# Quantification of Wigner Negativity Remotely Generated via Einstein-
Podolsky-Rosen Steering

Yu Xiang‡ State Key Laboratory for Mesoscopic Physics, School of Physics,
Frontiers Science Center for Nano-optoelectronics, $\&$ Collaborative
Innovation Center of Quantum Matter, Peking University, Beijing 100871, China
Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan,
Shanxi 030006, China  Shuheng Liu‡ State Key Laboratory for Mesoscopic
Physics, School of Physics, Frontiers Science Center for Nano-optoelectronics,
$\&$ Collaborative Innovation Center of Quantum Matter, Peking University,
Beijing 100871, China  Jiajie Guo  State Key Laboratory for Mesoscopic
Physics, School of Physics, Frontiers Science Center for Nano-optoelectronics,
$\&$ Collaborative Innovation Center of Quantum Matter, Peking University,
Beijing 100871, China  Qihuang Gong  State Key Laboratory for Mesoscopic
Physics, School of Physics, Frontiers Science Center for Nano-optoelectronics,
$\&$ Collaborative Innovation Center of Quantum Matter, Peking University,
Beijing 100871, China  Collaborative Innovation Center of Extreme Optics,
Shanxi University, Taiyuan, Shanxi 030006, China  Peking University Yangtze
Delta Institute of Optoelectronics, Nantong 226010, Jiangsu, China  Nicolas
Treps  Laboratoire Kastler Brossel, Sorbonne Université, CNRS, ENS-Université
PSL, Collège de France; 4 place Jussieu, F-75252 Paris, France  Qiongyi He
<EMAIL_ADDRESS>State Key Laboratory for Mesoscopic Physics, School of
Physics, Frontiers Science Center for Nano-optoelectronics, $\&$ Collaborative
Innovation Center of Quantum Matter, Peking University, Beijing 100871, China
Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan,
Shanxi 030006, China  Peking University Yangtze Delta Institute of
Optoelectronics, Nantong 226010, Jiangsu, China  Mattia Walschaers
Laboratoire Kastler Brossel, Sorbonne Université, CNRS, ENS-Université PSL,
Collège de France; 4 place Jussieu, F-75252 Paris, France

###### Abstract

Wigner negativity, as a well-known indicator of nonclassicality, plays an
essential role in quantum computing and simulation using continuous-variable
systems. Recently, it has been proven that Einstein-Podolsky-Rosen steering is
a prerequisite to generate Wigner negativity between two remote modes.
Motivated by the demand of real-world quantum network, here we investigate the
shareability of generated Wigner negativity in the multipartite scenario from
a quantitative perspective. By establishing a monogamy relation akin to the
generalized Coffman-Kundu-Wootters inequality, we show that the amount of
Wigner negativity cannot be freely distributed among different modes.
Moreover, for photon subtraction—one of the main experimentally realized non-
Gaussian operations—we provide a general method to quantify the remotely
generated Wigner negativity. With this method, we find that there is no direct
quantitative relation between the Gaussian steerability and the amount of
generated Wigner negativity. Our results pave the way for exploiting Wigner
negativity as a valuable resource for numerous quantum information protocols
based on non-Gaussian scenario.

Continuous-variable (CV) systems have attained impressive success in quantum
information processing cv . As an important platform that has been widely
studied, Gaussian systems and operations are extensively used in quantum
teleportation tele , quantum key distribution qkd , and quantum enhanced
sensing sensing1 ; sensing2 . These protocols come with the advantage of
deterministically producing resource states and being analytically tractable
due to the Gaussian properties of the states. However, non-Gaussian states and
operations have irreplaceable advantages in several CV protocols hybrid , such
as entanglement distillation distill1 ; distill2 , error correction error ,
secure quantum communication Nha_npj2019 , and the verification of Bell
nonlocality chenbell . Considerable progresses in controllable generation of
multimode non-Gaussian states have been made in recent experiments
npnongaussian ; prxnongaussian , which also provide support for the
implementation of universal CV quantum computation in the long term
universalqc .

For some non-Gaussian states the Wigner function can reach negative values.
This Wigner negativity has been seen as a necessary ingredient in CV quantum
computating and simulation to outperform classical devices Zhuangpra ;
Ferraropra ; Eisertprl . In the pursuit of networked quantum technologies it
is crucial to develop efficient methods to produce Wigner negativity in the
distant nodes. Recently, a scheme to remotely generate Wigner negativity was
proposed through Einstein-Podolsky-Rosen (EPR) steering WalschaersPRL ;
WalschaersPRXQ —a particular type of quantum correlation where local
measurements performed on one party can adjust (steer), instantaneously, the
state of the other remote party steering ; Howard07PRL ; ReidRMP . Based on
this kind of nonlocal effect, one can induce negativity in the steering mode
by applying a set of appropriate operations on the steered mode.

In consideration of the real-world quantum network in the multipartite
scenario, it is a worthwhile objective to deeply explore the remote generation
and distribution of Wigner negativity over many nodes in an entanglement-based
network. As an intermediate type of quantum correlations between entanglement
and Bell nonlocality, multipartite quantum steering genuine13  has received
extensive attention in recent developments of quantum information theory
cavalcanti17review ; otreview . It has been successfully implementated in CV
optical network ANUexp ; prlSu ; prlSu2020 ; yin , photonic network Spainexp ;
pan2015 ; pan2020 , and atomic ensembles bec2 . Inspired by the shareability
of EPR steering, known as monogamy reidPRA ; Yumonogamy ; Kimmonogamy ;
GSmonogamy ; Adesso16 ; SMmonogamy ; SMmonogamyexp , it is interesting to
explore how can the remotely generated Wigner negativity be distributed over
different modes? Is there any monogamy relations imposing quantitative
constraints on that negativity? And does stronger steerability generate more
negativity?

In this work we present a quantitative investigation of Wigner negativity that
is remotely created via multipartite EPR steering, in which non-Gaussian
operations performed on one steered node of quantum network produce Wigner
negativity in different distant nodes, as shown in Fig. 1. We first
investigate to what extent Wigner negativity can be shared by establishing a
monogamy relation. This constraints the degree of distributed negativity akin
to the Coffman-Kundu-Wootters (CKW) monogamy inequality for steerability
Yumonogamy . Then we focus on photon subtraction, a commonly used non-Gaussian
operation, and find a general measure for the amount of induced Wigner
negativity in the steering modes. This allows us to disprove the conjecture
that stronger steerability creates more negativity. Specifically, considering
the major experimentally realized CV EPR resources such as the two-mode EPR
states with phase-insensitive losses and the two-mode squeezed thermal states,
we show that the procedure for remote generation of Wigner negativity can be
significantly simplified when the covariance matrix (CM) is transformed into
its standard form. This provides a more insightful approach than previously
proposed procedures WalschaersPRL , and makes such states particularly
resourceful for remotely producing Wigner negativity.

Figure 1: Scheme of the remote generation of Wigner negativity through EPR
steering in a multipartite scenario. (a) The initial Gaussian steerable
system; (b) After some appropriate local operations on the steered mode hold
by Alice, the steering subsystem hold by Bob becomes non-Gaussian with Wigner
negativity.

