# Experimental demonstration of remotely creating Wigner negativity via
quantum steering

Shuheng Liu‡ State Key Laboratory for Mesoscopic Physics, School of Physics,
Frontiers Science Center for Nano-optoelectronics, Peking University, Beijing
100871, China  Dongmei Han‡ State Key Laboratory of Quantum Optics and Quantum
Optics Devices, Institute of Opto-Electronics, Shanxi University, Taiyuan,
030006, China  Collaborative Innovation Center of Extreme Optics, Shanxi
University, Taiyuan, Shanxi 030006, China  Na Wang  State Key Laboratory of
Quantum Optics and Quantum Optics Devices, Institute of Opto-Electronics,
Shanxi University, Taiyuan, 030006, China  Collaborative Innovation Center of
Extreme Optics, Shanxi University, Taiyuan, Shanxi 030006, China  Yu Xiang
State Key Laboratory for Mesoscopic Physics, School of Physics, Frontiers
Science Center for Nano-optoelectronics, Peking University, Beijing 100871,
China  Collaborative Innovation Center of Extreme Optics, Shanxi University,
Taiyuan, Shanxi 030006, China  Fengxiao Sun  State Key Laboratory for
Mesoscopic Physics, School of Physics, Frontiers Science Center for Nano-
optoelectronics, Peking University, Beijing 100871, China  Collaborative
Innovation Center of Extreme Optics, Shanxi University, Taiyuan, Shanxi
030006, China  Meihong Wang  State Key Laboratory of Quantum Optics and
Quantum Optics Devices, Institute of Opto-Electronics, Shanxi University,
Taiyuan, 030006, China  Collaborative Innovation Center of Extreme Optics,
Shanxi University, Taiyuan, Shanxi 030006, China  Zhongzhong Qin  State Key
Laboratory of Quantum Optics and Quantum Optics Devices, Institute of Opto-
Electronics, Shanxi University, Taiyuan, 030006, China  Collaborative
Innovation Center of Extreme Optics, Shanxi University, Taiyuan, Shanxi
030006, China  Qihuang Gong  State Key Laboratory for Mesoscopic Physics,
School of Physics, Frontiers Science Center for Nano-optoelectronics, 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, Jiangsu, China  Xiaolong
Su<EMAIL_ADDRESS>State Key Laboratory of Quantum Optics and Quantum Optics
Devices, Institute of Opto-Electronics, Shanxi University, Taiyuan, 030006,
China  Collaborative Innovation Center of Extreme Optics, Shanxi University,
Taiyuan, Shanxi 030006, China  Qiongyi He<EMAIL_ADDRESS>State Key
Laboratory for Mesoscopic Physics, School of Physics, Frontiers Science Center
for Nano-optoelectronics, 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, Jiangsu, China

###### Abstract

Non-Gaussian states with Wigner negativity are of particular interest in
quantum technology due to their potential applications in quantum computing
and quantum metrology. However, how to create such states at a remote location
remains a challenge, which is important for efficiently distributing quantum
resource between distant nodes in a network. Here, we experimentally prepare
optical non-Gaussian state with negative Wigner function at a remote node via
local non-Gaussian operation and shared Gaussian entangled state existing
quantum steering. By performing photon subtraction on one mode, Wigner
negativity is created in the remote target mode. We show that the Wigner
negativity is sensitive to loss on the target mode, but robust to loss on the
mode performing photon subtraction. This experiment confirms the connection
between the remotely created Wigner negativity and quantum steering. As an
application, we present that the generated non-Gaussian state exhibits
metrological power in quantum phase estimation.

Generation and manipulation of quantum states are crucial preconditions
underlying various quantum information tasks. Gaussian states which can be
generated deterministically have been widely applied in continuous variable
(CV) quantum information Samuel2005 ; Christian2012 ; wangxb2007 ; suxl2010 ;
jingjt2020 . On the other hand, CV non-Gaussian states are attracting
increasing interests due to the increasing entanglement contributed from
higher-order correlations akira2006 ; alexei2007 ; takahashi2010entanglement ,
and especially, the Wigner negativity of the non-Gaussian states has been
identified as an essential resource for reaching a quantum computation
advantage menicucci2006universal ; mari2012  and for error correction
lund2008error ; Victor2018 . Substantial progress has been made in
controllable generation of Wigner-negative states by performing non-Gaussian
operations, e.g., photon addition or subtraction on the previously prepared
Gaussian modes Wenger04 ; Zava04 ; Ourjoumtsev2006 ; Neergaard2006 ;
Valentina2007 ; kentaro2007 ; nicolas2020 , or directly by a higher-order
interaction such as three-photon spontaneous parametric down-conversion
Douady2004 ; Chang2020  or four-wave mixing with Kerr nonlinearity Leghtas2015
.

