1}$ and $p_{2}$ are positive reals such that $p_{1}+p_{2}=1$. Hence,
it is sufficient to prove the lower bound on $h(W)$ for the extremal Wigner-
positive states in order to have a proof over the full set.

Figure 2: Schematic view of a convex set. The black and red points form all
together the boundary of the convex set, while the red points are the extremal
points on this boundary (note the existence of isolated extremal points as
well as of a continuum of extremal points).

### Several classes of Wigner-positive states

To be more specific, we define several sets of Wigner-positive quantum states,
which will be useful in the rest of this paper. As we will see, conjecture
(10) is trivially verified for some of them, while it remains hard to prove
for others.

  * •

$\mathcal{Q}$ : Physical quantum states
It is the convex set of all single-mode quantum states. Their density operator
$\hat{\rho}$ satisfies the three physicality conditions: Hermiticity, positive
semidefiniteness and unit trace. Of course, they can have partly negative
Wigner functions.

  * •

$\mathcal{Q}_{+}$ : Wigner-positive quantum states
It is the subset of states in $\mathcal{Q}$ that have positive Wigner
functions. It is a convex set. All states within this set have a well-defined
Wigner entropy and are the subject of conjecture (10).

  * •

$\mathcal{G}$ : Gaussian states
It is the subset of states in $\mathcal{Q}_{+}$ that have a Gaussian Wigner
function. It does not form a convex set since the mixture of Gaussian states
does not need to be Gaussian, so we refer to its convex hull as
$\mathcal{G}_{c}$.

  * •

$\mathcal{C}$ : Classical states
According to Glauber’s definition, classical states are mixtures of coherent
states. They are characterized by a positive Glauber-Sudarshan $P$ function.
By definition, $\mathcal{C}$ is a convex set and
$\mathcal{C}\subset\mathcal{G}_{c}$ since coherent states are Gaussian states.

The extremal states of $\mathcal{C}$ and $\mathcal{G}_{c}$ are respectively
coherent states and Gaussian pure states. For these two sets, conjecture (10)
is trivially verified since the Wigner entropy of Gaussian pure states is
precisely $\ln\pi+1$ and since the entropy is concave. Unfortunately, the
convex closure of Gaussian states $\mathcal{G}_{c}$ is yet but a small
fraction of the set of Wigner-positive states $\mathcal{Q}_{+}$. As an
evidence of this, we construct a wider set of Wigner-positive states by
exploiting a technique relying on a balanced beam splitter (hence, we name
this set as $\mathcal{B}$).

  * •

$\mathcal{B}$ : Beam-splitter states
These are the states $\hat{\sigma}$ resulting from the setup depicted in Fig.
3. More precisely, a beam-splitter state $\hat{\sigma}$ denotes the reduced
output state of a beam splitter with transmittance $\eta=1/2$ fed by a tensor
product of two arbitrary states $\hat{\rho}_{A}$ and $\hat{\rho}_{B}$,

$\hat{\sigma}=\mathrm{Tr}_{2}\left[\hat{U}_{\frac{1}{2}}\left(\hat{\rho}_{A}\otimes\hat{\rho}_{B}\right)\hat{U}_{\frac{1}{2}}^{\dagger}\right].$
(22)

We show in Appendix B that state $\hat{\sigma}$ always possesses a positive
Wigner function, regardless of $\hat{\rho}_{A}$ and $\hat{\rho}_{B}$. The sole
condition is that the input state is a tensor product and the beam splitter is
balanced ($\eta=1/2$).

Figure 3: Beam-splitter state ${\hat{\sigma}}$ obtained at the output of a
balanced beam splitter (of transmittance $\eta=1/2$). If the input is an
arbitrary product state $\hat{\rho}_{A}\otimes\hat{\rho}_{B}$, then the
reduced state of the output ${\hat{\sigma}}$ is guaranteed to be Wigner
positive. This generalizes Fig. 1, where $\hat{\rho}_{B}=\ket{0}\\!\bra{0}$.
The set of beam-splitter states is denoted as $\mathcal{B}$. The convex hull
of these states, denoted as $\mathcal{B}_{c}$, is obtained by sending any
separable state (i.e., a mixture of product states) into a balanced beam
splitter and tracing over one of the output modes. The whole set
$\mathcal{B}_{c}$ is strictly included in the set of Wigner-positive states
$\mathcal{Q}_{+}$.

It can be shown with a simple argument that the set of Gaussian states
$\mathcal{G}$ is a subset of $\mathcal{B}$. Indeed, it is well known that the
product of two identical copies of a Gaussian state $\hat{\gamma}$ is
invariant under the action of a beam splitter (assuming the coherence vector
vanishes Weedbrook  _et al._ (2012)). We have the identity
$\hat{U}_{\eta}\left(\hat{\gamma}\otimes\hat{\gamma}\right)\hat{U}_{\eta}^{\dagger}=\hat{\gamma}\otimes\hat{\gamma}$,
where $\hat{\gamma}$ is any single-mode Gaussian state and $\hat{U}_{\eta}$ is
the unitary of a beam splitter with transmittance $\eta$. One can then easily
reconstruct the set of Gaussian states with the above setup, so it follows
that $\mathcal{G}\subset\mathcal{B}$. Note that it is easy to build beam-
splitter states as in Fig. 3 that are not Gaussian states; hence this is a
strict inclusion relation. The analog relation also applies to the respective
convex hull of these sets, namely, $\mathcal{G}_{c}\subset\mathcal{B}_{c}$.
Unfortunately, the set $\mathcal{B}_{c}$ does not coincide with
$\mathcal{Q}_{+}$ as we will see that there exist Wigner-positive states that
do not belong to $\mathcal{B}_{c}$ (see, e.g., the dark blue region in Fig.
6). In summary, we have the following chain of strict inclusion relations:

$\mathcal{Q}\supset\mathcal{Q}_{+}\supset\mathcal{B}_{c}\supset\mathcal{G}_{c}\supset\mathcal{C}\,$
(23)

as pictured in Fig. 4.

### Phase-invariant states in $\mathcal{Q}_{+}$

As it appears, the set $\mathcal{Q}_{+}$ of Wigner-positive states remains
hard to encompass and characterize efficiently. Therefore, in order to make a
concrete step towards the proof of conjecture (10), we restrict our attention
in this paper to a class of quantum states known as phase-invariant states.
Phase-invariant states have a Wigner function that is invariant under
rotation, so they are fully characterized by their radial Wigner function.
Such states have the advantage of being easily characterized in state space as
they can be written as mixtures of Fock states, which are eigenstates of the
harmonic oscillator. The wave function and the Wigner function of the
$n^{\mathrm{th}}$ Fock state (starting at $n=0$ for vacuum) are the following:

$\displaystyle\psi_{n}(x)$
$\displaystyle=\pi^{-\frac{1}{4}}2^{-\frac{n}{2}}\left(n!\right)^{-\frac{1}{2}}H_{n}(x)\exp\left(-\frac{x^{2}}{2}\right),$
(24) $\displaystyle W_{n}(x,p)$
$\displaystyle=\frac{1}{\pi}(-1)^{n}L_{n}\left(2x^{2}+2p^{2}\right)\exp\left(-x^{2}-p^{2}\right),$
(25)

where $H_{n}$ and $L_{n}$ are respectively the $n^{\mathrm{th}}$ Hermite and
Laguerre polynomials. A phase-invariant state is thus expressed as the mixture

$\hat{\rho}=\sum_{k=0}^{\infty}p_{k}\ket{k}\bra{k}$ (26)

with $\ket{k}$ denoting the $k^{\mathrm{th}}$ Fock state, so that it is fully
described by the probability vector $\mathbf{p}\in\mathbb{R}^{\mathbb{N}}$,
with components $p_{k}$. In order to be an acceptable probability
distribution, $\mathbf{p}$ must satisfy the physicality conditions

$p_{k}\geq 0\quad\forall k,\qquad\sum\limits_{k=0}^{\infty}p_{k}=1.$ (27)

We call $\mathbb{S}$ the restriction of $\mathbb{R}^{\mathbb{N}}$ satisfying
the physicality conditions (27). Any vector $\mathbf{p}$ that belongs to
$\mathbb{S}$ corresponds to a unique phase-invariant state in $\mathcal{Q}$.

Figure 4: Pictorial representation of the various sets considered here. The
full set of quantum states is denoted as $\mathcal{Q}$, while the set of
Wigner-positive states is denoted as $\mathcal{Q}_{+}$. Then,
$\mathcal{B}_{c}$ stands for the convex hull of the set $\mathcal{B}$ of beam-
splitter states, while $\mathcal{G}_{c}$ stands for the convex hull of the set
$\mathcal{G}$ of Gaussian states. Further, $\mathcal{C}$ stands for the set of
classical states. Within all these sets, we distinguish the states that are
phase invariant, which are characterized by a probability vector
$\mathbf{p}\in\mathbb{S}$. For states in $\mathcal{Q}_{+}$, the vector
$\mathbf{p}\in\mathbb{S}_{+}$, while for states in $\mathcal{B}_{c}$, the
vector $\mathbf{p}\in\mathbb{S}_{\mathrm{b}}$. To be rigorous, we note that it
is unknown whether the phase-invariant restriction of $\mathcal{B}_{c}$ might
also contain some states such that $\mathbf{p}\notin\mathbb{S}_{\mathrm{b}}$.
We have rigorously proven this is not the case for states up to two photons
only (see below). Note also that the areas of all the above sets should not be
understood quantitatively as they are arbitrary and only meant here to
illustrate the chain of inclusion.