Multimode CV systems.– We begin by briefly introducing the theoretical
framework of multimode CV quantum optics. The noninteracting quantized
electromagnetic field can be treated as a number $N$ of optical modes that
behave as quantum harmonic oscillators with different frequencies described by
$\hat{H}=\sum_{k=1}^{N}2\omega_{k}(\hat{a}_{k}^{\dagger}\hat{a}_{k}+\frac{1}{2})$.
Here, $\hat{a}_{k}$ and $\hat{a}_{k}^{\dagger}$ are the annihilation and
creation operators of a photon in mode $k$, satisfying the bosonic commutation
relation $[\hat{a}_{k},\hat{a}_{k^{\prime}}^{\dagger}]=\delta_{kk^{\prime}}$.
The corresponding quadrature phase operators for each mode are defined as
$\hat{x}_{k}=\hat{a}_{k}+\hat{a}_{k}^{\dagger}$ and
$\hat{p}_{k}=(\hat{a}_{k}-\hat{a}_{k}^{\dagger})/i$. Collecting the quadrature
operators for all the modes into a vector
$\hat{\boldsymbol{\xi}}\equiv(\hat{x}_{1},\hat{p}_{1},...,\hat{x}_{N},\hat{p}_{N})^{\top}$,
the CM $\sigma$ is given with elements
$\sigma_{ij}=\langle\hat{\xi}_{i}\hat{\xi}_{j}+\hat{\xi}_{j}\hat{\xi}_{i}\rangle/2-\langle\hat{\xi}_{i}\rangle\langle\hat{\xi}_{j}\rangle$.
If the system is prepared in a Gaussian state, the properties can be
completely determined by its CM. Otherwise, the first and second-order
statistical moments are not enough to characterize the non-Gaussian system,
and we must resort to a more complete description. Here we choose the Wigner
function as a preferred phase space representation for an arbitrary state with
density matrix $\hat{\rho}$,

$W({\boldsymbol{\xi}})=\int_{\mathbb{R}^{2N}}\frac{d^{2N}\boldsymbol{\alpha}}{(2\pi)^{2N}}\exp\left(-i{\boldsymbol{\xi}}^{\top}\boldsymbol{\Omega}\boldsymbol{\alpha}\right)\chi(\boldsymbol{\alpha}),$
(1)

where $\boldsymbol{\Omega}=\bigoplus_{1}^{N}\left(\begin{array}[]{cc}0&1\\\
-1&0\end{array}\right)$ is the symplectic form and the Wigner characteristic
function
$\chi(\boldsymbol{\alpha})=\text{Tr}[\hat{\rho}\exp(i\hat{\boldsymbol{\xi}}^{\top}\boldsymbol{\Omega}\boldsymbol{\alpha})]$.
A particular attribute of non-Gaussian states is the possibility for this
Wigner function to attain negative values that can be quantified as
$\mathcal{N}\equiv\int|W({\boldsymbol{\xi}})|d{\boldsymbol{\xi}}-1$ Kenfack .

Remote generation of Wigner negativity through multipartite EPR steering.– In
order to effectively generate and distribute Wigner negativity, an indirect
scheme was proposed based on EPR steering WalschaersPRL . In a two-mode
Gaussian system, when there exists steering from Bob to Alice, then an
appropriate local Gaussian transformation together with photon subtraction on
the steered mode $A$ can remotely generate Wigner negativity in the steering
mode $B$, i.e. $\mathcal{N}_{B}>0$. The bipartite Gaussian steerability can be
quantified by the parameter $\mathcal{G}^{B\rightarrow
A}=\max\\{0,\frac{1}{2}\ln\frac{\text{Det\leavevmode\nobreak\
}\sigma_{B}}{\text{Det\leavevmode\nobreak\ }\sigma_{AB}}\\}$, where
$\sigma_{B}$ and $\sigma_{AB}$ denote the CM for mode $B$, and the group
$(AB)$, respectively Adesso15 . This formalism was developed for arbitrary
conditional operations on an arbitrary number of modes, showing that EPR
steering is still necessary to prepare a Wigner-negative state in the steering
modes WalschaersPRXQ . The remotely generated Wigner negativity was not
quantified, nor are its multimode properties such as the shareability of the
negativity among steering modes understood. Especially, since EPR steering is
a prerequisite for remote preparation of Wigner negativity, one may
intuitively expect that stronger steerability in the initial Gaussian states
creates more Wigner negativity. With our quantitative investigation, we show
that conjecture is not the case.

Monogamy of remotely generated Wigner negativity.– First, we study the
multimode character of the remotely generated Wigner negativity by deriving
constraints on the distribution of this negativity among various modes in the
steering party $(B_{1}B_{2}\ldots B_{n})$ for a $(1+n)$-mode Gaussian state
$\sigma_{AB_{1}B_{2}\ldots B_{n}}$. As a fundamental property of EPR steering,
the CKW-type monogamy relation reveals that the sum of Gaussian steerability
between any two modes cannot exceed their intergroup steerability, i.e.,
$\mathcal{G}^{B_{1}B_{2}\ldots B_{n}\rightarrow
A}\geq\sum_{i=1}^{n}\mathcal{G}^{B_{i}\rightarrow A}$, which bounds the key
rate in quantum secret sharing Yumonogamy . In analogy with the steering
constraint, we establish a monogamy relation for the amount of the generated
Wigner negativity,

$\mathcal{N}_{B_{1}B_{2}\ldots B_{n}}(\mathcal{L}_{A|B_{1}B_{2}\ldots
B_{n}})\geq\sum_{i=1}^{n}\mathcal{N}_{B_{i}}(\mathcal{L}_{A|B_{i}}),$ (2)

where $\mathcal{N}_{B_{j}\ldots B_{k}}$ denotes the Wigner negativity created
in the set of modes $(B_{j}\ldots B_{k})$ by performing some appropriate
operations $\mathcal{L}_{A|B_{j}\ldots B_{k}}$ on subsystem $A$. Note that
generating negativities in different set of modes requires different optimal
operations on the steered mode $A$. For instance, inducing Wigner negativity
in the steering mode $B_{j}$, or $B_{k}$, or their joint $(B_{j}B_{k})$,
requires different local Gaussian transformations prior to a non-Gaussian
operation.

Figure 2: (a) Scheme of a pure three-mode Gaussian state generated by the
linear optical network. (b) Fixing $R_{1}:(1-R_{1})=50:50$ and input squeezing
levels $r=1$ (corresponding to $-8.7{\rm dB}$ quadrature noise), the initial
Gaussian steerability changes with a variable $R_{2}$. (c) After a single-
photon subtraction on mode $A$, the remotely generated Wigner negativity of
mode $B$, mode $C$ and the group $(BC)$, respectively.

