# Efficient detection of non-classicality of continuous variable states using
moments of Wigner function

Bivas Mallick<EMAIL_ADDRESS>S. N. Bose National Centre for Basic
Sciences, Block JD, Sector III, Salt Lake, Kolkata 700 106, India  Sudip
Chakrabarty<EMAIL_ADDRESS>S. N. Bose National Centre for Basic
Sciences, Block JD, Sector III, Salt Lake, Kolkata 700 106, India  Saheli
Mukherjee<EMAIL_ADDRESS>S. N. Bose National Centre for Basic
Sciences, Block JD, Sector III, Salt Lake, Kolkata 700 106, India  Ananda G.
Maity<EMAIL_ADDRESS>Networked Quantum Devices Unit, Okinawa
Institute of Science and Technology Graduate University, Onna-son, Okinawa
904-0495, Japan  A. S. Majumdar<EMAIL_ADDRESS>S. N. Bose National Centre
for Basic Sciences, Block JD, Sector III, Salt Lake, Kolkata 700 106, India

###### Abstract

States with negative Wigner function, a significant subclass of non-classical
states, serve as a valuable resource for various quantum information
processing tasks. Here, we provide a criterion for detecting such quantum
states exhibiting negative Wigner function. Our method relies on evaluating
moments of the Wigner function which involves computing simple functionals and
can be implemented in a real experiment without the need for full state
tomography or Wigner function reconstruction. We then provide explicit
examples to support our detection scheme. Further, we propose an experimental
method utilizing the continuous variable SWAP operator to realize these
moments in a real experiment.

##  I Introduction

Continuous variable (CV) quantum systems Braunstein and van Loock (2005), in
particular, optical systems Agarwal (2012); Gerry and Knight (2023), play a
significant role in the development of future quantum technologies by offering
a promising and powerful platform for efficiently engineering large and
scalable multimode quantum states. Traditionally, in CV systems, the states,
residing in infinite-dimensional Hilbert spaces, can effectively be
represented in terms of quasiprobability distribution over phase-space
$\\{q,p\\}$. Among these CV quantum states, a particular type of
quasiprobability distribution is characterized by Gaussian statistics and can
be succinctly described by their first and second statistical moments
Weedbrook  _et al._ (2012); Adesso  _et al._ (2014). While these states can
exhibit various intriguing quantum phenomena, they can be simulated
classically Mari and Eisert (2012) indicating that they are not sufficient on
their own for executing quantum information protocols. On the other hand,
there exist quantum states, including non-Gaussian states Albarelli  _et al._
(2018); Takagi and Zhuang (2018); Chabaud  _et al._ (2020); Adesso  _et al._
(2014); Chowdhury  _et al._ (2013), that offer genuinely non-classical
features, providing greater advantages in optical quantum communication,
quantum information processing, quantum metrology, quantum computation, and
many more tasks Giedke and Cirac (2002); Niset  _et al._ (2009); Bartlett  _et
al._ (2002); Adhikari  _et al._ (2008a, b).

The notion about the non-classicality of an optical quantum state was proposed
by Mandel (Mandel, 1979) in terms of the deviation of photon number statistics
from a Poisson distribution, which is a typical characteristic of the
classical states (e.g., coherent states) (Glauber, 1963). Since then, there
have been many different approaches to classify these non-classical states and
their behavior Dodonov (2002); Lee (1991). Typically, the non-classical states
of CV systems are defined as those whose Glauber-Sudarshan P representation
Sudarshan (1963); Glauber (1963) fails to meet the criteria of an ordinary
probability distribution Tan  _et al._ (2020); Titulaer and Glauber (1965).
Another significant subclass of these non-classical states includes those with
a negative Wigner function Wigner (1932); Kenfack and Życzkowski (2004);
Cormick  _et al._ (2006). The states with negative Wigner function (or Wigner
negativity for short) offer promising advantages in information processing
tasks such as quantum state distillation Giedke and Cirac (2002); Fiurášek
(2002); Eisert  _et al._ (2002), quantum error-correctionNiset  _et al._
(2009), quantum computing Bartlett  _et al._ (2002); Mari and Eisert (2012),
quantum computational speedup Galvao (2005) and so on. Experimental
demonstration of these Wigner negative states has been obtained in optical
cavities (Yoshikawa  _et al._ , 2013), trapped-ion (Flühmann  _et al._ ,
2019), superconducting circuits (Lu  _et al._ , 2021), optical and
optomechanical experiments with photon-number resolving detectors Innocenti
_et al._ (2022). These findings highlight states possessing Wigner negativity
as a powerful resource for implementing various quantum information processing
tasks in real systems.

However, detecting such resources is a prerequisite for exploring their
underlying applications in any legitimate information processing task. There
have been several advancements in this regard for the detection of Wigner
negativity. Some of these detection methods are based on constructing witness
operators (Horodecki  _et al._ , 1996; Terhal, 2001; Lewenstein  _et al._ ,
2001) for Wigner negative states, followed by proposals for its experimental
realization (Mari  _et al._ , 2011; Chabaud  _et al._ , 2021; Kiesel and
Vogel, 2012). The Wigner negativity can also be detected experimentally by
Wigner function reconstruction via direct or indirect methods. The direct
method involves obtaining the Wigner function directly at a specific point in
phase-space (Leibfried  _et al._ , 1996; Smithey  _et al._ , 1993; Banaszek
_et al._ , 1999; Lutterbach and Davidovich, 1997; Bertet  _et al._ , 2002;
Winkelmann  _et al._ , 2022) whereas in the indirect method, the quantum state
is reconstructed first and subsequently, the Wigner function is derived by a
mathematical transformation Lv  _et al._ (2017); Lvovsky  _et al._ (2001);
Flühmann and Home (2020). Additionally, Wigner negativity can also be detected
by numerical simulation techniques (Cimini  _et al._ , 2020).