Beyond above local preparation methods, remote state preparation (RSP) based
on the shared entanglement between distant nodes offers intrinsic security and
efficiency for creating desired quantum resources at a remote location
Lo2000remote ; Paris2003continuous ; pogorzalek2019secure ; optica2018 .
Compared to the well-known quantum teleportation, which transmits an unknown
state by sharing entanglement and performing joint Bell measurement, RSP
protocol only requires measurements acting on each individual mode. This
promises RSP various potential applications in quantum information processing,
such as on-demand preparation of single-photon states jeffery2004towards ;
peters2005remote , creating two-qubit hybrid entangled states
barreiro2010remote , generating and manipulating atomic quantum memories
remotely rosenfeld2007remote ; bao2012quantum . Toward networked quantum
technology it is, thus, crucial to find a way to prepare a remote quantum
state with a negative Wigner function.

Recently, it has been theoretically shown that a special kind of entanglement
known as Einstein-Podolsky-Rosen (EPR) steering Schrodinger35 ; wiseman2007 ;
reid2009 ; cava2017 ; Uola2020  is a necessary requirement for remotely
preparing Wigner-negative states mattiaPRL ; mattiaPRXquantum ; XiangarXiv .
EPR steering is a directional form of nonlocality, related to the Einstein
“spooky” paradox, that after performing local measurements on one of the
systems, it can apparently steer the state of the other distant system. Based
on this kind of nonlocal effect existing between distant systems, one can
remotely create a Wigner-negative state in the steering mode by subtracting a
photon from the steered mode mattiaPRL . At the same time, the shared
entanglement still maintains as the photon subtraction is a non-destructive
non-Gaussian operation. This connection has not been experimentally verified
yet, especially the quantitative relation when taking the practical channel
loss between spatially separated nodes into account. Thus, it is still quite
an open area for further investigations.

In this Letter, we experimentally demonstrate remote creation of non-Gaussian
state with negative Wigner function via local single-photon operation and
shared Gaussian EPR steering. Two optical modes $A$ and $B$ in a CV EPR
entangled state are sent to two distant stations controlled by Alice and Bob
respectively [Fig. 5(a)]. Once Alice successfully subtracts a photon from the
steered mode $A$, the Wigner function of the steering mode $B$ shows negative
values. We quantify the Wigner negativity by performing quantum tomography on
the conditional state of mode $B$, and validate the relation between the
initially shared Gaussian EPR steering and the remotely created Wigner
negativity. The dependence of Wigner negativity on channel loss is
investigated by transmitting Alice’s and Bob’s states through lossy channels,
respectively. The results show that the generated Wigner negativity is
sensitive to the loss in Bob’s channel but robust to the loss in Alice’s
channel. As an application, we show that the generated non-Gaussian state
exhibits metrological power in quantum phase estimation. Our work demonstrates
the feasibility of remote preparation of Wigner-negative state between
spatially separated stations, and confirms the connection between remotely
created Wigner negativity and quantum steering.

Figure 1: The principle and experimental setup. (a) Schematic of the remote
preparation of Wigner negativity. We first prepare a Gaussian EPR entangled
state and then transmit two entangled optical fields to two distant nodes
controlled by Alice and Bob, where the lossy channels are characterized by
$\eta_{A}$ and $\eta_{B}$, respectively. Then once Alice successfully performs
a single-photon subtraction from her mode, the remote Bob’s mode collapses to
a Wigner-negative state. (b) Experimental setup. Two acousto-optic modulators
(AOM) controlled by the periodically signals are used to chop the seed beam.
The NOPA is composed of a type-II KTP crystal and a concave mirror with 50 mm
radius. Lossy channel is simulated by the combination of a half wave plate
(HWP) and a polarization beamsplitter (PBS). The optical isolators are used to
avoid the back scattered light to the NOPA cavity. SNSPD: superconducting
nanowire single-photon detector, LO: local oscillator, MC: mode cleaner, OI:
optical isolator, LS: laser shutter, IF: interference filter, FPC: Fabry-Perot
cavity.