To prove the above inequality we use the fact that
$\mathcal{N}_{B_{j}}(\mathcal{L}_{A|B_{j}})>0$ and
$\mathcal{N}_{B_{k}}(\mathcal{L}_{A|B_{k}})>0$ ($j\neq k$) cannot be true
simultaneously, which is a consequence of another type of Gaussian steering
monogamy relation: modes $B_{j}$ and $B_{k}$ cannot simultaneously steer mode
$A$ under Gaussian measurements reidPRA ; Kimmonogamy . Assuming that mode
$B_{1}$ can steer mode $A$, negativity can be generated only in the Wigner
function of mode $B_{1}$ under Alice’s local operation
$\mathcal{L}_{A|B_{1}}$. We then just need to prove that
$\mathcal{N}_{B_{1}B_{2}\ldots B_{n}}(\mathcal{L}_{A|B_{1}B_{2}\ldots
B_{n}})\geq\mathcal{N}_{B_{1}}(\mathcal{L}_{A|B_{1}})$. The detailed proof is
deferred to the Supplemental Material Supp . An example is given in Fig. 2.
Here the local non-Gaussian operation we chose for Alice is single-photon
subtraction $\mathcal{S}$, which can be effectively realized in experiments
npnongaussian ; scisubtraction . In this system, a pure three-mode entangled
Gaussian state can be generated by the linear optical network with three
squeezed inputs, as illustrated in Fig. 2(a). When the first beam splitter is
fixed at $R_{1}:(1-R_{1})=50:50$ and the second beam splitter is adjustable,
the steerability distributed among three modes in the initial Gaussian state
and the corresponding Wigner negativities remotely created through photon
subtraction on mode $A$ are shown in Figs. 2(b) and (c), respectively. It is
clear that the two-mode and three-mode Gaussian steering
$\mathcal{G}^{B_{1}\rightarrow A}$, $\mathcal{G}^{B_{2}\rightarrow A}$ and
$\mathcal{G}^{B_{1}B_{2}\rightarrow A}$ are necessary to induce Wigner
negativity for individual mode $B_{1}$, $B_{2}$, and for the group
$(B_{1}B_{2})$, respectively. In addition, we observe that, even though
$\mathcal{N}_{B_{1}}(\mathcal{S}_{A|B_{1}})>0$ and
$\mathcal{N}_{B_{2}}(\mathcal{S}_{A|B_{2}})>0$ cannot be satisfied at the same
time, the joint Wigner negativity created on the group $(B_{1}B_{2})$ is
significantly higher than the negativity in either individual mode.

Quantification of the generated Wigner negativity.– Since remotely generated
Wigner negativity is both enabled and constrained by Gaussian steering, one
may intuitively expect that stronger steerability induces more Wigner
negativity. To show that –unexpectedly– this is not the case, we quantify the
amount of Wigner negativity in the steering modes, via the purities of initial
Gaussian states. We present an explicit study for some experimentally
prominent two-mode Gaussian states and show that purity, rather than
steerability, governs the amount of Wigner negativity that can be created.

It is well known that any two-mode Gaussian state can be transformed into a
standard form standard  through local linear unitary Bogoliubov operations
(LLUBOs), so that the CM $\sigma_{AB}$ reads

$\sigma_{AB,sf}=\left(\begin{array}[]{cc}\sigma_{A}&\gamma_{AB}\\\
\gamma_{AB}^{\top}&\sigma_{B}\\\
\end{array}\right)=\left(\begin{array}[]{cccc}a&0&c_{1}&0\\\ 0&a&0&c_{2}\\\
c_{1}&0&b&0\\\ 0&c_{2}&0&b\\\ \end{array}\right)$ (3)

with $a,\leavevmode\nobreak\ b\geq 1$ and $ab-c_{1(2)}^{2}\geq 0$. The two
local purities $\mu_{A(B)}\equiv 1/\sqrt{\text{Det\leavevmode\nobreak\
}\sigma_{A(B)}}=1/a(b)$ and the global purity $\mu_{AB}\equiv
1/\sqrt{\text{Det\leavevmode\nobreak\ }\sigma_{AB}}=1/[(ab-c_{1}^{2})(ab-
c_{2}^{2})]$, are invariant under LLUBOs pra73 . Furthermore, because LLUBOs
are local, the standard form $\sigma_{AB,sf}$ manifests equal quantity of
Gaussian steerability possessed in the initial states.

We now restrict our scope to two-mode Gaussian states $c_{1}=-c_{2}=c$, which
include states generated by parametric down-conversion. By focusing on the
experimentally relevant case where a photon subtracted from the steered mode,
we derive that the amount of remotely generated Wigner negativity
$\mathcal{N}_{B}$ is determined by the purities of initial Gaussian state
$\mu_{A}$, $\mu_{B}$ and $\mu_{AB}$:

$\mathcal{N}_{B}(\mathcal{S}_{A|B})=2\left[\frac{e^{\frac{\mu_{A}\mu_{B}-\mu_{AB}\mu_{A}}{\mu_{AB}-\mu_{A}\mu_{B}}}(\mu_{A}\mu_{B}-\mu_{AB})}{\mu_{AB}(\mu_{A}-1)}-1\right].$
(4)

Exchanging $\mu_{A}\leftrightarrow\mu_{B}$ we can obtain the result for the
other direction $\mathcal{N}_{A}(\mathcal{S}_{B|A})$. The derivation of the
above relation is detailed in the Supplemental Material Supp . EPR steering
and remotely induced Wigner negativity are both determined by the local and
global purities, but their dependence on these purities is very different. EPR
steering provides a necessary bridge to induce Wigner negativity, but it is
insufficient to unambiguously quantify the created Wigner negativity.

We explicitly show this point in Fig. 3, where we consider a two-mode EPR
state with one lossy channel on mode $A$, which is often used to demonstrate
one-way steering ANUexp ; prlSu ; OneWayNatPhot . This state is already in the
standard form (3), with $a=\eta_{A}(\cosh 2r-1)+1$, $b=\cosh 2r$, and
$c_{1}=-c_{2}=\sqrt{\eta_{A}}\sinh{2r}$, where $r$ is the squeezing parameter
and $\eta_{A}$ is the transmission efficiency. The asymmetric Gaussian
steerabilities in two directions are indicated in Fig. 3(b). We find that the
Gaussian steerability $\mathcal{G}^{A\rightarrow B}>0$ with a threshold level
of efficiency $\eta_{A}>0.5$ and becomes stronger with a higher squeezing
level. Thus, by performing a single-photon subtraction on the steered mode
$B$, the Wigner negativity $\mathcal{N}_{A}(\mathcal{S}_{B|A})$ can be
generated when $\eta_{A}>0.5$ as well and becomes larger with increasing
efficiency $\eta_{A}$, as shown in Fig. 3(c). The Gaussian steerability in the
other direction $\mathcal{G}^{B\rightarrow A}>0$ happens for any $\eta_{A}>0$
and enhances with a higher squeezing level. Interestingly, for this direction,
by performing a single-photon subtraction on the steered mode $A$, the
generated negativity $\mathcal{N}_{B}(\mathcal{S}_{A|B})$ does not vary with
$\eta_{A}$. From Eq. (4), we can obtain that
$\mathcal{N}_{B}(\mathcal{S}_{A|B})=2e^{-\mu_{B}/(\mu_{B}+1)}(\mu_{B}+1)-2$,
which is solely determined by the local purity $\mu_{B}$. It is unchanged
since no loss is considered in mode $B$. While the value of
$\mathcal{N}_{A}(\mathcal{S}_{B|A})$ depends on $\mu_{A,\leavevmode\nobreak\
B,\leavevmode\nobreak\ AB}$ and thus varies with $\eta_{A}$, as shown in Fig.
3(d). This highlights the asymmetry of the induced Wigner negativity and
suggests a way to remotely generate negativity that is robust to channel loss.
Although the states produced by a higher squeezing level (blue lines) possess
stronger Gaussian steerability, their purities are more sensitive to the loss
purityG . Therefore, the created Wigner negativity in the steering mode is
larger when the initial squeezing level is lower.