Driven by the motivation to utilize Wigner negativity as a resource, along
with recent efforts in designing useful tools for its detection, we present a
criterion for identifying the signature of Wigner negativity in an efficient
manner. Our approach, based on the moment criterion, enables the evaluation of
simple functionals, making it experimentally less demanding compared to
methods requiring full state tomography or Wigner function reconstruction
Lvovsky and Raymer (2009); D’Ariano and Presti (2001); D’Ariano  _et al._
(2003). Importantly, we demonstrate that the first three moments are
sufficient for identifying Wigner negativity. We provide explicit examples to
support our detection scheme and show that these moments can be efficiently
obtained in real experiments using the CV SWAP operator. Our criterion does
not require any prior knowledge about the state, in contrast with witness-
based detection schemes that are state-dependent. Additionally, the moment-
based approaches are advantageous in terms of the number of copies required to
determine Wigner negativity. The number of measurements needed for tomography
generally scales exponentially with the system size, whereas a polynomial
number of state copies suffices for the moment-based approaches.

The paper is organized as follows. In Sec. II, we present an overview of the
fundamental mathematical concepts related to continuous variable bosonic
systems along with an introduction to the Wigner function and its phase-space
representation. This section also revisits the moment criteria, previously
proposed for entanglement detection. In Sec. III, we introduce the concept of
the moments of the Wigner function and present our detection criteria along
with some explicit examples. An experimental proposal for calculating the
moments of the Wigner function using the CV SWAP operator is presented in Sec.
IV. Finally, in Sec. V, we summarize our main findings along with some future
perspectives.

##  II Preliminaries

A continuous variable (CV) system consisting of $N$ canonical bosonic modes is
characterized by a Hilbert space,

$\begin{split}\mathcal{H}=\bigotimes_{k=1}^{N}\mathcal{H}_{k}\end{split}$

where each $\mathcal{H}_{k}$, associated with a single mode, denotes a
separable, infinite-dimensional Hilbert space. The countable orthonormal basis
vectors are the elements of the single mode Fock basis
$\\{\ket{n}\\}_{n\in\mathbb{N}}$. $\mathcal{D(\mathcal{H})}$ represents the
set of positive operators with unit trace acting on $\mathcal{H}$ where the
density operator of a CV system resides. Let us denote the annihilation and
creation operators in mode $k$ by $\hat{a}_{k}$ and ${\hat{a}^{\dagger}_{k}}$
respectively, which satisfy the bosonic commutation relations,

$\begin{split}[\hat{a}_{k},\hat{a}^{\dagger}_{l}]=\delta_{kl}\mathbb{I}\hskip
0.85358pt,\hskip
8.53581pt[\hat{a}_{k},\hat{a}_{l}]=[\hat{a}^{\dagger}_{k},\hat{a}^{\dagger}_{l}]=0.\end{split}$

Their action on the Fock basis is defined as:

$\begin{split}&\hat{a}\ket{n}=\sqrt{n}\ket{n-1}\hskip 8.5359pt\text{for}\hskip
4.26773ptn\in{\mathbb{N}}^{*}\hskip 8.53581pt\text{with}\hskip
8.53581pt\hat{a}\ket{0}=0,\hskip 8.5359pt\\\
&\hat{{a^{\dagger}}}\ket{n}=\sqrt{n+1}\ket{n+1}\hskip
5.69046pt\text{for}\hskip 4.26773ptn\in{\mathbb{N}}.\end{split}$

The quadrature phase operators for mode $k$ are defined as,

$\hat{x}_{k}=\frac{(\hat{a}_{k}+\hat{a}^{\dagger}_{k})}{\sqrt{2}},\hskip
8.53581pt\hat{p}_{k}=\frac{(\hat{a}_{k}-\hat{a}^{\dagger}_{k})}{\sqrt{2}i},$
(1)

with $[\hat{x}_{k},\hat{p}_{k}]=i\mathbb{I}$. For convenience, we shall use
natural units for which $\hbar=1$ throughout the manuscript.

In the next subsection, we provide a brief introduction to the Wigner
function. Throughout the manuscript, we shall use the terminologies commonly
used in the field of quantum information.