The experimental setup is shown in Fig. 5(b). A continuous laser generates
$1080$ nm and $540$ nm laser beams simultaneously, which are used as the seed
and pump beams of a nondegenerate optical parametric amplifier (NOPA). An EPR
entangled state is generated from the NOPA when it is operated at the status
of deamplification liu2019 ; suddensu . Two modes of the EPR entangled state
are separated by a polarization beam splitter (PBS), and transmitted to Alice
and Bob through lossy channels, characterized by $\eta_{A}$ and $\eta_{B}$,
respectively. Alice then performs single-photon subtraction by splitting her
mode with a beamsplitter with around $4\%$ reflectivity and implementing
single-photon detection on it. The filter system used to select the degenerate
mode is composed of an interference filter with 0.6 nm bandwidth and a Fabry-
Perot cavity (FPC). When a photon is detected by the superconducting nanowire
single-photon detector (SNSPD), Bob measures his conditional state with a
homodyne detector.

Figure 2: The reconstructed Wigner function of Bob’s state remotely created by
performing single-photon subtraction from Alice’s field with two sets of input
squeezing levels: (a-c) $-$1.74/+2.08 dB and (d-f) $-$1.302/+1.407 dB, under
different transmission efficiencies of mode $B$:
$\eta_{B}=0.7,\leavevmode\nobreak\ 0.8,\leavevmode\nobreak\ 0.9$. Note that in
our experiment, $10\%$ detection loss is caused by the limited homodyne
detection efficiency including the quantum efficiency of the photo diode
($98\%$), the mode matching efficiency ($98\%$), and the clearance of homodyne
detection ($96\%$), which is corrected in the above plot.

The CV EPR entangled state shared between Alice and Bob can be described by
its covariance matrix with elements
$\sigma_{ij}=\langle\hat{\beta}_{i}\hat{\beta}_{j}+\hat{\beta}_{j}\hat{\beta}_{i}\rangle/2-\langle\hat{\beta}_{i}\rangle\langle\hat{\beta}_{j}\rangle$,
where
$\hat{\beta}\equiv(\hat{x}_{A},\hat{p}_{A},\hat{x}_{B},\hat{p}_{B})^{\top}$ is
the vector of the amplitude and phase quadratures of each mode. The quadrature
operators of Alice’s mode $A$ are denoted by
$\hat{x}_{A}=\hat{a}_{A}^{\dagger}+\hat{a}_{A},\hat{p}_{A}=i\left(\hat{a}_{A}^{\dagger}-\hat{a}_{A}\right)$
where $\hat{a}^{\dagger},\hat{a}$ are creation and annihilation operators
respectively. Same definition for Bob’s mode $B$. Thus the covariance matrix
is given by

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

where the submatrices $\sigma_{A}$ and $\sigma_{B}$ represent the statistical
features of the reduced states of subsystems $A$ and $B$, respectively;
$\gamma_{AB}$ provides cross correlations between the output optical modes. In
our experiment, the realized CV EPR resources with $c_{1}=-c_{2}=c$. The CM
elements of Eq. (1) can be retrieved from single mode measurements, i.e., by
simultaneously measuring the amplitude and phase quadratures of each of two
output modes JOB ; PRA69 ; PRA76 ; PRA87 ; OL46 ; EPJ05 ; PRL09 .

EPR steering in the direction from Bob to Alice is quantified by the parameter
$\mathcal{G}^{B\rightarrow
A}=\max\\{0,\frac{1}{2}\ln\frac{\text{Det\leavevmode\nobreak\
}\sigma_{B}}{\text{Det\leavevmode\nobreak\ }\sigma_{AB}}\\}$ Adesso15 , a
higher value means stronger steerability. Note that $\mathcal{G}^{B\rightarrow
A}>0$ has been theoretically proved as a sufficient and necessary resource to
remotely generate negative part of Wigner function of the steering mode $B$ by
a conditional operation applied on the steered mode $A$ mattiaPRL .