Figure 3: (a) Scheme of a two-mode squeezed vacuum state with one lossy
channel on mode $A$, where $R_{1}$ is a balanced beam splitter. (b) The
initial asymmetric Gaussian steerability with different squeezing levels $r=1$
(blue) and $r=0.85$ (red), corresponding to a quadrature noise reduction of
$-8.7{\rm dB}$ and $-7.4{\rm dB}$, respectively. Loss has a more significant
effect on the steering mode. (c) After the single-photon subtraction on the
steered mode, the amount of induced Wigner negativity on the steering mode
corresponding to Gaussian steerability given in (b). (d) The local and global
purities of the initial Gaussian states with different squeezing levels.

Moreover, for any globally pure state, i.e., $\mu_{AB}=1$, with
$\mu_{A}=\mu_{B}=\mu$, Eq. (4) is simplified to be

$\mathcal{N}_{B}(\mathcal{S}_{A|B})=\mathcal{N}_{A}(\mathcal{S}_{B|A})=2\left[e^{-\frac{\mu}{1+\mu}}(1+\mu)-1\right].$
(5)

We show that Eq. 5 bounds the remotely created Wigner negativity for arbitrary
mixed two-mode states. In Fig. 4, we plot 250000 dots which present the Wigner
negativities of randomly generated by photon subtraction from two-mode
Gaussian states. All dots are located below the red curve described by Eq. 5.
Furthermore, the CM for an arbitrary three-mode pure state
$\sigma_{A-(B_{1}B_{2})}^{\text{pure}}$ with respect to $A-(B_{1}B_{2})$
splitting is locally equivalent to

$\sigma_{A-(B_{1}B_{2}),sf}^{\text{pure}}=\left(\begin{array}[]{cccccc}a&0&c_{1}&0&0&0\\\
0&a&0&-c_{1}&0&0\\\ c_{1}&0&a&0&0&0\\\ 0&-c_{1}&0&a&0&0\\\ 0&0&0&0&1&0\\\
0&0&0&0&0&1\end{array}\right)$ (6)

through Williamson decomposition jpaGA , where $c_{1}=\sqrt{a^{2}-1}$. We can
then see that the system is expressed by a product of a two-mode squeezed
vacuum state tensor an uncorrelated vacuum mode, such that steering property
between mode $A$ and modes $(B_{1}B_{2})$ can be unitarily reduced to the two-
mode case. Thus, the generated Wigner negativity of the joint group
$(B_{1}B_{2})$ is equivalently indicated by the red curve in Fig. 4, which is
always higher than the negativity induced in the individual mode. Besides the
examples discussed above, we also analyze the asymmetric Gaussian steerability
and the properties of induced Wigner negativity for another important CV EPR
resource–two-mode squeezed thermal states He15 , which are detailed in the
Supplemental Material Supp .

Figure 4: The induced Wigner negativity $\mathcal{N}_{B}(\mathcal{S}_{A|B})$
for $\sim 250000$ randomly mixed two-mode Gaussian states (dots) are bounded
by Eq. (5) (red curve).

Finally, we stress that when the CM of a bipartite system is transformed in
the standard form and satisfies $c_{1}=-c_{2}=c$, photon subtraction on mode
$A$ can always generate Wigner negativity in mode $B$ as long as
$\mathcal{G}^{B\rightarrow A}>0$. The proof is deferred in the Supplemental
Material Supp . Thus producing EPR resource with CM in a standard form
significantly simplifies the procedure for remote generation of Wigner
negativity and makes the resulting non-Gaussian state readily available for
further applications. This complements the results of Ref. WalschaersPRL ,
where it was shown that an additional local Gaussian transformation prior to
photon subtraction is necessary to make EPR steering sufficient for remotely
generating Wigner negativity. This Gaussian transformation requires inline
squeezing and is experimentally challenging. Our results can be used as a
recipe to prepare resourceful Gaussian states for the remote generation of
Wigner negativity without the need for inline squeezing.

Conclusion.– We develop the scheme for remote generation of Wigner negativity
through EPR steering to multimode scenario, and show the presence of
constraints for distributing Wigner negativity over different modes. So far,
multipartite steering has been demonstrated in various Gaussian systems, e.g.,
linear optical networks ANUexp ; prlSu , quantum frequency comb yin , and
Bose-Einstein condensates bec2 . These experimental developments lay a
favorable foundation to implement remote generation of multipartite non-
Gaussian states through photon subtraction or other appropriate operations.
Furthermore, we present an intuitive and computable quantification of the
generated Wigner negativity for bipartite system in terms of the local and
global purities of initial Gaussian state. Our results deepen the
understanding of Wigner negativity as a resource and provide an important
framework of non-Gaussian quantum information theory.

Our work also triggers several new questions to stimulate further research.
For instance, as Gaussian steerability $\mathcal{G}^{A\rightarrow B_{1}}>0$
and $\mathcal{G}^{A\rightarrow B_{2}}>0$ can happen simultaneously, then by
performing a single-photon subtraction on each mode $B_{1}$, $B_{2}$, can we
achieve more significant increase of the negativity in mode $A$? In addition,
for this direction the Gaussian steerability still follows the CKW-type
monogamy constraint, however, this constraint does not hold any more for the
generated negativity. We have observed a violation in a pure three-mode state
Supp , i.e.,
$\mathcal{N}_{A}(\mathcal{L}_{B_{1}B_{2}|A})<\mathcal{N}_{A}(\mathcal{L}_{B_{1}|A})+\mathcal{N}_{A}(\mathcal{L}_{B_{2}|A})$.
Moreover, after non-Gaussian operations on the steered mode, the resulting
system cannot be fully captured by the second-order correlations given in CM.
To this day, relatively little is known about the characteristics of non-
Gaussian steering Yunon .

###### Acknowledgements.

This work is supported by the National Natural Science Foundation of China
(Grants No. 11975026, No. 61675007, and No. 12004011), the National Key R$\&$D
Program of China (Grants No. 2016YFA0301302, No. 2018YFB1107205, and
2019YFA0308702). Q.H. also thanks partial support from Beijing Natural Science
Foundation (Grant No. Z190005) and the Key R$\&$D Program of Guangdong
Province (Grant No. 2018B030329001). N.T. and M.W. received funding from the
European Union’s Horizon 2020 research and innovation programme under grant
agreement No 899587.

‡Y. Xiang and S. H. Liu contributed equally to this work.

*

## Appendix A: Monogamy of Wigner negativity remotely created through EPR
steering

Monogamy is known as a fundamental property of multipartite quantum
correlations, which has profound applications in quantum communication. In
this Section, we prove that the remotely created Wigner negativity cannot be
freely shared among different parties by establishing a monogamy relation in
direct analogy with the Coffman-Kundu-Wootters (CKW) inequality for EPR
steering Yumonogamy , as given by Eq. (2) in the main text.

Without loss of generality, let us focus on a tripartite scenario, in which
the steering party $B$ contains two modes $B_{1}$ and $B_{2}$, such that the
monogamy inequality takes the simpler form,

$\mathcal{N}_{B_{1}B_{2}}(\mathcal{L}_{A|B_{1}B_{2}})\geqslant\mathcal{N}_{B_{1}}(\mathcal{L}_{A|B_{1}})+\mathcal{N}_{B_{2}}(\mathcal{L}_{A|B_{2}}),$
(7)

when the subsystem $A$ can be steered by the group $(B_{1}B_{2})$. By taking
$A$ being a single mode and $B_{1},B_{2}$ comprising an arbitrary number of
modes, one can then apply iteratively above inequality to obtain the
corresponding general multipartite inequality (2) in the main text.