###  II.1 Wigner function

The Wigner function is a quasiprobability distribution function used in the
phase-space formulation of quantum mechanics, particularly for the study of CV
systems Cahill and Glauber (1969). Its ability to exhibit negativity
distinguishes its quantum behavior from classical. This function provides a
complete description of the quantum state by representing it in a combined
position and momentum space. Let ($x_{k},p_{k}$) be the position and momentum
quadratures of $k$-th mode, with the corresponding vectors represented by
$\vec{x}=(x_{1},x_{2},...,x_{N}),\leavevmode\nobreak\
\vec{p}=(p_{1},p_{2},...,p_{N})$. The Wigner function for an $N$-mode system
corresponding to density matrix $\hat{\rho}$ is defined as Case (2008)

$W(\vec{x},\vec{p})=\frac{1}{{(2\pi)}^{N}}\int\limits_{-\infty}^{\infty}\bra{\vec{x}+\frac{\vec{y}}{2}}\hat{\rho}\ket{\vec{x}-\frac{\vec{y}}{2}}e^{-i\vec{p}\cdot\vec{y}}\hskip
2.84544ptd^{N}y.$ (2)

Further, the Wigner function of a single mode system corresponding to a
density matrix $\hat{\rho}$ can be expressed as (Wigner, 1932):

$W(x,p)=\frac{1}{2\pi}\int\limits_{-\infty}^{\infty}\bra{x+\frac{y}{2}}\hat{\rho}\ket{x-\frac{y}{2}}e^{-ipy}\hskip
2.84544ptdy.$ (3)

For a pure state $\ket{\psi}$ with wave function $\psi(x)$, this Wigner
function simplifies to

$W(x,p)=\frac{1}{2\pi}\int\limits_{-\infty}^{\infty}\psi(x+\frac{y}{2})\psi^{*}(x-\frac{y}{2})e^{-ipy}\hskip
2.84544ptdy.$ (4)

As a consequence of the above definition, it can be seen that the Wigner
function is a real-valued function. Additionally, if $W(x,p)$ is normalized,
then

$\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}W(x,p)\hskip
2.84544ptdxdp=1.$ (5)

The Wigner function resembles a probability distribution but can have negative
values. Therefore, it cannot be measured directly in real experiments.
However, its marginals are valid probability distributions that can be
realized by homodyne detection Lvovsky and Raymer (2009). The marginal
probability distributions can be obtained by integrating the Wigner function
over the momentum and position space, respectively i.e.,

$\displaystyle|\psi(x)|^{2}=\int\limits_{-\infty}^{\infty}W(x,p)dp,$ (6)
$\displaystyle|\phi(p)|^{2}=\int\limits_{-\infty}^{\infty}W(x,p)dx,$ (7)

where, $\phi(p)$ is the momentum wave function.

Now, to successfully develop a phase-space formalism using the Wigner
function, it is necessary to establish a mapping between functions in the
quantum phase-space formulation and operators in the Hilbert space within the
Schrödinger picture. This mapping is achieved through the Weyl transform
$\tilde{A}$ of an operator $\hat{A}$, defined as follows Case (2008):

$\tilde{A}(x,p)=\int\limits_{-\infty}^{\infty}e^{-ipy}\langle
x+\frac{y}{2}|\hat{A}(\hat{x},\hat{p})|x-\frac{y}{2}\rangle\hskip
2.84544ptdy.$ (8)

The expectation value of the operator $\hat{A}$ is,

$\langle A\rangle=\int\limits_{-\infty}^{\infty}W(x,p)\tilde{A}(x,p)\hskip
2.84544ptdxdp.$ (9)

Thus, the expectation value of the operator $\hat{A}$ is achieved, by
averaging $\tilde{A}(x,p)$ over phase-space using the quasiprobability density
$W(x,p)$ that characterizes the state.

In CV systems, states can be classified into Gaussian and non-Gaussian based
on their Wigner function. A state is considered Gaussian if its Wigner
function is Gaussian, and hence positive. But the converse is not true in
general. According to Hudson theorem Hudson (1974); Soto and Claverie (1983),
pure states with positive Wigner function are always Gaussian. However, this
statement does not hold true for mixed states Mandilara  _et al._ (2009);
Filip and Mišta Jr (2011). There exist non-Gaussian states with a positive
Wigner function. The negativity of the Wigner function is regarded as a figure
of merit of non-classicality.

###  II.2 Partial transpose moments

Entanglement is a particular type of quantum correlation that forms the basis
of many information processing tasks. However, the task of entanglement
detection is in general, an NP-hard problem. One of the efficient and widely
used criteria for detecting bipartite entanglement in discrete variable (DV)
systems is based on the positive partial transposition (PPT) criterion. This
criterion examines whether the partial transposed state $\rho^{T_{B}}_{AB}$
(where partial transposition is taken with respect to the subsystem B) is
positive semidefinite. A violation of this criterion indicates that the given
state $\rho_{AB}$ is entangled. This criterion serves as a necessary and
sufficient condition for ${\mathbb{C}}^{2}\otimes{\mathbb{C}}^{2}$,
${\mathbb{C}}^{2}\otimes{\mathbb{C}}^{3}$ and
${\mathbb{C}}^{3}\otimes{\mathbb{C}}^{2}$ systems. However, directly computing
all the eigenvalues of $\rho^{T_{B}}_{AB}$ is infeasible in real experiments
due to computational complexity. To overcome this challenge, the concept of
moments of the partially transposed density matrix (PT-moments) was introduced
by Calabrese et al. in 2012 Calabrese  _et al._ (2012). These moments provide
a way to study correlations in many-body systems in relativistic quantum field
theory. The eigenvalues $\\{\lambda_{i}\\}$ of $\rho^{T_{B}}_{AB}$ are related
to the roots of its characteristic polynomial