Now, we turn to the phase-invariant states in $\mathcal{Q}_{+}$. In order to
check that the phase-invariant state that is characterized by a vector
$\mathbf{p}\in\mathbb{S}$ is Wigner-positive, we need to verify that the
corresponding mixture of Fock states has a positive Wigner function everywhere
in phase space. This is done by using Eq. (25), so that the Wigner-positivity
condition on $\mathbf{p}$ reads as

$\sum\limits_{k=0}^{\infty}p_{k}\,(-1)^{k}\,L_{k}(t)\geq 0\qquad\forall t\geq
0,$ (28)

where we define $t=2x^{2}+2p^{2}$. Let us also define the usual radial
parameter $r=\sqrt{x^{2}+p^{2}}$, so that each value of $t$ corresponds to a
specific value of $r$ through the relation $t=2r^{2}$. When condition (28) is
fulfilled for some $t$, the Wigner function is non-negative at $r=\sqrt{t/2}$.
We call $\mathbb{S}_{+}$ the restriction of $\mathbb{S}$ satisfying the
Wigner-positivity conditions (28), so that any vector $\mathbf{p}$ in
$\mathbb{S}_{+}$ is associated with a unique phase-invariant Wigner-positive
state in $\mathcal{Q}_{+}$.

The characterization of $\mathbb{S}_{+}$ can be operated as follows. Each
value of $t$ in Eq. (28) gives the equation of a hyperplane dividing
$\mathbb{S}$ in two halves [$\mathbf{p}$ must be located on one side of the
hyperplane to guarantee that $W(r)\geq 0$ for the corresponding $r$]. Two
hyperplanes associated respectively to $t$ and $t+\mathrm{d}t$ intersect in a
(lower-dimensional) hyperplane which is at the boundary of the convex set
$\mathbb{S}_{+}$. When $t$ goes from $0$ to $\infty$, the collection of all
these intersections forms a locus of points which determines the curved
boundary of $\mathbb{S}_{+}$. Mathematically, the condition that a point
$\mathbf{p}\in\mathbb{S}$ belongs to the curved boundary of $\mathbb{S}_{+}$
is equivalent to the following condition:

$\exists t\quad\text{such
that}\quad\begin{cases}\sum\limits_{k=0}^{\infty}p_{k}\,(-1)^{k}\,L_{k}(t)=0\\\\[11.99998pt]
\sum\limits_{k=0}^{\infty}p_{k}\,(-1)^{k}\,\dfrac{\mathrm{d}}{\mathrm{d}t}L_{k}(t)=0\end{cases}$
(29)

Note that since $\mathbb{S}_{+}$ is convex, all the points in its curved
boundary are extremal points. However, other isolated extremal points may
exist, as illustrated in Fig. 2.

Figure 5: Beam-splitter state ${\hat{\sigma}}(m,n)$ obtained at the output of
a balanced beam splitter (of transmittance $\eta=\frac{1}{2}$) that is fed
with Fock states of $m$ and $n$ photons. The states ${\hat{\sigma}}(m,n)$ are
Wigner-positive phase-invariant states; hence they belong to the set
$\mathbb{S}_{+}$.

### Phase-invariant beam-splitter states in $\mathcal{B}$

The above considerations reflect the fact that characterizing the set of
phase-invariant Wigner-positive states (associated with
$\mathbf{p}\in\mathbb{S}_{+}$) remains complex. For this reason, we consider a
subset of states that are built by using a balanced beam splitter, following
the same idea as for the construction of set $\mathcal{B}$ but injecting
phase-invariant Fock states at the input. As pictured in Fig. 5, we define the
beam-splitter state $\hat{\sigma}(m,n)$ as the reduced output state of a
balanced beam splitter fed by $m$ and $n$ photons at its two inputs, that is,

$\hat{\sigma}(m,n)=\mathrm{Tr}_{2}\left[\hat{U}_{\frac{1}{2}}\left(\ket{m}\bra{m}\otimes\ket{n}\bra{n}\right)\hat{U}_{\frac{1}{2}}^{\dagger}\right].$
(30)

Thus, any state $\hat{\sigma}(m,n)$ is Wigner positive and phase invariant. It
is a mixture of Fock states with mixture coefficients given in Appendix B. We
denote as $\mathbb{S}_{\mathrm{b}}$ the set of probability vectors
$\mathbf{p}$ corresponding to all mixtures of states $\hat{\sigma}(m,n)$. It
is clear that $\mathbb{S}_{\mathrm{b}}\subset\mathbb{S}_{+}\subset\mathbb{S}$,
as depicted in Fig. 4 and discussed below.

Interestingly, the Wigner function associated with any state
$\hat{\sigma}(m,n)$ happens to have a minimum value that reaches precisely
zero [except for $\hat{\sigma}(0,0)$, which is simply the vacuum state]. In
fact, it is shown in Appendix B that whenever $m\neq n$, the Wigner function
of $\hat{\sigma}(m,n)$ always cancels at the origin in phase space. This
suggests that the states $\hat{\sigma}(m,n)$ are the extremal states of the
set of phase-invariant Wigner-positive states (those associated with
$\mathbb{S}_{+}$). However, as we will show in the following example, the
situation is more tricky as this set also admits other extremal states that
are not of the form $\hat{\sigma}(m,n)$. Hence, we will see that
$\mathbb{S}_{\mathrm{b}}\subset\mathbb{S}_{+}$ is a strict inclusion and there
exist phase-invariant Wigner-positive states that cannot be written as
mixtures of beam-splitter states $\hat{\sigma}(m,n)$.

### Example : restriction to two photons

Figure 6: Two-dimensional representation of $\mathbb{S}^{2}$ (large white
triangle) and $\mathbb{S}^{2}_{+}$ (blue zone, including dark and light blue).
The points $a$, $b$, $c$, and $d$ correspond to the beam-splitter states whose
convex closure yields $\mathbb{S}^{2}_{\mathrm{b}}$, represented as the light
blue zone. The dark blue zone stands for the subset of phase-invariant Wigner-
positive states that cannot be expressed as mixtures of beam-splitter states.
As discussed in Sec. IV, the triangle $a$-$b$-$e$ encompasses the set of
passive states while the triangle $a$-$b$-$d$ encompasses the states whose
Wigner function coincides with the Husimi $Q$ function of a state.

Let us denote by $\mathbb{S}^{n}$ and $\mathbb{S}^{n}_{+}$ the restriction of
respectively $\mathbb{S}$ and $\mathbb{S}_{+}$ that have components $p_{k}=0$
for $k>n$. As an example, let us consider the set $\mathbb{S}^{2}$, which
corresponds to mixtures of Fock states up to $n=2$, that is,

$\hat{\rho}=(1-p_{1}-p_{2})\ket{0}\bra{0}+p_{1}\ket{1}\bra{1}+p_{2}\ket{2}\bra{2}$
(31)

with $p_{1},p_{2}\geq 0$ and $p_{1}+p_{2}\leq 1$. We are interested in the
Wigner-positive subset of $\mathbb{S}^{2}$, namely, $\mathbb{S}^{2}_{+}$.
Restricting ourselves to $n=2$ makes it possible to represent
$\mathbb{S}^{2}_{+}$ in a two-dimensional plane with coordinates $p_{1}$ and
$p_{2}$ (see Fig. 6). The mathematical description of $\mathbb{S}^{2}_{+}$ was
also given in Bröcker and Werner (1995), but we analyze it here from a
physical perspective, through the prism of quantum optics. Since the beam
splitter conserves the total photon number, we know that only the states
$\hat{\sigma}(m,n)$ such that $m+n\leq 2$ belong to $\mathbb{S}^{2}_{+}$.
These states are expressed as

$\begin{split}\hat{\sigma}_{a}&\equiv\hat{\sigma}(0,0)=\hskip
8.99994pt\ket{0}\bra{0}\\\
\hat{\sigma}_{b}&\equiv\hat{\sigma}(1,0)=\frac{1}{2}\ket{0}\bra{0}+\frac{1}{2}\ket{1}\bra{1}\\\
\hat{\sigma}_{c}&\equiv\hat{\sigma}(1,1)=\frac{1}{2}\ket{0}\bra{0}+\frac{1}{2}\ket{2}\bra{2}\\\
\hat{\sigma}_{d}&\equiv\hat{\sigma}(2,0)=\frac{1}{4}\ket{0}\bra{0}+\frac{1}{2}\ket{1}\bra{1}+\frac{1}{4}\ket{2}\bra{2}\end{split}$
(32)

and their corresponding Wigner functions are displayed in Figs. 7 and 8. We
observe that the minimum value of the Wigner functions always reaches zero
(except for the vacuum state ${\hat{\sigma}}_{a}$), which reflects that these
are extremal states of the set of Wigner-positive phase-invariant states
(associated with $\mathbb{S}^{2}_{+}$).

This is confirmed in Fig. 6, where the four beam-splitter states are
represented by points $a$, $b$, $c$, and $d$: they are indeed extremal points
of the convex set $\mathbb{S}^{2}_{+}$, which appears as the blue zone
(including light and dark blue). However, as we will see, they are not the
only extremal points of $\mathbb{S}^{2}_{+}$. The complete characterization of
$\mathbb{S}^{2}_{+}$ can be done by using the Wigner-positivity conditions
(28) and (29). The derivation is done in Appendix C and leads to the following
conditions on $p_{1}$ and $p_{2}$:

$\begin{cases}p_{1}\leq\dfrac{1}{2}\\\\[10.00002pt]
p_{2}\leq\dfrac{1}{4}+\dfrac{1}{4}\sqrt{1-4p_{1}^{2}}\end{cases}$ (33)

Any state in the form (31) is Wigner positive if and only if its components
$p_{1}$ and $p_{2}$ satisfy conditions (33).