To do so, recall the Gaussian steering monogamy relations from Refs. reidPRA ;
Kimmonogamy  that it is impossible for modes $B_{1}$ and $B_{2}$ to
simultaneously steer the mode $A$ under Gaussian measurements, i.e.,
$\mathcal{N}_{B_{1}}(\mathcal{L}_{A|B_{1}})>0$ and
$\mathcal{N}_{B_{2}}(\mathcal{L}_{A|B_{2}})>0$ cannot be true simultaneously.
Granted that $B_{1}$ can steer mode $A$, the monogamy relation (7) reduces to
$\mathcal{N}_{B_{1}B_{2}}(\mathcal{L}_{A|B_{1}B_{2}})\geqslant\mathcal{N}_{B_{1}}(\mathcal{L}_{A|B_{1}})$
(or the analogous expression with swapped $B_{1}\leftrightarrow B_{2}$).
Because the Wigner negativity is nonincreasing under partial trace on the
steering party $(B_{1}B_{2})$ Ferraropra , we have

$\mathcal{N}_{B_{1}B_{2}}(\mathcal{L}_{A|B_{1}})\geqslant\mathcal{N}_{B_{1}}(\mathcal{L}_{A|B_{1}}).$
(8)

Note that the left and right sides of above inequality are given by the same
resulting state $\rho_{B_{1}B_{2}}$ after a local operation
$\mathcal{L}_{A|B_{1}}$ on the steered mode $A$. The proof is straightforward
Ferraropra , as given by

$\displaystyle\mathcal{N}_{B_{1}}(\mathcal{L}_{A|B_{1}})$
$\displaystyle=\mathcal{N}[\operatorname{Tr}_{B_{2}}\rho_{B_{1}B_{2}}]$ (9)
$\displaystyle=\int
d\boldsymbol{r}_{B_{1}}\left|W[\operatorname{Tr}_{B_{2}}[\rho_{B_{1}B_{2}}]](\boldsymbol{r}_{B_{1}})\right|-1$
$\displaystyle=\int d\boldsymbol{r}_{B_{1}}\left|\int
d\boldsymbol{r}_{B_{2}}W[\rho_{B_{1}B_{2}}](\boldsymbol{r}_{B_{1}},\boldsymbol{r}_{B_{2}})\right|-1$
$\displaystyle\leqslant\int d\boldsymbol{r}_{B_{1}}\int
d\boldsymbol{r}_{B_{2}}\left|W[\rho_{B_{1}B_{2}}](\boldsymbol{r}_{B_{1}},\boldsymbol{r}_{B_{2}})\right|-1$
$\displaystyle=\mathcal{N}[\rho_{B_{1}B_{2}}]=\mathcal{N}_{B_{1}B_{2}}(\mathcal{L}_{A|B_{1}}),$

where $W[\rho_{B_{1}B_{2}}](\boldsymbol{r})$ represents the Wigner
distribution of state $\rho_{B_{1}B_{2}}$. Afterwards it is promptly verified
that there always exists an optimized local operation
$\mathcal{L}_{A|B_{1}B_{2}}$ can generated the largest Wigner negativity
$\mathcal{N}_{B_{1}B_{2}}(\mathcal{L}_{A|B_{1}B_{2}})$ to make
$\mathcal{N}_{B_{1}B_{2}}(\mathcal{L}_{A|B_{1}B_{2}})\geq\mathcal{N}_{B_{1}B_{2}}(\mathcal{L}_{A|B_{1}})\geq\mathcal{N}_{B_{1}}(\mathcal{L}_{A|B_{1}})$.

## Appendix B: Quantifying the remotely created Wigner negativity

It is of particular interest to us is whether stronger steerability in the
initial Gaussian states induces more Wigner negativity, as it is enabled and
constrained by Gaussian steering. To answer this, we need first quantify the
amount of Wigner negativity. In this part, we aim to derive the qualitative
measure of Wigner negativity Eq. (4) in the main text by focusing on the
experimentally relevant case where a photon subtracted from the steered mode
in two-mode Gaussian states $c_{1}=-c_{2}=c$, which includes the major
experimentally realized CV EPR states such as the two-mode squeezed vacumm
states with phase-insensitive losses and the two-mode squeezed thermal states
(STS).

Let us recall that, any two-mode Gaussian state can be transformed through
LLUBOs to the standard form

$\sigma_{AB,sf}=\left(\begin{array}[]{cc}\sigma_{A}&\gamma_{AB}\\\
\gamma_{AB}^{\top}&\sigma_{B}\\\
\end{array}\right)=\left(\begin{array}[]{cccc}a&0&c_{1}&0\\\ 0&a&0&c_{2}\\\
c_{1}&0&b&0\\\ 0&c_{2}&0&b\\\ \end{array}\right).$ (10)

Using the formula derived from Ref. WalschaersPRL , we obtain the reduced
Wigner function of the steering mode $B$ after the local Gaussian
transformation $R_{A|B}$ combined with a single-photon subtraction applied on
the steered mode $A$,

$\displaystyle W_{B}\left(\beta_{B}\right)$
$\displaystyle=\frac{\exp\left\\{-\frac{1}{2}\left(\beta_{B},\sigma_{B}^{-1}\beta_{B}\right)\right\\}}{2\pi\sqrt{\operatorname{Det}\sigma_{B}}\left[\operatorname{Tr}\left(R_{A|B}^{\top}\sigma_{A}R_{A|B}\right)-2\right]}\times$
(11) $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\left[\beta_{B}^{\top}{\sigma_{B}^{-1}}^{\top}\gamma_{AB}^{\top}R_{A|B}R_{A|B}^{\top}\gamma_{AB}\sigma_{B}^{-1}\beta_{B}+\operatorname{Tr}(R_{A|B}^{\top}V_{A\mid
B}R_{A|B})-2\right]$
$\displaystyle=\frac{\exp\left\\{-\frac{1}{2}\left(\beta_{B},\sigma_{B}^{-1}\beta_{B}\right)\right\\}}{2\pi\sqrt{\operatorname{Det}\sigma_{B}}\left[\operatorname{Tr}\left(\nu\sigma_{A}V_{A\mid
B}^{-1}\right)-2\right]}\times$ $\displaystyle\leavevmode\nobreak\
\leavevmode\nobreak\
\left(\nu\beta_{B}^{\top}{\sigma_{B}^{-1}}^{\top}\gamma_{AB}^{\top}V_{A\mid
B}^{-1}\gamma_{AB}\sigma_{B}^{-1}\beta_{B}+2\nu-2\right),$

where $\vec{\beta}_{B}$ is the coordinate in a multimode phase spaces of
subsystem $B$, $V_{A\mid
B}=\sigma_{A}-\gamma_{AB}\sigma_{B}^{-1}\gamma_{AB}^{\top}$ is the Schur
complement of $\sigma_{B}$ and $\nu$ is the corresponding symplectic
eigenvalue. The Schur complement $V_{A\mid B}$ can be decomposed through
Williamson decomposition via $V_{A\mid B}=\nu S_{A|B}^{\top}S_{A|B}$, where
$S_{A|B}$ is the corresponding symplectic matrix and a local Gaussian
transformation $R_{A|B}=S_{A|B}^{-1}$. When it comes to our particular
interest subclass $c_{1}=-c_{2}=c$, it is easy to find out that the Schur
complement is a multiple of identity matrix so that there is no need to
perform an additional local Gaussian operation. Then we get