$\text{Det}(\rho^{T_{B}}_{AB}-\lambda I)=\sum_{n}a_{n}{\lambda}^{n}$ (10)

where, $a_{n}$ are the functions of PT moments $(P_{n})$ which are defined as

$P_{n}=\text{Tr}[{(\rho_{AB}}^{T_{B}})^{n}]$ (11)

for $n=1,2,3,...$. In this way, the PT moments encapsulate information about
$\\{\lambda_{i}\\}$ offering a way to express the eigenvalues in terms of
experimentally accessible quantities. In recent years, several studies have
demonstrated the advantages and experimental realization of PT moments Gray
_et al._ (2018); Carteret (2005); Bartkiewicz  _et al._ (2015); Elben  _et
al._ (2020); Yu  _et al._ (2021); Mallick  _et al._ (2024); Neven  _et al._
(2021). Note that, $P_{1}=\text{Tr}[\rho_{AB}^{T_{A}}]=1$. Furthermore, it can
be shown that $P_{2}=\text{Tr}[(\rho_{AB}^{T_{A}})^{2}]$ is related to the
purity of the state. Therefore, the first non-trivial moment $P_{3}$ captures
additional information related to the partial transposition beyond the purity
of the state. Using the first three moments, a straightforward yet effective
criterion for detecting entanglement has been proposed Elben  _et al._ (2020).
This criterion suggests that if a state $\rho_{AB}$ is PPT, then
${P_{2}}^{2}\leq P_{3}P_{1}$. Consequently, if a state violates this
condition, it must be entangled, forming the $p_{3}$-PPT criterion. The
$p_{3}$-PPT criterion and positive partial transposition (PPT) criterion are
equivalent for the case of Werner state, and hence, the $p_{3}$-PPT criterion
is a necessary and sufficient criterion for bipartite entanglement detection
of Werner state.

##  III Detection of Wigner negativity

In this section, we present our main results for detecting Wigner negativity
using moments of the Wigner function. We also provide explicit examples that
support our proposed theorem, aiming to firmly establish its validity.

Definition 1: Let, $W$ be the Wigner function of a two-mode quantum state. We
define the $n$-th order $W$-moments $(w_{n})$ as:

$\begin{split}w_{n}=\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}W^{n}\hskip
5.69046pt{dx}_{1}{dx}_{2}dp_{1}dp_{2}\end{split}$ (12)

with $n$ being an integer. With the above definition, we are now ready to
propose our criterion for detecting Wigner negativity as the following
theorem.

Theorem 1: If the Wigner function $(W)$ for a two-mode quantum state is
positive, then

${w_{2}}^{2}\leq w_{3}$ (13)

where $w_{2}$ and $w_{3}$ are defined in (12).

###### Proof.

Let the Wigner function $(W)$ be a positive, real-valued, normalized
distribution. Let us now consider the ${\mathcal{L}}_{p}$ space for $p\geq 1$,
which is defined as Rudin (1987); Kubrusly (2007)

$\begin{split}&||W||_{\mathcal{L}_{P}}=\left(\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}|W|^{p}\hskip
4.26773pt{dx}_{1}{dx}_{2}dp_{1}dp_{2}\right)^{\frac{1}{p}}.\end{split}$ (14)

Hölder’s inequality for ${\mathcal{L}}_{p}$ space: Let $1\leq p<\infty$ and
$1<q<\infty$ along with $\frac{1}{p}+\frac{1}{q}=1$. If
$f\in{\mathcal{L}}_{p}$ and $g\in{\mathcal{L}}_{q}$, then
$f.g\in{\mathcal{L}}_{1}$. Moreover,

$||fg||_{{\mathcal{L}}_{1}}\leq||f||_{{\mathcal{L}}_{p}}||g||_{{\mathcal{L}}_{q}}.$
(15)

For $p=q=2$, Hölder’s inequality reduces to Cauchy-Schwarz inequality.

In our case, we put $p=3$, $q=\frac{3}{2}$ in Hölder’s inequality and taking
$f=g=W$, we get

$||W.W||_{{\mathcal{L}}_{1}}\leq||W||_{{\mathcal{L}}_{3}}||W||_{{\mathcal{L}}_{\frac{3}{2}}}.$
(16)

Now,

$\begin{split}||W||_{{\mathcal{L}}_{\frac{3}{2}}}&=\left(\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}|W|^{\frac{3}{2}}\hskip
4.26773pt{dx}_{1}{dx}_{2}dp_{1}dp_{2}\right)^{\frac{2}{3}}=\left(\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}|W|.|W|^{\frac{1}{2}}\hskip
4.26773pt{dx}_{1}{dx}_{2}dp_{1}dp_{2}\right)^{\frac{2}{3}}\\\
&\leq\left[\left(\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}|W|^{2}\hskip
4.26773pt{dx}_{1}{dx}_{2}dp_{1}dp_{2}\right)^{\frac{1}{2}}\left(\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}|W|\hskip
4.26773pt{dx}_{1}{dx}_{2}dp_{1}dp_{2}\right)^{\frac{1}{2}}\right]^{\frac{2}{3}}\\\
&\hskip 284.52756pt\text{[using Cauchy-Schwarz inequality]}\\\
&=\left[\left(\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}|W|^{2}\hskip
4.26773pt{dx}_{1}{dx}_{2}dp_{1}dp_{2}\right)^{\frac{1}{3}}\left(\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}|W|\hskip
4.26773pt{dx}_{1}{dx}_{2}dp_{1}dp_{2}\right)^{\frac{1}{3}}\right]\end{split}$
(17)

i.e.,

$||W||_{{\mathcal{L}}_{\frac{3}{2}}}\leq{||W||^{\frac{2}{3}}_{{\mathcal{L}}_{2}}}{||W||^{\frac{1}{3}}_{{\mathcal{L}}_{1}}}$
(18)