Figure 7: Wigner functions of the four beam-splitter states
$\hat{\sigma}_{a}$, $\hat{\sigma}_{b}$, $\hat{\sigma}_{c}$, and
$\hat{\sigma}_{d}$, denoted respectively as $W_{a}(x,p)$, $W_{b}(x,p)$,
$W_{c}(x,p)$, and $W_{d}(x,p)$. These four states are Wigner-positive phase-
invariant states, but, in addition, their Wigner functions touch precisely
zero (except for the vacuum state $\hat{\sigma}_{a}$) as is more evident from
Fig. 8. This fact reflects that these are extremal states of the set of
Wigner-positive phase-invariant states (associated with $\mathbb{S}^{2}_{+}$).
Figure 8: Radial Wigner functions of the four phase-invariant beam-splitter
states $\hat{\sigma}_{a}$, $\hat{\sigma}_{b}$, $\hat{\sigma}_{c}$, and
$\hat{\sigma}_{d}$, denoted respectively as $W_{a}(r)$, $W_{b}(r)$,
$W_{c}(r)$, and $W_{d}(r)$. As advertised, the minimum value of these Wigner
functions touches zero [except for the vacuum state $\hat{\sigma}_{a}$, for
which $W_{a}(r)\to 0$ as $r\to\infty$].  Figure 9: Expressing the positivity
of the radial Wigner function $W(r)$ for increasing values of $r$ corresponds
to a continuum of straight lines, which are all tangents of ellipse (35). As
an illustration, we plot as dashed lines the tangents associated with $W(r)=0$
for $r=0$, $r=1/\sqrt{2}$, $r=1$, $r=\sqrt{2}$, and $r\to\infty$. For
instance, expressing $W(0)\geq 0$ implies $p_{1}\leq 1/2$, while expressing
$W(1)\geq 0$ implies $p_{2}\leq 1/2$. For $r>1$, the positivity condition
becomes redundant, and, at the limit $r\to\infty$, it gives $p_{2}\geq 0$,
which is equivalent to the physicality condition.

Several observations can be made from Fig. 6. First, state
${\hat{\sigma}}_{a}$, which coincides with the vacuum state, is a trivial
extremal state of $\mathbb{S}^{2}_{+}$ even if its Wigner function does not
reach zero. As already mentioned, ${\hat{\sigma}}_{b}$, ${\hat{\sigma}}_{c}$,
and ${\hat{\sigma}}_{d}$ are other extremal states of $\mathbb{S}^{2}_{+}$, as
witnessed by the fact that their Wigner function vanishes at some location in
phase space. The convex set $\mathbb{S}^{2}_{+}$ has three facets. Two of them
correspond to the physicality conditions (27), i.e., $p_{1}\geq 0$ and
$p_{2}\geq 0$. The third one corresponds to condition (28) where we have set
$t=0$, which gives us $p_{0}+p_{2}\geq 1/2$ or equivalently $p_{1}\leq 1/2$.
Note that the points in these facets belong to the boundary of
$\mathbb{S}^{2}_{+}$ but are not extremal. This can be easily understood for
the third facet corresponding in Fig. 6 to the segment connecting
${\hat{\sigma}}_{b}$ to ${\hat{\sigma}}_{d}$, which both admit a zero of their
Wigner function at the same location (i.e., the origin). Note also that, in
general, the set $\mathbb{S_{+}}$ always has a facet corresponding to

$\sum\limits_{k\ \mathrm{even}}p_{k}=\dfrac{1}{2},$ (34)

which expresses the positivity of the Wigner function at $r=0$ (recall that
$t=2r^{2}$). As pictured in Fig. 9, expressing the positivity of the radial
Wigner function for increasing values of $r$ yields a continuum of straight
lines, whose locus of intersecting points forms an ellipse centered in
$(0,1/4)$, namely,

$\left(\frac{p_{1}}{1/2}\right)^{2}+\left(\frac{p_{2}-1/4}{1/4}\right)^{2}=1\,.$
(35)

The resulting constraints on $p_{1}$ and $p_{2}$ for all $r$’s are summarized
by Eq. (33).

Overall, Fig. 6 shows that the subspace $\mathbb{S}^{2}_{\mathrm{b}}$, which
is spanned by the extremal states ${\hat{\sigma}}_{a}$, ${\hat{\sigma}}_{b}$,
${\hat{\sigma}}_{c}$, and ${\hat{\sigma}}_{d}$, covers a large region of
$\mathbb{S}^{2}_{+}$ (indicated in light blue) so any point in this region can
thus be generated by a convex mixture of them. However, $\mathbb{S}^{2}_{+}$
also includes a small region (indicated in dark blue) that is located under
the ellipse defined by Eq. (35) and above the straight line $c$-$d$. This
region is thus outside the polytope $\mathbb{S}^{2}_{\mathrm{b}}$ generated by
the $\hat{\sigma}$ states, which confirms that $\mathbb{S}^{2}_{+}$ also
admits a continuum of extremal points along this ellipse.

Note finally that it is not a trivial observation to see that
$\mathbb{S}^{2}_{\mathrm{b}}$ coincides with the two-photon phase-invariant
restriction of $\mathcal{B}_{c}$ (i.e., the phase-invariant states with up to
two photons within the convex hull of beam-splitter states of $\mathcal{B}$).
Indeed, $\mathbb{S}^{2}_{\mathrm{b}}$ is defined as the convex hull of beam-
splitter states built from (phase-invariant) Fock states in Fig. 5 with up to
two photons, that is, the convex hull of
$\\{\hat{\sigma}_{a},\hat{\sigma}_{b},\hat{\sigma}_{c},\hat{\sigma}_{d}\\}$.
Since it is possible to create beam-splitter states in the setup of Fig. 3
that are phase-invariant starting from two input states that are not phase
invariant (e.g., two squeezed states with orthogonal squeezing produce a
thermal state), it might a priori be possible to build states within the two-
photon phase-invariant restriction of $\mathcal{B}_{c}$ that do not belong to
$\mathbb{S}^{2}_{\mathrm{b}}$. However, a simple argument convinces us
otherwise. First, notice that we may restrict to pure input states without
loss of generality. Since the output is a mixture with up to two photons, we
must consider input states that are either in the form

$\ket{\psi}=\ket{0}\otimes\left(a_{0}\ket{0}+a_{1}\ket{1}+a_{2}\ket{2}\right),$
(36)

or

$\ket{\psi}=\left(b_{0}\ket{0}+b_{1}\ket{1}\right)\otimes\left(c_{0}\ket{0}+c_{1}\ket{1}\right).$
(37)

In case (36), the first input is the vacuum, which is phase invariant, so that
the output state is phase invariant only if the second input state is also
phase invariant. This is easy to understand given that the output Wigner
function is a (scaled) convolution of the two input Wigner functions. In case
(37), a straightforward calculation shows us that the output state is phase
invariant only if at least one of the coefficients $b_{0}$, $b_{1}$, $c_{0}$,
or $c_{1}$ vanishes. This implies that one of the two input states must be
phase invariant, which in turns implies that the other input must be phase-
invariant too in order to ensure the phase invariance of the output. As a
result, the two-photon phase-invariant restriction of $\mathcal{B}_{c}$
coincides with the set $\mathbb{S}^{2}_{\mathrm{b}}$ (it is unknown, however,
whether this remains true for more than two photons, that is, whether the
phase-invariant restriction of $\mathcal{B}_{c}$ corresponds to the set
$\mathbb{S}_{\mathrm{b}}$ in general). Since we have found phase-invariant
Wigner-positive states outside $\mathbb{S}^{2}_{\mathrm{b}}$, this confirms
that $\mathcal{B}_{c}$ is strictly included in $\mathcal{Q}_{+}$, as
advertised earlier (see Fig. 4).

##  IV Conjectured lower bound

The conjectured lower bound on the Wigner entropy reads

$h\left(W_{\\!\hat{\rho}}\right)\geq\ln\pi+1\qquad\forall\hat{\rho}\in\mathcal{Q}_{+}$
(38)

Note that an extended version for the Wigner-Rényi entropy is also discussed
in Appendix A. We wish to prove Eq. (38) for all Wigner-positive states in
$\mathcal{Q}_{+}$ but it appeared in Sec. III that this set is hard to
characterize. In this section, we will expose the central result of our paper,
namely, a proof of this conjecture over a subset of phase-invariant Wigner-
positive states with thermodynamical relevance that are called passive states.
As a side result, we will exhibit an unexpectedly simple relation between the
extremal passive states and the beam-splitter states $\hat{\sigma}(m,n)$,
which guides us to test the conjecture over the much larger set
$\mathbb{S}_{\mathrm{b}}$ of phase-invariant Wigner-positive states.