$W_{B}\left(\rho\right)=\frac{e^{-\frac{\rho^{2}}{2b}}\left(2b^{2}(a-1)-2bc^{2}+c^{2}\rho^{2}\right)}{4\pi
b^{3}(a-1)},$ (12)

which is circular symmetric. It is straightforward to calculate Wigner
negativity using integral,

$\mathcal{N}_{B}(\mathcal{S}_{A|B})=\frac{2c^{2}e^{\frac{b(a-1)}{c^{2}}-1}}{b(a-1)}-2.$
(13)

By expressing $\mathcal{N}_{B}(\mathcal{S}_{A|B})$ in terms of purity
$\mu_{AB}=1/(ab-c^{2}),\leavevmode\nobreak\ \mu_{A}=1/a,\leavevmode\nobreak\
\mu_{B}=1/b$, Eq. (13) becomes

$\mathcal{N}_{B}(\mathcal{S}_{A|B})=2\left[\frac{e^{\frac{\mu_{A}\mu_{B}-\mu_{AB}\mu_{A}}{\mu_{AB}-\mu_{A}\mu_{B}}}(\mu_{A}\mu_{B}-\mu_{AB})}{\mu_{AB}(\mu_{A}-1)}-1\right].$
(14)

Figure A1: (a) Scheme of a two-mode squeezed thermal state with asymmetric
thermal noise $n_{A}$ and $n_{B}=0$. (b) The initial asymmetric Gaussian
steerability with fixed squeezing level of $r=0.6$ (corresponding to $5.2{\rm
dB}$ quadrature noise), where thermal noise has a more significant effect on
the steered mode. (c) Corresponding to (b), after the single-photon
subtraction on one side, the remotely generated Wigner negativity of the other
side. (d) The local and global purities of the initial Gaussian states.

Figure 3 in the main text showed the case of two-mode squeezed vacuum state
with one lossy channel on one mode. Here, we particularly show the results for
the two-mode squeezed thermal states. The CM elements of these states are
$a=(n_{A}+n_{B}+1)\cosh(2r)+(n_{A}-n_{B})$,
$b=(n_{A}+n_{B}+1)\cosh(2r)-(n_{A}-n_{B})$,
$c_{1}=-c_{2}=(n_{A}+n_{B}+1)\sinh(2r)$, where $n_{A}$, $n_{B}$ are the
average number of thermal photons for each subsystem He15 . We set the thermal
noise only on mode $A$ and leave $n_{B}=0$, as illustrated in Fig. A1(a). The
asymmetric Gaussian steerability in two directions varying with $n_{A}$ is
denoted in Fig. A1(b), and as a consequence the induced Wigner negativity on
the steering mode by applying single-photon subtraction on the steered mode is
quantified in Fig. A1(c). Note that the effect of thermal noise on the steered
mode is more significant than that on the steering mode, which is opposite to
the effect of losses on two modes in the main text. As there exists a thermal
barrier in the direction $\mathcal{G}^{B\rightarrow A}$, and correspondingly,
a nonzero $\mathcal{N}_{B}(\mathcal{S}_{A|B})$ can exist only when
$n_{A}<0.6816$. In the opposite direction, $\mathcal{G}^{A\rightarrow B}>0$
and thus $\mathcal{N}_{A}(\mathcal{S}_{B|A})>0$ for arbitrarily large value of
thermal noise $n_{A}$. The values of remotely created Wigner negativity are
quantitatively determined by the purities of initial states as given in Eq. 14
and plotted in Fig. A1(d).

## Appendix C: Necessity of the additional local Gaussian transformation
$R_{A|B}$

Here, we prove that the additional local Gaussian transformation required
prior to photon subtraction in Ref. WalschaersPRL  to optimize the induced
Wigner negativity is no longer needed if and only if the CM of two-mode
Gaussian states in the standard form $\sigma_{AB,sf}$ satisfies
$c_{1}=-c_{2}=c$. This makes preparation resourceful Gaussian states for the
remote generation of Wigner negativity without the need for inline squeezing.

From the standard form $\sigma_{AB,sf}$, the Schur complement of $\sigma_{B}$
is given by

$V_{A\mid B}=\left(\begin{array}[]{cc}a-c_{1}^{2}/b&0\\\
0&a-c_{2}^{2}/b\end{array}\right),$ (15)

whose symplectic eigenvalue is $v=\sqrt{(a-c_{1}^{2}/b)(a-c_{2}^{2}/b)}$. When
there exists Gaussian steering $\mathcal{G}^{B\rightarrow A}$, the symplectic
eigenvalue $v$ must be smaller than 1 Adesso15 . Without any local Gaussian
transformation $R_{A|B}$ prior to the photon subtraction on mode $A$, the
condition for $W_{B}\left(\beta_{B}\right)<0$ should be
$\operatorname{tr}\left[V_{A\mid B}\right]<2$. Note that every CM
$\sigma_{AB}$ that corresponds to a physical quantum state has to satisfy the
bona fide condition $a-c_{1}^{2}/b>0$ and $a-c_{2}^{2}/b>0$ Adesso15 , then we
have

$\operatorname{Tr}\left[V_{A\mid
B}\right]=(a-\frac{c_{1}^{2}}{b})+(a-\frac{c_{2}^{2}}{b})\geqslant
2\sqrt{(a-\frac{c_{1}^{2}}{b})(a-\frac{c_{2}^{2}}{b})}=2v.$ (16)

The above inequality can be saturated if and only if $c_{1}^{2}=c_{2}^{2}$.
With this condition, $\operatorname{tr}\left[V_{A\mid B}\right]<2$ is
equivalent to $v<1$, i.e., the photon subtraction on mode $A$ can always
generate Wigner negativity in mode $B$ as long as $\mathcal{G}^{B\rightarrow
A}>0$ without any prior local Gaussian transformation. Otherwise, if
$c_{1}^{2}\neq c_{2}^{2}$, then $\operatorname{tr}\left[V_{A\mid
B}\right]>2v$, which means an additional local Gaussian transformation
$R_{A|B}$ is necessary to make EPR steering sufficient for remotely generating
Wigner negativity.

## Appendix D: The CKW-type monogamy constraint to the distribution of Wigner
negativity created in the other direction

Steering is a directional form of nonlocality, related to the Einstein
“spooky” paradox, which is fundamentally defined differently to entanglement.
In previous work Yumonogamy , we have derived the CKW-type monogamy
inequalities for multipartite Gaussian steering in two directions. We then
wonder whether the CKW-type monogamy inequality holds for the distribution of
Wigner negativity created in the opposite direction of that studied in the
main text as well as in (S1), i.e.,
$\mathcal{N}_{A}(\mathcal{L}_{B_{1}B_{2}|A})\geq\mathcal{N}_{A}(\mathcal{L}_{B_{1}|A})+\mathcal{N}_{A}(\mathcal{L}_{B_{2}|A})$.