Putting the value of (18) in (16), we get,

$||W||_{{\mathcal{L}}_{1}}\leq||W||_{{\mathcal{L}}_{3}}{||W||^{\frac{2}{3}}_{{\mathcal{L}}_{2}}}{||W||^{\frac{1}{3}}_{{\mathcal{L}}_{1}}}.$
(19)

Again,

$\begin{split}||W.W||_{{\mathcal{L}}_{1}}&=\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}|W.W|\hskip
4.26773pt{dx}_{1}{dx}_{2}dp_{1}dp_{2}\\\ &={||W||^{2}_{{\mathcal{L}}_{2}}}.\\\
\end{split}$ (20)

Therefore, from Eq.(21), we get

${||W||^{2}_{{\mathcal{L}}_{2}}}\leq||W||_{{\mathcal{L}}_{3}}{||W||^{\frac{2}{3}}_{{\mathcal{L}}_{2}}}||W||^{\frac{1}{3}}_{{\mathcal{L}}_{1}}$
(21)

Taking $3$rd power on both sides, we get

${||W||^{6}_{{\mathcal{L}}_{2}}}\leq{||W||^{3}_{{\mathcal{L}}_{3}}}{||W||^{2}_{{\mathcal{L}}_{2}}}{||W||^{1}_{{\mathcal{L}}_{1}}},$
(22)

i.e.

${||W||^{4}_{{\mathcal{L}}_{2}}}\leq{||W||^{3}_{{\mathcal{L}}_{3}}}{||W||^{1}_{{\mathcal{L}}_{1}}}$
(23)

From the normalization condition of Wigner function $(W)$, we get

$||W||_{\mathcal{L}_{1}}=\left(\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}|W|^{1}\hskip
4.26773pt{dx}_{1}{dx}_{2}dp_{1}dp_{2}\right)^{1}=1.$ (24)

Since, we have assumed that the Wigner function is positive, i.e. $W=|W|$,
Therefore, from Eq.(23), we get

${||W||^{4}_{{\mathcal{L}}_{2}}}\leq{||W||^{3}_{{\mathcal{L}}_{3}}},$ (25)

i.e.

${w_{2}}^{2}\leq w_{3},$ (26)

which completes our proof. ∎
Note that, our detection criteria can be generalized for $n$-mode quantum
systems ($n\geq 1$) in a similar fashion. We would now like to present some
examples in support of Theorem 1.

###  III.1 Examples

Example 1: Two-mode squeezed vacuum state (TMSV) : This state is obtained by
squeezing the two-mode vacuum state $\ket{0,0}$ using the two-mode squeezing
operator
$S(\zeta)=\exp(\zeta\hat{a}_{1}^{\dagger}\hat{a}_{2}^{\dagger}-{\zeta}^{*}\hat{a}_{1}\hat{a}_{2})$
where $\zeta=r\exp(i\phi)$ and $r>0$ is the squeezing parameter. This state
can be produced in a nondegenerate optical parametric amplifier (NOPA), and
hence is represented as:

$\begin{split}\ket{NOPA}=|\zeta\rangle=S(\zeta)\ket{0,0}=\sqrt{1-\lambda^{2}}\sum_{n-0}^{\infty}\lambda^{n}\ket{n,n}\end{split}$
(27)

where, $\lambda=\tanh{r}\in[0,1]$, and
$|m,n\rangle=|m\rangle\otimes|n\rangle$, where ( $|m\rangle$, $|n\rangle$ are
the usual Fock states). The Wigner function corresponding to this state is
given byChowdhury  _et al._ (2014); Agarwal (2012); Maity  _et al._ (2017),

$\begin{split}W_{\zeta}[x_{1},p_{1},x_{2},p_{2}]=&\frac{1}{\pi^{2}}\text{exp}[-2\sinh{[2r]}(x_{1}x_{2}-p_{1}p_{2})\\\
&-\cosh{[2r]}({x_{1}}^{2}+{x_{2}}^{2}+{p_{1}}^{2}+{p_{2}}^{2})].\end{split}$
(28)

The Wigner function of the two-mode squeezed vacuum state is always positive
Agarwal (2012). Therefore, from Theorem 1, we should have
$\omega_{2}^{2}\leq\omega_{3}$. Now, we calculate the second $(\omega_{2})$
and third order moments $(\omega_{3})$ of the Wigner function as follows:

$\begin{split}\omega_{2}^{2}&=\left(\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}W_{\zeta}^{2}\hskip
4.26773pt{dx}_{1}dp_{1}{dx}_{2}dp_{2}\right)^{2}\\\
&=\frac{1}{16\pi^{4}}.\end{split}$ (29)
$\begin{split}\omega_{3}&=\left(\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}W_{\zeta}^{3}\hskip
4.26773pt{dx}_{1}dp_{1}{dx}_{2}dp_{2}\right)\\\
&=\frac{1}{9\pi^{4}}\end{split}$ (30)

Clearly, $\omega_{2}^{2}\leq\omega_{3}$, which validates our proposed
criterion.