Before doing so, let us discuss the implication of the conjecture in the
restricted subspace of phase-invariant Wigner-positive states associated with
$\mathbb{S}^{2}_{+}$. First, as a consequence of Eq. (6), we know that the
Wigner functions of ${\hat{\sigma}}_{a}$, ${\hat{\sigma}}_{b}$, and
${\hat{\sigma}}_{d}$ coincide respectively with the Husimi $Q$ functions of
$\ket{0}$, $\ket{1}$, and $\ket{2}$. Hence, the (proven) Wehrl conjecture
applied to $\ket{0}$, $\ket{1}$, and $\ket{2}$ implies that the Wigner entropy
of ${\hat{\sigma}}_{a}$, ${\hat{\sigma}}_{b}$, and ${\hat{\sigma}}_{d}$ is
indeed lower bounded by $\ln\pi+1$. Further, this naturally extends to the
subspace spanned by ${\hat{\sigma}}_{a}$, ${\hat{\sigma}}_{b}$, and
${\hat{\sigma}}_{d}$, corresponding to the triangle $a$-$b$-$d$ in Fig. 6.
Thus, the states that are located in the blue region but do not belong to this
triangle are Wigner-positive states whose Wigner function cannot be expressed
as a physical $Q$ function. This underlies the fact that conjecture (10) is
stronger than the Wehrl conjecture. In particular, let us prove that the
Wigner function of state ${\hat{\sigma}}_{c}$ cannot be written as the $Q$
function of a physical state. Reasoning by contradiction, assume there exists
an input state $\hat{\rho}$ in the setup of Fig. 1 such that the resulting
output state is ${\hat{\sigma}}_{c}$. First, since the transformation on
$\hat{\rho}$ is a (scaled) convolution with a (Gaussian) rotation-invariant
function, the Wigner function of $\hat{\rho}$ must necessarily be rotation
invariant in order to get the rotation-invariant Wigner function associated
with ${\hat{\sigma}}_{c}$. Thus, $\hat{\rho}$ must be phase invariant, that
is, a mixture of Fock states. Second, since ${\hat{\sigma}}_{c}$ does not
contain more than two photons, it is clear that $\hat{\rho}$ can only be a
mixture of $\ket{0}$, $\ket{1}$, and $\ket{2}$. However, the output state
corresponding to any such mixture precisely belongs to the triangle
$a$-$b$-$d$, which does not contain $c$. Hence, there is no state
$\hat{\rho}$.

### Passive states

Passive states are defined in quantum thermodynamics as the states from which
no work can be extracted through unitary operations Pusz and Woronowicz
(1978). If $\hat{\rho}_{p}$ is the density operator of a passive state, then
the following relation holds true for any unitary operator $\hat{U}$:

$\mathrm{Tr}\left[\hat{\rho}_{p}\hat{H}\right]\leq\mathrm{Tr}\left[\hat{U}\hat{\rho}_{p}\hat{U}^{\dagger}\hat{H}\right],$
(39)

where $\hat{H}$ is the Hamiltonian of the system. Passive states are useless
in the sense that it is not possible to decrease their energy by applying a
unitary (since a unitary conserves the entropy, any work extraction should
come with a decrease of internal energy). It can be shown that passive states
are decreasing mixtures of energy eigenstates, in the sense that if the
eigenstates are labeled with increasing energy, the associate probabilities
must be decreasing Lenard (1978). In the present paper, we are considering
eigenstates of the harmonic oscillator, which are the Fock states. A passive
state is then written as

$\hat{\rho}_{p}=\sum_{k=0}^{\infty}p_{k}\ket{k}\bra{k}\qquad\mathrm{with~{}}p_{k}\geq
p_{k+1}.$ (40)

Among the set of passive states, extremal passive states are defined as
equiprobable mixtures of the low-energy eigenstates up to some threshold. We
refer to the $n^{\mathrm{th}}$ extremal passive state as
$\hat{\varepsilon}_{n}$ and to its Wigner function as $E_{n}$. They are
defined as follows:

$\begin{split}\hat{\varepsilon}_{n}&=\dfrac{1}{n+1}\sum\limits_{k=0}^{n}\ket{k}\bra{k}\\\\[10.00002pt]
E_{n}(x,p)&=\dfrac{1}{n+1}\sum\limits_{k=0}^{n}W_{k}(x,p)\end{split}$ (41)

The states $\hat{\varepsilon}_{n}$ are called extremal 222Note that the
extremal passive states $\hat{\varepsilon}_{n}$ are very different from the
extremal Wigner-positive states, such as the states $\hat{\sigma}(m,n)$: the
extremality of $\hat{\varepsilon}_{n}$ pertains to a set of states that are
defined in state space, while the extremality of $\hat{\sigma}(m,n)$ pertains
to a distinct set of states that are defined in phase space. in the sense that
any passive state $\hat{\rho}_{p}$ can be expressed as a unique convex mixture
of extremal passive states, namely,

$\hat{\rho}_{p}=\sum\limits_{k=0}^{\infty}e_{k}\,\hat{\varepsilon}_{k}\,,$
(42)

where $p_{k}$ and $e_{k}$ are probabilities that are linked through the
relation $e_{k}=(k+1)\left(p_{k}-p_{k+1}\right)$.

In the special case of phase-invariant states within the restricted space with
up to two photons, the set of passive states corresponds to the triangle
$a$-$b$-$e$ in Fig. 6, which belongs to $\mathbb{S}^{2}_{+}$ as expected. Of
course, $a$, $b$, and $e$ correspond respectively to the extremal passive
states $\hat{\varepsilon}_{0}$, $\hat{\varepsilon}_{1}$, and
$\hat{\varepsilon}_{2}$.

### Proof of the conjecture for passive states

Let us prove the lower bound (38) for the subset of passive states
$\hat{\rho}_{p}$. First, note that passive states are known to be Wigner
positive Bastiaans (1983), a fact that will become clear from Eq. (43). Thus,
their Wigner entropy is well defined. Second, notice that, as a consequence of
the concavity of entropy, it is sufficient to prove the conjecture for all
extremal passive states $\hat{\varepsilon}_{n}$.

The main tool that we will use to carry out our proof is a formula that we
have derived from an identity involving Laguerre and Hermite polynomials Szegő
(1939), making a nontrivial link between the Wigner functions and wave
functions of the first $n$ Fock states. It reads as follows (to the best of
our knowledge, it has never appeared as such in the literature):

$\sum\limits_{k=0}^{n}W_{k}(x,p)=\sum\limits_{k=0}^{n}\psi_{k}(x)^{2}\,\psi_{n-k}(p)^{2},$
(43)

where $W_{k}$ and $\psi_{k}$ are respectively the Wigner function and wave
function of the $k^{\text{th}}$ Fock state as defined in Eqs. (24) and (25).
As a by-product, note that Eq. (43) immediately implies that all extremal
passive states $\hat{\varepsilon}_{n}$ admit a positive Wigner function; hence
the Wigner function of an arbitrary passive state is necessarily positive.
More details on the derivation of Eq. (43) can be found in Appendix D.

Let us denote the $x$ and $p$ probability densities of the $n^{\text{th}}$
Fock state as $\rho_{n}(x)=|\psi_{n}(x)|^{2}$ and
$\rho_{n}(p)=|\psi_{n}(p)|^{2}$. Their corresponding Shannon differential
entropy is defined as
$h\left(\rho_{k}(x)\right)=-\int\rho_{k}(x)\ln\rho_{k}(x)\,\mathrm{d}x$ and
$h\left(\rho_{k}(p)\right)=-\int\rho_{k}(p)\ln\rho_{k}(p)\,\mathrm{d}p$. In
the following, we refer to these quantities as $h\left(\rho_{k}\right)\equiv
h\left(\rho_{k}(x)\right)=h\left(\rho_{k}(p)\right)$. We are now ready to
lower bound the Wigner entropy of the $n^{\text{th}}$ extremal passive state
$\hat{\varepsilon}_{n}$ by using Eq. (43):

$\begin{split}h\left(E_{n}\right)&=h\left(\dfrac{1}{n+1}\sum\limits_{k=0}^{n}W_{k}(x,p)\right)\\\
&=h\left(\dfrac{1}{n+1}\sum\limits_{k=0}^{n}\psi_{k}(x)^{2}\psi_{n-k}(p)^{2}\right)\\\
&\geq\dfrac{1}{n+1}\sum\limits_{k=0}^{n}h\big{(}\rho_{k}(x)\rho_{n-k}(p)\big{)}\\\
&=\dfrac{1}{n+1}\sum\limits_{k=0}^{n}\big{(}h\left(\rho_{k}\right)+h\left(\rho_{n-k}\right)\big{)}\\\
&=\dfrac{2}{n+1}\sum\limits_{k=0}^{n}h\left(\rho_{k}\right)\\\
&\geq\ln\pi+1\end{split}$ (44)

The first inequality in Eq. (44) results from the concavity of the entropy.
Then, we use the fact that the entropy of a product distribution is the sum of
the marginal entropies. Finally, we apply the entropic uncertainty relation of
Białynicki-Birula and Mycielski Białynicki-Birula and Mycielski (1975) on Fock
states, namely, $2\,h\left(\rho_{k}\right)\geq\ln\pi+1$, $\forall k$. We have
thus proven the conjecture for all extremal passive states and this proof
naturally extends to the whole set of passive states. ∎

Let us now prove that a slightly tighter lower bound can be derived for the
Wigner entropy of passive states by exploiting Eq. (42), namely the fact that
these states can be expressed as convex mixtures of extremal passive states
$\hat{\varepsilon}_{n}$ (in place of decreasing mixtures of Fock states). We
denote the Wigner function of the passive state $\hat{\rho}_{p}$ as
$W_{P}(x,p)$ and bound its Wigner entropy as follows:

$\begin{split}h\left(W_{P}\right)&=h\left(\sum\limits_{k=0}^{\infty}e_{k}\,E_{k}(x,p)\right)\\\
&\geq\sum\limits_{k=0}^{\infty}e_{k}\,h\big{(}E_{k}(x,p)\big{)}\\\
&=\sum\limits_{k=0}^{\infty}(k+1)\left(p_{k}-p_{k+1}\right)h\big{(}E_{k}(x,p)\big{)}\\\
&\geq\sum\limits_{k=0}^{\infty}(k+1)\left(p_{k}-p_{k+1}\right)\dfrac{2}{k+1}\sum\limits_{j=0}^{k}h\left(\rho_{j}\right)\\\
&=2\sum\limits_{k=0}^{\infty}\sum\limits_{j=0}^{k}\left(p_{k}-p_{k+1}\right)h\left(\rho_{j}\right)\\\
&=2\sum\limits_{j=0}^{\infty}\sum\limits_{k=j}^{\infty}\left(p_{k}-p_{k+1}\right)h\left(\rho_{j}\right)\\\
&=2\sum\limits_{j=0}^{\infty}p_{j}\,h\left(\rho_{j}\right).\qed\end{split}$
(45)

The first inequality in (45) comes from the concavity of entropy over the
convex set of extremal states, while the second inequality is obtained from
Eq. (44). The final expression is a stronger lower bound on the Wigner entropy
of any passive state which reads as

$h\left(\sum\limits_{k=0}^{\infty}p_{k}\,W_{k}\right)\geq
2\sum\limits_{k=0}^{\infty}p_{k}\,h\left(\rho_{k}\right)$ (46)

and is valid as soon as the probabilities $p_{k}$ are decreasing, that is,
$p_{k}\geq p_{k+1}$.

It is tempting to extrapolate that the bound (46) remains valid beyond the set
of passive states. We know indeed that there exist phase-invariant Wigner-
positive states that are not passive states (in Fig. 6, these are the states
within the light blue region that do not belong to the triangle $a$-$b$-$e$).
As long as the coefficients $p_{k}$ are such that the corresponding state is
Wigner positive, it has a well-defined Wigner entropy and we may expect that
the lower bound (46) applies. Unfortunately, our numerical simulations have
shown that relation (46) does not hold in general for nonpassive (Wigner-
positive) states. Of course, we conjecture that relation (38) does hold for
such states and we have not found any counterexample.

### Relation between the extremal passive states and the beam-splitter states

Let us now highlight an interesting relation between extremal passive states
$\hat{\varepsilon}_{n}$ and the beam-splitter states $\hat{\sigma}(m,n)$ that
we defined in Sec. III. To this purpose, we consider a mixed quantum state of
two modes (or harmonic oscillators) which we denote as $\hat{\tau}_{n}$. It is
defined as an equal mixture of all two-mode states with a total photon number
(or energy) equal to $n$, namely,

$\hat{\tau}_{n}=\dfrac{1}{n+1}\sum\limits_{k=0}^{n}\ket{k}\bra{k}\otimes\ket{n-k}\bra{n-k}.$
(47)

This state is maximally mixed over the set of states with total energy $n$, so
that it is invariant under any unitary transformation that preserves the total
energy. In particular, it is invariant under the action of a balanced beam
splitter, which implies the identity $\hat{U}_{1/2}\ \hat{\tau}_{n}\
\hat{U}_{1/2}^{\dagger}=\hat{\tau}_{n}$. After partial tracing over the second
mode, we obtain

$\mathrm{Tr}_{2}\left[\hat{\tau}_{n}\right]=\dfrac{1}{n+1}\sum\limits_{k=0}^{n}\ket{k}\bra{k}\,,$
(48)

which is simply the extremal state $\hat{\varepsilon}_{n}$. Alternatively,
exploiting the invariance under $\hat{U}_{1/2}$ and recalling the definition
of the beam-splitter states $\hat{\sigma}(m,n)$, we have

$\mathrm{Tr}_{2}\left[\hat{\tau}_{n}\right]=\dfrac{1}{n+1}\sum\limits_{k=0}^{n}\hat{\sigma}(k,n-k).$
(49)

This establishes an interesting link between the extremal passive states and
the beam-splitter states, namely,

$\hat{\varepsilon}_{n}=\dfrac{1}{n+1}\sum\limits_{k=0}^{n}\hat{\sigma}(k,n-k).$
(50)

Expressed in terms of Wigner function, this translates as

$\sum\limits_{k=0}^{n}W_{k}(x,p)=\sum\limits_{k=0}^{n}S_{(k,n-k)}(x,p),$ (51)

where $S_{(m,n)}$ denotes the Wigner function of $\hat{\sigma}(m,n)$.

It is instructive to compare Eq. (51) with Eq. (43). Extremal passive states
$\hat{\varepsilon}_{n}$ are defined as mixtures of Fock states [see Eq. (40)],
which possess each a nonpositive Wigner function (except for the vacuum). This
is at the heart of the difficulty of proving the conjecture: we cannot give a
meaning to the Wigner entropy of a Fock state (except for the vacuum), so the
convex decomposition of a state into Fock states cannot be used to bound its
Wigner entropy. In this context, both Eqs. (43) and (51) have the crucial
interest to provide the decomposition of the Wigner function of an extremal
passive state into a sum of positive functions. However, with Eq. (43), these
positive functions do not correspond to physical Wigner functions. Numerical
simulations indeed show that in general $\psi_{k}(x)^{2}\psi_{n-k}(p)^{2}$ is
not a physically acceptable Wigner function (it is positive but does not
correspond to a positive-semidefinite density operator). On the contrary, Eq.
(51) exhibits the decomposition of an extremal passive state into states
$\hat{\sigma}(m,n)$, which are Wigner-positive quantum states as we have
shown.

The set spanned by the states $\hat{\sigma}(m,n)$ associated with
$\mathbb{S}_{\mathrm{b}}$ is obviously bigger than the set of passive states
and offers a nice playground for testing our conjecture. Indeed, each state
$\hat{\sigma}(m,n)$ is Wigner positive so it has a well-defined Wigner
entropy. Figure 10 displays the Wigner entropy of the states
$\hat{\sigma}\left(m,n\right)$ as computed numerically up to $m,n=30$. As
expected, the minimum Wigner entropy $\ln\pi+1$ is reached for the vacuum
state $\sigma\left(0,0\right)=\ket{0}\bra{0}$, so it follows that the
conjecture holds for whole set $\mathbb{S}_{\mathrm{b}}$ due the concavity of
the entropy. Of course, this is based on numerical evidence since we do not
have an analytical proof that $h(S_{(m,n)})\geq\ln\pi+1$. Further, although
the set $\mathbb{S}_{\mathrm{b}}$ is much bigger than the set of passive
states, it still does not encompass the whole set of phase-invariant Wigner-
positive states $\mathbb{S}_{+}$, as evidenced by Fig. 6.

Figure 10: Wigner entropy of the beam-splitter states $\hat{\sigma}(m,n)$ as
computed numerically for $m,n=0,1,\dots,\,30$. It appears that the Wigner
entropy increases monotonically for increasing values of $m$ and $n$.

##  V Conclusion

We have promoted the Wigner entropy of a quantum state as a distinct
information-theoretical measure of its uncertainty in phase space. Although it
is, by definition, restricted to Wigner-positive states, the fact that such
states form a convex set makes it a useful physical quantity. Since it is a
concave functional of the state, we naturally turn to its lower bound over the
convex set of Wigner-positive states. We conjecture that this lower bound is
$\ln\pi+1$, which is the value taken on by the Wigner entropy of all Gaussian
pure states. The latter then play the role of minimum Wigner-uncertainty
states.

This conjecture is consistent with the Hudson theorem, whereby all Wigner-
positive pure states must be Gaussian states, thus states reaching the value
$\ln\pi+1$. The conjecture also implies a lower bound on the sum of the
marginal entropies of $x$ and $p$; hence it results in a tightening of the
entropic uncertainty relation due to Bialynicki-Birula and Mycielski that is
very natural from the point of view of Shannon information theory (the Wigner
entropy accounts for $x$-$p$ correlations since it is the joint entropy of $x$
and $p$). Of course, it also implies the Heisenberg uncertainty relation
formulated in terms of variances of $x$ and $p$. Furthermore, the conjecture
implies (but is stronger than) Wehrl conjecture, notoriously proven by Lieb.
It is supported by several elements. First, we have provided in Sec. IV an
analytical proof for a subset of phase-invariant Wigner-positive states,
namely, the passive states. Second, this was complemented by a semianalytical
seminumerical proof for the larger set of phase-invariant states associated
with $\mathbb{S}_{\mathrm{b}}$. Third, we also carried out an extensive
numerical search for counterexamples in $\mathcal{Q}_{+}$ but could not find
any.

Given that the Wigner entropy is only properly defined for Wigner-positive
states, we have also been led to investigate the structure of such states in
Sec. III. We have put forward an extensive technique to produce Wigner-
positive states using a balanced beam splitter. In particular, we have focused
on the beam-splitter states $\hat{\sigma}(m,n)$ and have highlighted their
connection with the (smaller) set of passive states and (larger) set of phase-
invariant Wigner-positive states. We have also found an unexpectedly simple
relation between the states $\hat{\sigma}(m,n)$ and the extremal passive
states.