We present a three-mode entangled Gaussian state that is similar to the case
shown in Fig. 2(a) in the main text, but with the first beamsplitter being
adjustable $R_{1}:(1-R_{1})$ and the second fixed as a balanced one. In
particular, when the first beamsplitter is adjusted at $R_{1}=1/3$, the
initial Gaussian state is produced as a GHZ-like state. For the direction
where mode $A$ acts as the steering party to steer the modes
$B_{1},\leavevmode\nobreak\ B_{2}$, the Gaussian steerability distributed
among three modes and the corresponding Wigner negativities remotely created
by a sing-photon subtraction on the individual or joint modes $B_{1},B_{2}$
are denoted in Figs. A2(a) and (b). It is clear that the two-mode and three-
mode Gaussian steerability $\mathcal{G}^{A\rightarrow B_{1}(B_{2})}>0$ and
$\mathcal{G}^{A\rightarrow B_{1}B_{2}}>0$ are necessary to induce negativities
$\mathcal{N}_{A}(\mathcal{S}_{B_{1}(B_{2})|A})$ and
$\mathcal{N}_{A}(\mathcal{S}_{B_{1}B_{2}|A})$ in the Wigner functions of the
steering mode $A$, respectively. Interestingly, we observe that
$\mathcal{N}_{A}(\mathcal{S}_{B_{1}B_{2}|A})<\mathcal{N}_{A}(\mathcal{S}_{B_{1}|A})+\mathcal{N}_{A}(\mathcal{S}_{B_{2}|A})$
when $R_{1}$ approaches to $1$ (Fig. A2(d)), even though the Gaussian
steerability still follows the monogamy constraint, i.e.,
$\mathcal{G}^{A\rightarrow B_{1}B_{2}}-\mathcal{G}^{A\rightarrow
B_{1}}-\mathcal{G}^{A\rightarrow B_{2}}>0$ presented in Fig. A2(c). This
settles an open question for the shareability of generated Wigner negativity
in this direction.

Figure A2: Scheme of a pure three-mode Gaussian state generated by the linear
optical network similar to Fig. 2 in main text but with varying $R_{1}$ and
fixed $R_{2}:(1-R_{2})=50:50$. The input squeezing levels $r=1$ corresponding
to $-8.7{\rm dB}$ quadrature noise. (a) The initial Gaussian steerabilities
varying with $R_{1}$. (b) The corresponding transferred Wigner negativities in
the steering mode $A$ by applying single-photon subtraction on the individual
mode $B_{1}(B_{2})$, or their joint $(B_{1}B_{2})$, respectively. (c) The
residual tripartite Gaussian steering $\mathcal{G}^{A\rightarrow
B_{1}B_{2}}-\mathcal{G}^{A\rightarrow B_{1}}-\mathcal{G}^{A\rightarrow
B_{2}}\geq 0$ which is stemming from the reverse CKW-type monogamy. (d) The
residual tripartite Wigner negativity remotely created in mode $A$,
$\mathcal{N}_{A}(\mathcal{S}_{B_{1}B_{2}|A})-\mathcal{N}_{A}(\mathcal{S}_{B_{1}|A})-\mathcal{N}_{A}(\mathcal{S}_{B_{2}|A})$,
which is not alway larger than zero.