Example 2: Single photon-subtracted squeezed vacuum (SPSSV) : Let us now
consider an explicit example of a non-Gaussian state that is obtained by
subtracting photons from a Gaussian state. The subtraction of $n$ photons from
the state $|\zeta\rangle$ (Eq. (27)) may be represented as,

$|\zeta_{-n}\rangle=[a\otimes I+(-1)^{k}I\otimes b]^{n}|\zeta\rangle$ (31)

where $k\in\\{0,1\\}$. It is also assumed that the specific mode from which
the photon is subtracted remains unknown. SPSSV state is generated by
subtracting a single photon from any mode of a two-mode squeezed vacuum state
represented as Biswas and Agarwal (2007),

$\begin{split}\ket{\xi_{-1}}=&\frac{1}{2\sinh{r}^{2}}\sqrt{1-\lambda^{2}}\sum_{n=0}^{\infty}\lambda^{n}\sqrt{n}[\ket{n-1,n}\\\
&+(-1)^{k}\ket{n,n-1}]\end{split}$ (32)

with $k\in[0,1]$ and $\lambda=\tanh{r},\hskip 2.84544ptr>0$.

The Wigner function of SPSSV state Agarwal (2012), can be represented as

$\begin{split}W_{1}[x_{1},p_{1},x_{2},p_{2}]=&\frac{1}{\pi^{2}}\text{exp}[2\sinh{[2r]}(x_{1}x_{2}-p_{1}p_{2})\\\
&-\cosh{[2r]}({x_{1}}^{2}+{x_{2}}^{2}+{p_{1}}^{2}+{p_{2}}^{2})]\\\
&\times[-\sinh{[2r]}((p_{1}-p_{2})^{2}-(x_{1}-x_{2})^{2})\\\
&+\cosh{[2r]}((p_{1}-p_{2})^{2}+(x_{1}-y_{1})^{2})-1].\end{split}$

It is well known that the state exhibits Wigner negativity Agarwal (2012),
therefore from our proposed criteria we should have
$\omega_{2}^{2}>\omega_{3}$. Now, we compute the second $(\omega_{2})$ and
third order moments $(\omega_{3})$ of the Wigner function as follows:

$\begin{split}\omega_{2}^{2}&=\left(\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}W_{1}^{2}\hskip
4.26773pt{dx}_{1}dp_{1}{dx}_{2}dp_{2}\right)^{2}\\\
&=\frac{1}{16\pi^{4}}\end{split}$ (33)
$\begin{split}\omega_{3}&=\left(\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}W_{1}^{3}\hskip
4.26773pt{dx}_{1}dp_{1}{dx}_{2}dp_{2}\right)\\\
&=\frac{1}{81\pi^{4}}\end{split}$ (34)

Clearly, $\omega_{2}^{2}>\omega_{3}$ which again supports Theorem 1.

Example 3: NOON State : This is an example of a two-mode state where $N$
photons can be found either in the mode $a$ or in the mode $b$. The NOON state
is a maximally path entangled number state with the form

$\begin{split}|\psi\rangle=\frac{1}{\sqrt{2}}(|N\rangle_{a}|0\rangle_{b}+e^{i\phi}|0\rangle_{a}|N\rangle_{b}).\end{split}$
(35)

The Wigner function for this state with $\phi=\pi$, in terms of dimensionless
quadratures ${x_{1},p_{1}}$ and ${x_{2},p_{2}}$ is given by Agarwal (2012),

$\begin{split}W_{N}[x_{1},p_{1},x_{2},p_{2}]=&\frac{1}{2\pi^{2}N!}e^{-({x_{1}}^{2}+{x_{2}}^{2}+{p_{1}}^{2}+{p_{2}}^{2})}\\\
&\times[-2^{N}\\{(x_{1}+ip_{1})^{N}(x_{2}-ip_{2})^{N}\\\
&+(x_{1}-ip_{1})^{N}(x_{2}+ip_{2})^{N}\\}\\\
&+(-1)^{N}N!\\{L_{N}[2({x_{1}}^{2}+{p_{1}}^{2})]\\\
&+L_{N}[2({x_{2}}^{2}+{p_{2}}^{2})]\\}].\end{split}$ (36)

Here $L_{N}(x)$ is the Laguerre polynomial.
This state exhibits Wigner negativity for any $N\geq 1$. Therefore, for $N\geq
1$, we should have ${w_{2}}^{2}-w_{3}>0$ which is evident from Fig. 1. The
explicit values of the moments of Wigner function and $w_{2}^{2}-w_{3}$ are
presented in table 1 of appendix A.

Figure 1: Plot of $\Delta$ = $w_{2}^{2}-w_{3}$ with the photon number $(N)$
for NOON states upto $N=5$.

Example 4: Fock States : Another important class of non-Gaussian states is the
Fock states. Let $|n\rangle\equiv{\frac{a^{\dagger n}}{\sqrt{n!}}}|0\rangle$
be the ${n}$-th Fock state of a quantum harmonic oscillator. The associated
Wigner function in dimensionless variables is given by Agarwal (2012),

$\begin{split}{W_{|n\rangle}(x,p)={\frac{(-1)^{n}}{\pi}}e^{-(x^{2}+p^{2})}L_{n}{\big{(}}2(p^{2}+x^{2}){\big{)}}}\end{split}$
(37)

where ${L_{n}(x)}$ denotes the ${n}$-th Laguerre polynomial.

The vacuum state $|0\rangle$ is a Gaussian state, hence it possesses a
positive Wigner function. All other Fock states are Wigner negative.
Therefore, for $n\geq 1$, we should have ${w_{2}}^{2}-w_{3}>0$ which is again
evident from Fig. 2. The moments of the Wigner function for the first five
Fock states are provided in table 2 of appendix A.