The Wigner entropy enjoys various reasonable properties; in particular it is
invariant over all symplectic transformations in phase space or equivalently
all Gaussian unitaries in state space. Its excess with respect to $\ln\pi+1$
is an asymptotic measure of the number of random bits that are needed to
generate a sample of the Wigner function from the vacuum state. More
generally, since the Wigner entropy is the Shannon differential entropy of the
Wigner function, viewed as a genuine probability distribution, it inherits all
its key features. For example, we may easily extend to Wigner entropies the
celebrated entropy power inequality Cover and Thomas (1991), which relates to
the entropy of the convolution of probability distributions. Consider the
setup of Fig. 3 where the input state is again a product state
$\hat{\rho}_{A}\otimes\hat{\rho}_{B}$ but the beamsplitter now has an
arbitrary transmittance $\eta$, so that the output state reads

$\hat{\sigma}=\mathrm{Tr}_{2}\left[\hat{U}_{\eta}\left(\hat{\rho}_{A}\otimes\hat{\rho}_{B}\right)\hat{U}_{\eta}^{\dagger}\right].$
(52)

Let us restrict to the special case where both $\hat{\rho}_{A}$ and
$\hat{\rho}_{B}$ are Wigner-positive states, which of course implies that
$\hat{\rho}_{A}\otimes\hat{\rho}_{B}$ is Wigner positive as well as
${\hat{\sigma}}$ (even if $\eta\neq 1/2$). Thus, $\hat{\rho}_{A}$, and
$\hat{\rho}_{B}$, and ${\hat{\sigma}}$ all have a well-defined Wigner entropy,
which we denote respectively as $h_{A}$, $h_{B}$, and $h_{\textrm{out}}$.
Since the beam splitter effects the affine transformation
$x_{\textrm{out}}=\sqrt{\eta}\,x_{A}+\sqrt{1-\eta}\,x_{B}$ and
$p_{\textrm{out}}=\sqrt{\eta}\,p_{A}+\sqrt{1-\eta}\,p_{B}$ in phase space, we
may simply treat this as a convolution formula for probability distributions.
Hence, the entropy power inequality directly applies to the Wigner entropy.
Defining the Wigner entropy-power 333We define the Wigner entropy-power $N$ of
a (Wigner-positive) state as the entropy power of the Wigner function of the
state. Here, the entropy power is defined following Shannon information theory
for a pair of continuous variables $x$ and $p$, namely $N=(2\pi
e)^{-1}\textrm{e}^{h(x,p)}$, where $h$ stands for the Shannon differential
entropy.  of the two input states as

$N_{A}=(2\pi e)^{-1}\textrm{e}^{h_{A}},\quad N_{B}=(2\pi
e)^{-1}\textrm{e}^{h_{B}},$ (53)

and the Wigner entropy-power of the output state as

$N_{\textrm{out}}=(2\pi e)^{-1}\textrm{e}^{h_{\textrm{out}}},$ (54)

we obtain the Wigner entropy-power inequality

$N_{\textrm{out}}\geq\eta\,N_{A}+(1-\eta)N_{B}.$ (55)

This is equivalent to a nontrivial lower bound on the Wigner entropy of the
output state $\hat{\sigma}$, namely, $h(W_{\hat{\sigma}})\geq
h(W_{\hat{\sigma}_{G}})$, where $\hat{\sigma}_{G}$ denotes the Gaussian output
state obtained if each input state is replaced by the phase-invariant Gaussian
state (i.e., thermal states) with the same Wigner entropy. This illustrates
the physical significance of the Wigner entropy.

Defining the Wigner entropy for Wigner-positive states might also be a good
starting point for investigating the states that are not within
$\mathcal{Q}_{+}$ and whose Wigner function admits a negative region, hence
indicating their nonclassicality and potential computational advantage (we
recall that Wigner-positive states are efficiently simulatable classically
Mari and Eisert (2012)). Just as the characterization of separable states
helps understand the advantage offered by entanglement and leads to a resource
theory of entanglement, we may envisage building a resource theory of Wigner
negativity based on Wigner entropies along the lines of the resource theory of
quantum non-Gaussianity Zhuang  _et al._ (2018); Albarelli  _et al._ (2018),
going beyond witnesses of Wigner negativity Chabaud  _et al._ (2021).

Finally, a natural extension of the present work is to consider more than a
single harmonic oscillator (or bosonic mode) as we expect that all properties
of the Wigner entropy and especially conjecture (10) will generalize. Further,
following the lines of a recent work Floerchinger  _et al._ (2021), we might
investigate the detection of entanglement in continuous-variable states by
defining a Wigner conditional entropy and Wigner mutual information. Let us
mention that this work is part of a broader project. The key observation is
that the Wigner function of Wigner-positive states can be interpreted as a
true probability distribution. Hence, we can take advantage of this
observation and adapt all standard tools of probability theory (here, we have
applied Shannon information theory to define the Wigner entropy). In this
context, the theory of majorization Marshall  _et al._ (1979) has proved to be
another powerful tool, and it notably allows to formulate a generalization of
Wehrl conjecture Lieb (2002). In a forthcoming paper Van Herstraeten  _et al._
(2021), we use the theory of majorization to state a stronger conjecture on
the uncertainty content of Wigner functions. This enables us, for instance, to
demonstrate analytically the lower bound on $h(W)$ for all phase-invariant
Wigner-positive states in $\mathbb{S}^{2}_{+}$, including the dark blue
region.

### Note added

We have learned that our method for generating positive Wigner functions with
a 50:50 beam splitter as explained in Appendix B has recently also been
described in Becker  _et al._ (2021).

### Acknowledgments

The authors warmly thank Christos Gagatsos, Anaelle Hertz, Michael G. Jabbour
and Karol Życzkowski for helpful discussions on this subject. Z.V.H.
acknowledges a fellowship from the FRIA foundation (F.R.S.-FNRS). N.J.C.
acknowledges support by the F.R.S.-FNRS under Project No. T.0224.18 and by the
EC under project ShoQC within ERA-NET Cofund in Quantum Technologies
(QuantERA) program.

##  Appendix A Wigner-Rényi entropy

The Shannon differential entropy is an uncertainty measure that belongs to a
broader family, known as Rényi differential entropies. Just as we defined the
Wigner entropy of a Wigner-positive state as the Shannon differential entropy
of its Wigner function, it is natural to define the Wigner-Rényi entropy of a
Wigner-positive state as

$h_{\alpha}\left(W\right)=\frac{1}{1-\alpha}\ln\left(\iint[W(x,p)]^{\alpha}\,\mathrm{d}x\,\mathrm{d}p\right),$
(56)

where $\alpha\neq 1$ is a real non-negative parameter.

Interestingly, some values of $\alpha$ are endowed with a special meaning.
Denoting as $\mathrm{supp}(W)$ the part of the domain of $W$ where $W>0$ and
denoting as $\nu$ the Lebesgue measure, we have
$h_{0}(W)=\ln(\nu[\mathrm{supp}(W)])$ when the parameter $\alpha=0$. This
diverges when applied to any Wigner function $W$ since the size of the support
of $W$ is infinite. In the limit $\alpha\rightarrow 1$, $h_{\alpha}$ tends to
the Shannon differential entropy, so that $h_{1}(W)$ coincides with $h(W)$.
The Rényi entropy with parameter $\alpha=2$ is sometime called the collision
entropy, and, applied to a Wigner function $W$, it is related to the purity of
the corresponding state. Denoting the purity as
$\mu=\mathrm{Tr}\left[\hat{\rho}^{2}\right]=2\pi\iint[W(x,p)]^{2}\,\mathrm{d}x\,\mathrm{d}p$,
we have the relation $h_{2}(W)=\ln(2\pi/\mu)$. Finally, the case
$\alpha\rightarrow\infty$ can be related to the maximum value of $W$ as
$h_{\infty}(W)=-\ln[\max_{x,p}W(x,p)]$.

Note that, following the same reasoning as in Sec. II, we observe that the
Wigner-Rényi entropy is invariant under symplectic transformations in phase
space (i.e., Gaussian unitaries in state space). The Wigner-Rényi entropy of
the vacuum state (or any pure Gaussian state) gives

$h_{\alpha}(W_{0})=\ln\pi+\frac{\ln\alpha}{\alpha-1}.$ (57)

Then, in the same spirit as conjecture (10), we conjecture that the Wigner-
Rényi entropy of any Wigner-positive state is lower bounded by the value it
takes for the vacuum:

$h_{\alpha}\left(W_{\\!\hat{\rho}}\right)\geq
h_{\alpha}\left(W_{0}\right)\qquad\forall\hat{\rho}\in\mathcal{Q}_{+}$ (58)

Of course, it coincides with conjecture (10) when $\alpha\rightarrow 1$. Let
us examine this Wigner-Rényi conjecture for other special values of the
parameter $\alpha$, namely,

$\displaystyle h_{2}(W)$ $\displaystyle\geq\ln 2\pi,$ (59) $\displaystyle
h_{\infty}(W)$ $\displaystyle\geq\ln\pi.$ (60)

For $\alpha=2$, the fact that the purity $\mu$ is upper bounded by $1$ implies
Eq. (59). Also, the Wigner function of any state is upper bounded by $1/\pi$,
which implies Eq. (60) for $\alpha\to\infty$. Furthermore, for $\alpha=0$, the
Wigner-Rényi conjecture implies that the support of any Wigner function is
unbounded, which is a well-known fact. These elements support the validity of
the Wigner-Rényi conjecture and especially conjecture (10) when
$\alpha\rightarrow 1$.