## References

  * (1) S. L. Braunstein and P. van Loock, Quantum information with continuous variables, Rev. Mod. Phys. 77, 513–577 (2005).
  * (2) S. L. Braunstein and H. J. Kimble, Teleportation of continuous quantum variables, Phys. Rev. Lett. 80, 869 (1998).
  * (3) F. Grosshans and P. Grangier, Continuous variable quantum cryptography using coherent states, Phys. Rev. Lett. 88, 057902 (2002).
  * (4) S. H. Tan, B. I. Erkmen, V. Giovannetti, S. Guha, S. Lloyd, L. Maccone, S. Pirandola, and J. H. Shapiro, Quantum illumination with Gaussian states, Phys. Rev. Lett. 101, 253601 (2008).
  * (5) Q. Zhuang, Z. Zhang, and J. H. Shapiro, Optimum mixed-state discrimination for noisy entanglement-enhanced sensing, Phys. Rev. Lett. 118, 040801 (2017).
  * (6) U. L. Andersen, J. S. Neergaard-Nielsen, P. van Loock, and A. Furusawa. Hybrid discrete- and continuous-variable quantum information, Nat. Phys. 11, 713–719 (2015).
  * (7) J. Eisert, S. Scheel, and M. B. Plenio, Distilling Gaussian states with Gaussian operations is impossible, Phys. Rev. Lett. 89, 137903 (2002).
  * (8) H. Takahashi, J. S. Neergaard-Nielsen, M. Takeuchi, M. Takeoka, K. Hayasaka, A. Furusawa, and M. Sasaki, Entanglement distillation from Gaussian input states, Nat. photonics 4, 178-181 (2010).
  * (9) J. Niset, J. Fiurášek, and N. J. Cerf, No-go theorem for Gaussian quantum error correction, Phys. Rev. Lett. 102, 120501 (2009).
  * (10) J. Lee, J. Park, and H. Nha, Quantum non-Gaussianity and secure quantum communication, npj Quantum Information 5, 49 (2019).
  * (11) Z. B. Chen, J. W. Pan, G. Hou, and Y. D. Zhang, Maximal violation of Bell’s inequalities for continuous variable systems, Phys. Rev. Lett. 88, 040406 (2002).
  * (12) Y. S. Ra, A. Dufour, M. Walschaers, C. Jacquard, T. Michel, C. Fabre, and N. Treps, Non-Gaussian quantum states of a multimode light field, Nat. Phys. 16, 144–147 (2020).
  * (13) C. W. S. Chang, C. Sabín, P. Forn-Díaz, F. Quijandría, A. M. Vadiraj, I. Nsanzineza, G. Johansson, and C. M. Wilson, Observation of three-photon spontaneous parametric down-conversion in a superconducting parametric cavity, Phys. Rev. X 10, 011011 (2020).
  * (14) N. C. Menicucci, P. Van Loock, M. Gu, C. Weedbrook, T. C. Ralph, M. A. Nielsen, Universal quantum computation with continuous-variable cluster states. Phys. Rev. Lett. 97, 110501 (2006).
  * (15) R. Takagi and Q. Zhuang, Convex resource theory of non-Gaussianity, Phys. Rev. A 97, 062337 (2018).
  * (16) F. Albarelli, M. G. Genoni, M. G. A. Paris, and A. Ferraro, Resource theory of quantum non-Gaussianity and Wigner negativity, Phys. Rev. A 98, 052350 (2018).
  * (17) A. Mari and J. Eisert, Positive Wigner functions render classical simulation of quantum computation efficient, Phys. Rev. Lett. 109, 230503 (2012).
  * (18) M. Walschaers, and N. Treps, Remote generation of Wigner negativity through Einstein-Podolsky-Rosen steering, Phys. Rev. Lett. 124, 150501 (2020).
  * (19) M. Walschaers, V. Parigi, and N. Treps, Practical framework for conditional non-Gaussian quantum state preparation, PRX Quantum 1, 020305 (2020)
  * (20) E. Schrödinger, Discussion of probability relations between separated systems. Proc. Camb. Phil. Soc. 31, 555–563 (1935).
  * (21) H. M. Wiseman, S. J. Jones, and A. C. Doherty, Steering, entanglement, nonlocality, and the Einstein-Podolsky-Rosen paradox, Phys. Rev. Lett. 98, 140402 (2007).
  * (22) M. D. Reid, P. D. Drummond, W. P. Bowen, E. G. Cavalcanti, P. K. Lam, H. A. Bachor, U. L. Andersen, and G. Leuchs, Colloquium: The Einstein-Podolsky-Rosen paradox: From concepts to applications, Rev. Mod. Phys. 81, 1727–1751 (2009).
  * (23) Q. Y. He and M. D. Reid, Genuine multipartite Einstein-Podolsky-Rosen steering, Phys. Rev. Lett. 111, 250403 (2013).
  * (24) D. Cavalcanti and P. Skrzypczyk, Quantum steering: a review with focus on semidefinite programming, Rep. Prog. Phys. 80, 024001 (2017).
  * (25) R. Uola, A. C. S. Costa, H. Chau Nguyen, O. Gühne, Quantum steering, Rev. Mod. Phys. 92, 015001 (2020).
  * (26) S. Armstrong, M. Wang, R. Y. Teh, Q. H. Gong, Q. Y. He, J. Janousek, H. A. Bachor, M. D. Reid, and P. K. Lam, Multipartite Einstein-Podolsky-Rosen steering and genuine tripartite entanglement with optical networks, Nat. Phys. 11, 167–172 (2015).
  * (27) X. W. Deng, Y. Xiang, C. Tian, G. Adesso, Q. Y. He, Q. H. Gong, X. L. Su, C. D. Xie, and K. C. Peng, Demonstration of monogamy relations for Einstein-Podolsky-Rosen steering in Gaussian cluster states, Phys. Rev. Lett. 118, 230501 (2017).
  * (28) M. H. Wang, Y. Xiang, H. J. Kang, D. M. Han, Y. Liu, Q. Y. He, Q. H. Gong, X. L. Su,and K. C. Peng, Deterministic distribution of multipartite entanglement and steering in a quantum network by separable states, Phys. Rev. Lett. 125, 260506 (2020).
  * (29) Y. Cai, Y. Xiang, Y. Liu, Q. Y. He, and N. Treps, Versatile multipartite Einstein-Podolsky-Rosen steering via a quantum frequency comb, Phys. Rev. Research 2, 032046(R) (2020).
  * (30) D. Cavalcanti, P. Skrzypczyk, G. H. Aguilar, R. V. Nery, P. H. Souto Ribeiro, and S. P. Walborn, Detection of entanglement in asymmetric quantum networks and multipartite quantum steering, Nat. Commun. 6, 7941 (2015).
  * (31) C.-M. Li, K. Chen, Y.-N. Chen, Q. Zhang, Y.-A. Chen, and J.-W. Pan, Genuine high-order Einstein-Podolsky-Rosen steering, Phys. Rev. Lett. 115, 010402 (2015).
  * (32) H. Lu, C.-Y. Huang, Z.-D. Li, X.-F. Yin, R. Zhang, T.-L. Liao, Y.-A. Chen, C.-M. Li, and J.-W. Pan, Counting classical nodes in quantum networks, Phys. Rev. Lett. 124, 180503 (2020).
  * (33) P. Kunkel, M. Prüfer, H. Strobel, D. Linnemann, A. Frölian, T. Gasenzer, M. Gärttner, M. K. Oberthaler, Spatially distributed multipartite entanglement enables EPR steering of atomic clouds, Science 360, 413–416 (2018).
  * (34) M. D. Reid, Monogamy inequalities for the Einstein-Podolsky-Rosen paradox and quantum steering, Phys. Rev. A 88, 062108 (2013).
  * (35) S-W. Ji, M. S. Kim and H. Nha, Quantum steering of multimode Gaussian states by Gaussian measurements: Monogamy relations and the Peres conjecture, J. Phys. A: Math. Theor. 48, 135301 (2015).
  * (36) Y. Xiang, I. Kogias, G. Adesso, and Q. Y. He, Multipartite Gaussian steering: Monogamy constraints and quantum cryptography applications, Phys. Rev. A 95, 010101(R) (2017).
  * (37) G. Adesso and R. Simon, Strong subadditivity for log-determinant of covariance matrices and its applications, J. Phys. A: Math. Theor. 49, 34LT02 (2016).
  * (38) L. Lami, C. Hirche, G. Adesso, and A. Winter, Schur Complement Inequalities for Covariance Materices and Monogamy of Quantum Correlations, Phys. Rev. Lett. 117, 220502 (2016).
  * (39) S. Cheng, A. Milne, M. J. W. Hall, and H. M. Wiseman, Volume monogamy of quantum steering ellipsoids for multiqubit systems, Phys. Rev. A 94, 042105 (2016).
  * (40) C. Zhang, S. Cheng, L. Li, Q.-Y. Liang, B.-H. Liu, Y.-F. Huang, C.-F. Li, G.-C. Guo, M. J. W. Hall, H. M. Wiseman, and G. J. Pryde, Experimental validation of quantum steering ellipsoids and tests of volume monogamy relations, Phys. Rev. Lett. 122, 070402 (2019).
  * (41) A. Kenfack and K. Życzkowski, Negativity of the Wigner function as an indicator of non-classicality, J. Opt. B: Quantum Semiclass. Opt. 6, 396–404 (2004).
  * (42) I. Kogias, A. R. Lee, S. Ragy, and G. Adesso, Quantification of Gaussian quantum steering, Phys. Rev. Lett. 114, 060403 (2015).
  * (43) See Supplemental Material for details of the all proofs and additional example.
  * (44) V. Parigi, A. Zavatta, M. Kim, and M. Bellini, Probing quantum commutation rules by addition and subtraction of single photons to/from a light field, Science 317, 1890–1893 (2007).
  * (45) L. M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, Inseparability criterion for continuous variable systems, Phys. Rev. Lett. 84, 2722 (2000).
  * (46) G. Adesso, A. Serafini, and F. Illuminati, Multipartite entanglement in three-mode Gaussian states of continuous-variable systems: Quantification, sharing structure, and decoherence, Phys. Rev. A 73, 032345 (2006).
  * (47) V. Händchen, T. Eberle, S. Steinlechner, A. Samblowski, T. Franz, R. F. Werner, and R. Schnabel, Observation of one-way Einstein-Podolsky-Rosen steering, Nat. Photon. 6, 596–599 (2012).
  * (48) M. G. A. Paris, F. Illuminati, A. Serafini, and S. De Siena, Purity of Gaussian states: Measurement schemes and time evolution in noisy channels, Phys. Rev. A 68, 012314 (2003).
  * (49) G. Adesso, and F. Illuminati, Entanglement in continuous-variable systems: recent advances and current perspectives, J. Phys. A: Math. Theor. 40, 7821–7880 (2007).
  * (50) Q. Y. He, Q. H. Gong, and M. D. Reid, Classifying directional Gaussian entanglement, Einstein-Podolsky-Rosen steering, and discord, Phys. Rev. Lett. 114, 060402 (2015).
  * (51) Y. Xiang, B. Xu, L. Mišta Jr., T. Tufarelli, Q. Y. He, and G. Adesso, Investigating Einstein-Podolsky-Rosen steering of continuous-variable bipartite states by non-Gaussian pseudospin measurements, Phys. Rev. A 96, 042326 (2017).