Figure 2: $\Delta$ = $w_{2}^{2}-w_{3}$ is plotted versus the number of the
Fock state up to $n=5$. The figure clearly shows Wigner negativity (solid
blue) for all states except $n=0$ (solid red).

##  IV Experimental proposal for calculating moments of the Wigner function

In this section, we propose a method for experimentally measuring the second
and third order moments of the Wigner function. Our approach involves
measuring these moments using only the SWAP operator between the individual
constituents of $m$ copies of an unknown two-mode quantum state. The SWAP gate
for continuous variable systems can be written as Volkoff and Subaşı (2022)

$\text{SWAP}=\text{exp}\left[{{i\pi}{(\frac{{\hat{a}^{\dagger}_{1}}-{\hat{a}^{\dagger}_{2}}}{\sqrt{2}})(\frac{{\hat{a}_{1}}-{\hat{a}_{2}}}{\sqrt{2}})}}\right]$
(38)

where, $\hat{a}_{k}$ and ${\hat{a}^{\dagger}_{k}}$ are the $k$-th mode
creation and annihilation operators, for $k=1,2$. The creation and
annihilation operators in the canonical basis can be written as

$\begin{split}\hat{a}_{k}=\frac{(\hat{x}_{k}+i\hat{p}_{k})}{\sqrt{2}}\hskip
5.69054pt,\hskip
5.69054pt{\hat{a}_{k}}^{\dagger}=\frac{(\hat{x}_{k}-i\hat{p}_{k})}{\sqrt{2}}\end{split}$
(39)

with $[\hat{x}_{k},\hat{p}_{k}]=i\mathbb{I}$. Using Eqs.(38) and (39), we get

$\text{SWAP}=\text{exp}\left[{\frac{i\pi}{4}}{\left({{(\hat{x}_{1}-\hat{x}_{2})^{2}+(\hat{p}_{1}-\hat{p}_{2})^{2}-2\mathbb{I}}}\right)}\right].$
(40)

Now, we can do the Weyl transform of this two-mode SWAP operator to get the
two-mode SWAP function $(f_{\text{SWAP}})$,

$\begin{split}f_{\text{SWAP}}&=\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}dy_{1}dy_{2}\bigg{[}e^{-i(p_{1}y_{1}+p_{2}y_{2})}\\\
&\times\left\langle
x_{1}+\frac{y_{1}}{2},x_{2}+\frac{y_{2}}{2}\left|{\text{SWAP}}\right|x_{1}-\frac{y_{1}}{2},x_{2}-\frac{y_{2}}{2}\right\rangle\bigg{]}.\end{split}$
(41)

Realization of the second order moment $(w_{2})$: The second order moment of
the Wigner function of an unknown quantum state $\ket{{\psi}}_{AB}$ is defined
as

$w_{2}=\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}W^{2}\hskip
2.84544pt{dx}_{1}{dp}_{1}dx_{2}dp_{2}.$ (42)

To calculate the second order moment $(w_{2})$, we take two copies of
$\ket{{\psi}}_{AB}$. Let $C_{2}=A_{1}B_{1}A_{2}B_{2}$ be the CV registrar
acting on the $2$-mode registers $A$ and $B$ consisting of isomorphic modes
$A_{1},A_{2}$ and $B_{1},B_{2}$ respectively. The first step of the protocol
is to express $w_{2}$ as an expectation value of the SWAP operator, similar to
the procedure in Gray  _et al._ (2018), i.e.,

$w_{2}=\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}(W^{\otimes
2})\hskip 2.84544ptf_{\text{SWAP}}\hskip
5.69046pt{dx}_{1}{dp}_{1}dx_{2}dp_{2}$ (43)

Now, Alice and Bob both apply the SWAP operator on their respective subsystems
of $\ket{{\psi}}_{AB}$ repeatedly, until the expectation value is obtained.

Realization of the third order moment $(w_{3})$: The third order moment of the
Wigner function of an unknown quantum state $\ket{{\psi}}_{AB}$ is defined as

$w_{3}=\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}W^{3}\hskip
2.84544pt{dx}_{1}{dp}_{1}dx_{2}dp_{2}.$ (44)

The third order moment is calculated using three copies of $\ket{{\psi}}_{AB}$
and by implementing the CV registrar $(C_{3}=A_{1}B_{1}A_{2}B_{2}A_{3}B_{3})$
on the $2$-mode registers A and B consisting of isomorphic modes
$A_{1},A_{2},A_{3}$ and $B_{1},B_{2},B_{3}$ respectively. Let,
${\mathbb{P}}^{(3)}=\prod_{i=1}^{3}\text{SWAP}_{A_{i},A_{i+1}}(\prod_{i=1}^{3}\text{SWAP}_{B_{i},B_{i+1}})$
be the CV permutation operator acting on the register $A$ ($B$) (where $3+1$
is taken modulo $3$). We express $w_{3}$ as the expectation value of the
permutation operator as follows

$w_{3}=\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}(W^{\otimes
3})\hskip 2.84544pt{f_{\mathbb{P}^{(3)}}}\hskip
5.69046pt{dx}_{1}{dp}_{1}dx_{2}dp_{2}$ (45)

where
${f_{\mathbb{P}^{(3)}}}=\prod_{i=1}^{3}f_{\text{SWAP}_{A_{i},A_{i+1}}}(\prod_{i=1}^{3}f_{\text{SWAP}_{B_{i},B_{i+1}}})$
is the function corresponding to CV permutation operator acting on the
register $A(B)$. Measuring $w_{3}$ necessarily involves the following steps :

  * •

Prepare $3$ copies of $\ket{{\psi}}_{AB}$.