##  Appendix B A balanced beam-splitter produces Wigner-positive states

In this Appendix, we show that when a balanced beam splitter is fed by a two-
mode separable input, then its reduced single-mode output is Wigner positive.
We consider the following setup:

$\hat{\sigma}=\mathrm{Tr}_{B}\left[\hat{U}_{\frac{1}{2}}\left(\hat{\rho}_{A}\otimes\hat{\rho}_{B}\right)\hat{U}_{\frac{1}{2}}^{\dagger}\right],$
(61)

where $\hat{U}_{\eta}$ is the unitary operator of the beam splitter,

$\hat{U}_{\eta}=\exp\left(\theta\left(\hat{a}^{\dagger}\hat{b}-\hat{a}\hat{b}^{\dagger}\right)\right),$
(62)

such that $\eta=\cos^{2}\theta$ and $0\leq\theta\leq\pi/2$. We are going to
show that $\hat{\sigma}$ is Wigner positive when $\eta=1/2$:

$W_{\hat{\sigma}}(x,p)\geq 0\qquad\forall x,p.$ (63)

Our proof has two parts. In the first part we show that the action of a beam
splitter in phase space corresponds to a convolution between the Wigner
functions of the two inputs. In the second part, we show that the convolution
of two Wigner functions corresponds in state space to the overlap between two
density operators, which is always a non-negative quantity. Finally, we give
some further observations regarding the $\hat{\sigma}(m,n)$ states which are
built using that setup.

###  B.1 Convolution in phase space

The action of a beam splitter on the mode operators is described by the
following transformation:

$\begin{pmatrix}\hat{a}^{\prime}\\\
\hat{b}^{\prime}\end{pmatrix}=\begin{pmatrix}\sqrt{\eta}&\sqrt{1-\eta}\\\
-\sqrt{\eta}&\sqrt{\eta}\end{pmatrix}\begin{pmatrix}\hat{a}\\\
\hat{b}\end{pmatrix}.$ (64)

From this, we derive the expression of the old quadrature operators
$\hat{x}_{A}$, $\hat{x}_{B}$, $\hat{p}_{A}$, and $\hat{p}_{B}$ as a function
of the new quadrature operators $\hat{x}_{A}^{\prime}$,
$\hat{x}_{B}^{\prime}$, $\hat{p}_{A}^{\prime}$, and $\hat{p}_{B}^{\prime}$:

$\begin{cases}\hat{x}_{A}&=\sqrt{\eta}\ \hat{x}_{A}^{\prime}-\sqrt{1-\eta}\
\hat{x}_{B}^{\prime},\\\\[5.0pt] \hat{p}_{A}&=\sqrt{\eta}\
\hat{p}_{A}^{\prime}-\sqrt{1-\eta}\ \hat{p}_{B}^{\prime},\\\\[5.0pt]
\hat{x}_{B}&=\sqrt{1-\eta}\ \hat{x}_{A}^{\prime}+\sqrt{\eta}\
\hat{x}_{B}^{\prime},\\\\[5.0pt] \hat{p}_{B}&=\sqrt{1-\eta}\
\hat{p}_{A}^{\prime}+\sqrt{\eta}\ \hat{p}_{B}^{\prime}.\end{cases}$ (65)

The two-mode Wigner function of the output $W^{\prime}$ is then obtained
through the following relation:

$\begin{split}W^{\prime}(x_{A}^{\prime},p_{A}^{\prime},x_{B}^{\prime},p_{B}^{\prime})&=W(x_{A},p_{A},x_{B},p_{B})\\\
&=W_{A}(x_{A},p_{A})W_{B}(x_{B},p_{B}),\end{split}$ (66)

where $W_{A}$ and $W_{B}$ are the Wigner functions of respectively
$\hat{\rho}_{A}$ and $\hat{\rho}_{B}$. Computing the Wigner function of
$\hat{\sigma}$ is done by integrating
$W^{\prime}(x_{A}^{\prime},p_{A}^{\prime},x_{B}^{\prime},p_{B}^{\prime})$ over
the variables $x_{B}^{\prime}$, $p_{B}^{\prime}$:

$\begin{split}W_{\hat{\sigma}}\left(x_{A}^{\prime},p_{A}^{\prime}\right)=\iint\mathrm{d}x_{B}^{\prime}\mathrm{d}p_{B}^{\prime}\\\
\times W_{A}\left(\sqrt{\eta}\ x_{A}^{\prime}-\sqrt{1-\eta}\
x_{B}^{\prime},\sqrt{\eta}\ p_{A}^{\prime}-\sqrt{1-\eta}\
p_{B}^{\prime}\right)\\\ \times W_{B}\left(\sqrt{1-\eta}\
x_{A}^{\prime}+\sqrt{\eta}\ x_{B}^{\prime},\sqrt{1-\eta}\
p_{A}^{\prime}+\sqrt{\eta}\ p_{B}^{\prime}\right).\end{split}$ (67)

That expression finds a natural writing by introducing the new variables
$x^{\prime\prime}$, $p^{\prime\prime}$:

$\begin{cases}x^{\prime\prime}=\eta\
x_{A}^{\prime}-\sqrt{\eta(1-\eta)}x_{B}^{\prime},\\\\[10.00002pt]
p^{\prime\prime}=\eta\
p_{A}^{\prime}-\sqrt{\eta(1-\eta)}p_{B}^{\prime},\end{cases}$ (68)

so that it reduces to

$\begin{split}W_{\hat{\sigma}}\left(x_{A}^{\prime},p_{A}^{\prime}\right)=\iint\mathrm{d}x^{\prime\prime}\mathrm{d}p^{\prime\prime}\frac{1}{\eta}W_{A}\left(\frac{x^{\prime\prime}}{\sqrt{\eta}},\frac{p^{\prime\prime}}{\sqrt{\eta}}\right)\\\
\times\hskip
5.0pt\frac{1}{1-\eta}W_{B}\left(\frac{x_{A}^{\prime}-x^{\prime\prime}}{\sqrt{1-\eta}},\frac{p_{A}^{\prime}-x^{\prime\prime}}{\sqrt{1-\eta}}\right).\end{split}$
(69)

Equation (69) shows that the action of a beam splitter on a product input
corresponds to a convolution between $W_{A}$ and $W_{B}$ rescaled,
respectively, by a factor $\sqrt{\eta}$ and $\sqrt{1-\eta}$. In the case of a
balanced beam splitter, the value of the parameter $\eta$ is $1/2$, and the
rescaling values are equal. Introducing the new variables
$\tilde{x}=\sqrt{2}x^{\prime\prime}$ and $\tilde{p}=\sqrt{2}p^{\prime\prime}$,
we can then write the previous expression as

$\begin{split}W_{\hat{\sigma}}\left(x_{A}^{\prime},p_{A}^{\prime}\right)=2\iint\mathrm{d}\tilde{x}\mathrm{d}\tilde{p}W_{A}\left(\tilde{x},\tilde{p}\right)\\\
\times\hskip
5.0ptW_{B}\left(\sqrt{2}x_{A}^{\prime}-\tilde{x},\sqrt{2}p_{A}^{\prime}-\tilde{p}\right).\end{split}$
(70)

###  B.2 State space picture

Equation (70) can be expressed in state space formalism. To that purpose, we
recall the usual rotation and displacement operators acting on mode $\hat{b}$:

$\begin{split}\hat{R}(\varphi)&=\exp(-i\varphi\hat{b}^{\dagger}\hat{b}),\\\
\hat{D}(\alpha)&=\exp\left(\alpha\hat{b}^{\dagger}-\alpha^{\ast}\hat{b}\right).\end{split}$
(71)

Let us define $\hat{\rho}^{\prime}_{B}$ as the result of a particular
combination of $\hat{R}$ and $\hat{D}$ acting on $\hat{\rho}_{B}$:

$\hat{\rho}^{\prime}_{B}=\hat{D}(x_{\alpha}+ip_{\alpha})\hat{R}(\pi)\hat{\rho}_{B}\hat{R}^{\dagger}(\pi)\hat{D}^{\dagger}(x_{\alpha}+ip_{\alpha}).$
(72)

The Wigner function of $\hat{\rho}^{\prime}_{B}$ can be expressed from the
Wigner function of $\hat{\rho}_{B}$ as follows:

$\begin{split}\hat{\rho}_{B}\quad&\longleftrightarrow\quad W_{B}(x,p)\\\
\hat{\rho}^{\prime}_{B}\quad&\longleftrightarrow\quad
W_{B}(\sqrt{2}x_{\alpha}-x,\sqrt{2}p_{\alpha}-p).\end{split}$ (73)

The last ingredient of our proof is the expression of the overlap between two
quantum states, which is always non-negative. That quantity can be expressed
equivalently in the formalism of state space or phase space:

$\mathrm{Tr}\left[\hat{\rho}_{1}\hat{\rho}_{2}\right]=2\pi\iint\mathrm{d}x\mathrm{d}pW_{1}(x,p)W_{2}(x,p)\geq
0.$ (74)

Combining Eqs. (70), (73), and (74) gives us the expression of
$W_{\hat{\sigma}}$ as an overlap between two quantum states:

$W_{\hat{\sigma}}(x_{A}^{\prime},p_{A}^{\prime})=\frac{1}{\pi}\mathrm{Tr}\left[\hat{\rho}_{A}\hat{\rho}^{\prime}_{B}\right]\geq
0,$ (75)

where the $x_{A}^{\prime},p_{A}^{\prime}$ dependence is hidden in
$\hat{\rho}_{B}^{\prime}$. This concludes our proof, and we have shown that
$\hat{\sigma}$ is Wigner positive. The proof naturally expands from product
inputs to separable inputs since the mixing of Wigner-positive states remains
Wigner positive.