  * •

Apply SWAP operator in a forward sequence on Alice’s side
$\text{SWAP}_{A_{i},A_{i+1}}$ between neighboring copies for $i=1,2$.

  * •

Apply adjacent SWAP operator in a backward sequence on Bob’s side
$\text{SWAP}_{B_{i},B_{i+1}}$ between neighboring copies, ranging from $b=3$
to $b=2$.

  * •

Repeat these steps until the expectation value is obtained.

The computation of these moments are experimentally less demanding and, unlike
tomography, do not require complete knowledge of the state itself.

##  V Conclusions

Non-classical states that exhibit properties beyond classical physics, are
essential for various quantum information processing tasks Niset  _et al._
(2009); Eisert  _et al._ (2002); Bartlett  _et al._ (2002); Galvao (2005).
However, before utilizing them for any information processing tasks, it is
crucial to detect the signature of that specific non-classicality pertaining
to each continuous variable state. In this paper, we have focused on detecting
a particular type of such non-classicality known as Wigner negativity, based
on moments of the Wigner function which involve evaluating simple functionals.
Our results demonstrate that calculating only the first three moments of the
Wigner function is sufficient to identify such non-classical traits reliably.
Unlike tomography or witness-based approaches, our method does not require
complete knowledge of the state or reconstruction of the Wigner function
Chabaud  _et al._ (2021); Lvovsky and Raymer (2009); D’Ariano and Presti
(2001); D’Ariano  _et al._ (2003). Instead, it relies on classical shadows of
the state. We provide several examples to support our detection scheme and
propose an experimental method for calculating these moments. Our proposal
suggests that these moments can be computed efficiently in real experiments
using SWAP operator only, establishing the protocol’s experimental
feasibility.

Our study uncovered several potential open problems and directions for future
analysis. Specifically, akin to entanglement detection Gray  _et al._ (2018);
Elben  _et al._ (2020); Neven  _et al._ (2021); Yu  _et al._ (2021), a moment-
based detection scheme could prove highly relevant for identifying Wigner
negativity in multi-qubit systems, requiring exponentially fewer samples than
standard tomography. Exploring and developing a detailed derivation and
analysis of the moment-based Wigner negativity detection scheme for multi-
qubit or high-dimensional optical states would be an intriguing extension of
our current study, which we leave open for future research. Moreover,
extending such moment-based detection schemes to other types of non-
classicality is also crucial. Utilization of these schemes for continuous
variable (CV) states seems promising in effectively characterizing near-term
devices, particularly those consisting of a large number of particles and
optical setups.

##  VI Acknowledgements

BM acknowledges DST INSPIRE fellowship program for financial support.

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##  Appendix A Calculation of moments of Wigner function for NOON and Fock
states

Here, we present two tables demonstrating the effectiveness of our detection
criteria for detecting Wigner negativity in NOON states and Fock states
respectively. The tables provide explicit values of the second and third order
moments of the Wigner function ($w_{2}$ and $w_{3}$ respectively) for
different numbers of photons that are used for Fig. 1 and Fig. 2 respectively.

$N$ |  $w_{2}$ |  $w_{3}(\times 10^{-4})$ |  $w_{2}^{2}-w_{3}(\times 10^{-4})$
---|---|---|---
1 |  $0.025330$ |  $1.26741$ |  $5.14883$
2 |  $0.025330$ |  $0.140823$ |  $6.27542$
3 |  $0.025330$ |  $-0.488533$ |  $6.90477$
4 |  $0.025330$ |  $0.592074$ |  $5.82416$
5 |  $0.025330$ |  $0.271104$ |  $6.14693$

Table 1: Computation of $w_{2}$, $w_{3}$ and $w_{2}^{2}-w_{3}$ for the number
of photons, $N\in\\{1,2,3,4,5\\}$ for NOON states.

  * •

NOON states: From the table, it is evident that the values of
$w_{2}^{2}-w_{3}$ are always positive for the five four NOON states, all of
which posses Wigner negativity.

$n$ |  $w_{2}$ |  $w_{3}$ |  $w_{2}^{2}-w_{3}$
---|---|---|---
$0$ |  $0.159155$ |  $0.033774$ |  $-0.008443$
$1$ |  $0.159155$ |  $0.003753$ |  $0.021578$
$2$ |  $0.159155$ |  $0.010424$ |  $0.014906$
$3$ |  $0.159155$ |  $0.002723$ |  $0.022607$
$4$ |  $0.159155$ |  $0.006548$ |  $0.018782$
$5$ |  $0.159155$ |  $0.002222$ |  $0.023108$

Table 2:  Computation of $w_{2}$, $w_{3}$ and $w_{2}^{2}-w_{3}$ for the photon
number, $n\in\\{0,1,2,3,4,5\\}$ for Fock states.

  * •

Fock states: The table reveals that the value of $w_{2}^{2}-w_{3}$ is negative
for $n=0$, which is a Gaussian state (positive Wigner function), and positive
for other Fock states which exhibit Wigner negativity.

Hence, these results are in agreement with our detection criterion.