###  B.3 Beam-splitter $\hat{\sigma}(m,n)$ states

Beam-splitter $\hat{\sigma}(m,n)$ states are defined in Sec. III and play a
particular role in this paper. We give here the expression of their density
operator decomposed onto the basis of Fock states:

$\begin{split}\hat{\sigma}(m,n)=\left(m!n!2^{m}2^{n}\right)^{-1}\sum\limits_{z=0}^{m+n}\sum\limits_{i=\max(0,z-n)}^{\min(z,m)}\sum\limits_{j=\max(0,z-n)}^{\min(z,m)}\\\
\binom{m}{i}\binom{n}{z-i}\binom{m}{j}\binom{n}{z-j}(-1)^{i+j}z!(m+n-z)!\ket{z}\bra{z},\end{split}$
(76)

where $\ket{z}\bra{z}$ is a projector onto the $z^{\text{th}}$ Fock state.

Taking advantage of Eqs. (72) and (75), we can easily show that the Wigner
function of $\hat{\sigma}(m,n)$ (that we write $S_{(m,n)}$) cancels at the
origin when $m\neq n$. Indeed, since Fock states are invariant under rotation,
it follows that

$S_{(m,n)}(0,0)=\frac{1}{\pi}\mathrm{Tr}\left[\ket{m}\bra{m}\ket{n}\bra{n}\right]=\frac{1}{\pi}\delta_{mn},$
(77)

where $\delta_{mn}$ is the Kronecker delta.

##  Appendix C Wigner-positivity conditions for phase-invariant mixtures up to
2 photons

In this Appendix, we derive the conditions that apply to a mixture of the
first three Fock states (0, 1, and 2) such that it has a non-negative Wigner
function. We are considering any state that can be written as

$\hat{\rho}=\left(1-p_{1}-p_{2}\right)\ket{0}\bra{0}+p_{1}\ket{1}\bra{1}+p_{2}\ket{2}\bra{2},$
(78)

where $p_{1},p_{2}\geq 0$ and $p_{1}+p_{2}\leq 1$. We first identify a
condition on $p_{1}$ and $p_{2}$ that is equivalent to the Wigner positivity
of $\hat{\rho}$. Then we compute the geometrical locus of points in the
$(p_{1},p_{2})$ plane that ensures Wigner positivity, and extremal Wigner
positivity.

###  C.1 Equivalent condition of Wigner-positivity

The Wigner function of a Fock state is radial and reads as

$W_{n}(r)=\frac{1}{\pi}(-1)^{n}\exp(-r^{2})L_{n}(2r^{2}),$ (79)

where we use the non-negative radial parameter $r=\sqrt{x^{2}+p^{2}}$. We
recall the three first Laguerre polynomials:

$\begin{split}L_{0}(x)&=1,\\\ L_{1}(x)&=-x+1,\\\
L_{2}(x)&=\frac{1}{2}x^{2}-2x+1.\end{split}$ (80)

Using this, we can express the Wigner function of $\hat{\rho}$ as

$W(r)=\dfrac{1}{\pi}\exp(-r^{2})\left(2p_{2}r^{4}+\left(2p_{1}-4p_{2}\right)r^{2}+1-2p_{1}\right).$
(81)

We want to identify the set of possible values of $p_{1},p_{2}$ such that
$\hat{\rho}$ is Wigner positive. Introducing the parameter $t=2r^{2}$, we can
write an equivalent condition to $W(r)\geq 0$ $\forall r\geq 0$ as

$\frac{1}{2}p_{2}t^{2}+(p_{1}-2p_{2})t+1-2p_{1}\geq 0\qquad\forall t\geq 0.$
(82)

###  C.2 Locus of Wigner-positivity in the $(p_{1},p_{2})$ plane

Equation (82) is a second-order polynomial with a non-negative coefficient
associated to $t^{2}$. We want to have non-negative values for all $t\geq 0$.
This is possible either if its discriminant $\Delta$ is nonpositive
($\Delta\leq 0$), or if both its roots correspond to $t\leq 0$.

Let us examine the latter possibility first. For a second-order polynomial
equation defined by $at^{2}+bt+c=0$, the sum of its roots is $-b/a$ and their
product is $c/a$. The two roots are nonpositive if their sum is nonpositive
and their product is non-negative. Applied to Eq. (82), this gives the
following conditions:

$\begin{cases}p_{1}\geq 2p_{2}\\\\[10.00002pt]
p_{1}\leq\frac{1}{2}.\end{cases}$ (83)

Condition (83) describes a locus which is the intersection of two half planes.
We now check the discriminant condition. The discriminant is equal to
$\Delta=4p_{2}^{2}-2p_{2}+p_{1}^{2}$, so that the condition $\Delta\leq 0$ can
be written as

$\left(\frac{p_{1}}{1/2}\right)^{2}+\left(\frac{p_{2}-1/4}{1/4}\right)^{2}\leq
1.$ (84)

Condition (84) describes an ellipse. Note that the union of the sets
determined by conditions (83) and (84) alongside with the physicality
conditions can be summarized as

$\begin{cases}p_{1}\leq\frac{1}{2}\\\\[10.00002pt]
p_{2}\leq\frac{1}{4}+\frac{1}{4}\sqrt{1-4p_{1}^{2}},\end{cases}$ (85)

with the additional constraint that $p_{1},p_{2}\geq 0$. Figure 11 illustrates
the geometrical locus associated to the different conditions.

Figure 11: Geometrical locus of Wigner-positivity within the $(p_{1},p_{2})$
plane, corresponding to the boundary of the dark blue region
$\mathbb{S}^{2}_{+}$ satisfying Eq. (85). The boundary points and curve of
$\mathbb{S}^{2}_{+}$ that are extremal are shown in bold. The ellipse ($A$)
corresponds to the region where $(p_{1},p_{2})$ is such that Eq. (82) is never
negative. The semi-infinite triangular region ($B$) corresponds to values of
$(p_{1},p_{2})$ such that Eq. (82) becomes negative only for negative values
of $t$. The dashed lines forming a triangle define the physicality limits,
that is, $p_{1},p_{2}\geq 0$ and $p_{1}+p_{2}\leq 1$. The union of ($A$) and
($B$) that belongs to the physicality triangle yields the dark blue region
$\mathbb{S}^{2}_{+}$.

###  C.3 Locus of extremal Wigner-positive states

Let us define $P(t)$ as the second-order polynomial described in Eq. (82). We
refer to the first-order derivative of $P(t)$ with respect to $t$ as
$P^{\prime}(t)$:

$\begin{split}P(t)&=\frac{1}{2}p_{2}t^{2}+(p_{1}-2p_{2})t+1-2p_{1},\\\\[10.00002pt]
P^{\prime}(t)&=p_{2}t+p_{1}-2p_{2}.\end{split}$ (86)

The locus of extremal Wigner-positive states is the set of $(p_{1},p_{2})$
such that

$\exists t\geq 0\quad\text{such that}\quad\begin{cases}P(t)=0\\\
P^{\prime}(t)=0.\end{cases}$ (87)

The condition $P^{\prime}(t)=0$ is satisfied at $t=2-p_{1}/p_{2}$. Injecting
that value of $t$ in $P(t)=0$ gives us the following equation:

$4p_{2}^{2}-2p_{2}+p_{1}^{2}=0.$ (88)

with the additional constraint that $2p_{2}\geq p_{1}$, since $t\geq 0$. This
describes an arc of an ellipse, which we can parametrize as follows:

$\begin{cases}p_{1}=\dfrac{1}{2}\sqrt{1-a^{2}}\\\\[10.00002pt]
p_{2}=\dfrac{1}{4}(a+1),\end{cases}$ (89)

where the parameter $a$ goes from $0$ to $1$. Injecting that parametrization
in Eq. (81) yields the following expression:

$W_{a}(r)=\dfrac{1}{\pi}\exp\left(-r^{2}\right)\dfrac{1}{2}(a+1)\left(r^{2}-1+\sqrt{\dfrac{1-a}{1+a}}\right)^{2}.$
(90)

$W_{a}(r)$ is the radial Wigner function of the extremal Wigner-positive
states located on the arc of the ellipse appearing in bold in Fig. 11.

##  Appendix D Convex decomposition of extremal passive states into positive
functions

The Wigner positivity of extremal passive states, and by extension of the
whole set of passive states, is often taken as a known fact Bastiaans (1983).
However, what is less known is that is in fact only the weakening of a
stronger mathematical relationship. Indeed, going back to the origin of this
result, we find the following identity in Szegő (1939):

$\begin{split}2^{n}n!\sum\limits_{k=0}^{n}(-1)^{k}L_{k}(2x^{2}+2y^{2})\qquad\qquad\qquad\\\
\qquad\qquad\qquad=\sum\limits_{k=0}^{n}\begin{pmatrix}n\\\
k\end{pmatrix}\left[H_{k}(x)\right]^{2}\left[H_{n-k}(y)\right]^{2},\end{split}$
(91)

where $L_{k}$ and $H_{k}$ are respectively the $k^{\text{th}}$ Laguerre and
Hermite polynomials. From Eq. (91), it readily appears that the left-hand side
is non-negative since the right-hand side is a sum of squared functions. This
naturally implies that the Wigner function $E_{n}$ of extremal passive states
$\hat{\varepsilon}_{n}$ [see Eqs. (25) and (41)] is non-negative.

However, it is possible to get an equality out of Eq. (91). Indeed,
multiplying both sides by $\exp(-x^{2}-y^{2})$ and rearranging the
normalization factors, we can make appear the Wigner function of Fock states
[Eq. (25)] in the left-hand side and the wave function of Fock states [Eq.
(24)] in the right-hand side. Formulated in these terms, Eq. (91) becomes then
Eq. (43).

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