To confirm this connection, we then subtract a single photon from the steered
mode $A$ and observe its nonlocal effects on the steering mode $B$. After a
successful single-photon subtraction, the Wigner function of the reduced
quantum state of mode $B$ is expressed as

$\displaystyle W^{A-}_{B}(\beta_{B})=$
$\displaystyle\frac{\exp\left\\{-\frac{1}{2}\left(\beta_{B},\sigma_{B}^{-1}\beta_{B}\right)\right\\}}{2\pi\sqrt{\operatorname{Det}\sigma_{B}}\left[\operatorname{Tr}\left(\sigma_{A}\right)-2\right]}$
(2)
$\displaystyle\times\left[\beta_{B}^{\top}\sigma_{B}^{-1^{\top}}\gamma_{AB}^{\top}\gamma_{AB}\sigma_{B}^{-1}\beta_{B}+\operatorname{Tr}(V_{A\mid
B})-2\right],$

where $\beta_{B}=(x_{B},\leavevmode\nobreak\ p_{B})^{\top}$ is the vector of
possible measurement outcomes of the quadrature operators, and $V_{A\mid
B}=\sigma_{A}-\gamma_{AB}\sigma_{B}^{-1}\gamma_{AB}^{\top}$ is the Schur
complement of $\sigma_{A}$. The non-classical features of the reduced quantum
state can be characterized by the negativity of the above Wigner function,
defined as the doubled volume of the integrated negative part
kenfack2004negativity ,

$\mathcal{N}_{B}^{A-}=\frac{2c^{2}e^{\frac{m(n-1)}{c^{2}}-1}}{m(n-1)}-2.$ (3)

The derivation is detailed in supp , where we also express the created Wigner
negativity as a function of the local and global purities of initial Gaussian
EPR entangled state before photon subtraction, confirming the quantitative
relation given in a recent theoretical work XiangarXiv .

Figure 3: The steerability $\mathcal{G}^{B\rightarrow A}$ of the initial
Gaussian entangled states and the remotely created Wigner negativity in the
steering mode $B$ after subtracting a single photon from the steered mode $A$
as functions of the transmission efficiency $\eta_{B}$ shown in (a, b), or
$\eta_{A}$ in (c, d), respectively. In (a, b), we also examine two sets of
input squeezing levels, $-$1.74/+2.08 dB (red lines) and $-$1.302/+1.407 dB
(blue lines). Error bars represent $\pm$ one standard deviation and are
obtained based on the statistics of the measured noise variances and density
matrices.

When taking the dark counts of the single-photon detector into account, the
Wigner function given in Eq. (2) is then changed to $W_{B,\text{s}}^{A-}=\xi
W_{B}^{A-}+(1-\xi)W_{B}$ darkcount , where $\xi$ is the ratio of the true
counts from single-photon subtraction on $A$ to the total detector clicks,
$W_{B}^{A-}$ is the ideal Wigner function of photon-subtracted state and
$W_{B}$ is the initial Gaussian state which corresponds to the failure event
of single-photon subtraction. Thus, the negativity
$\mathcal{N}_{B,\text{s}}^{A-}$ of Wigner function $W_{B,\text{s}}^{A-}$ when
considering dark counts reads

$\mathcal{N}_{B,\text{s}}^{A-}=\frac{2c^{2}\xi
e^{\frac{m(n-1)}{c^{2}\xi}-1}}{m(n-1)}-2.$ (4)

The experimental details of the click rate (generation rate) and $\xi$ can be
found in supp .

In the experiment, we first prepare a Gaussian entangled state from a NOPA and
retrieve its covariance matrix from single mode measurements supp . In
particular, the amplitude and phase quadratures of optical fields $A$ and $B$
are measured by homodyne detectors in the time domain, where the signals of
detectors pass through two low-pass filters with bandwidth of $60$ MHz and are
recorded simultaneously by a digital storage oscilloscope at the sampling rate
of $500$ KS/s. Two different entangled Gaussian states characterized by
different squeezing levels are generated at the source by injecting $50$ mW
and $30$ mW pump beams into the NOPA respectively, which results in different
purities after a lossy evolution from the crystal downward the detection ra1 ;
ra2 ; ra3 .

To investigate the effects of squeezing level and purity on the remote
creation of Wigner negativity, we analyze the elements of the measured CM in
terms of
$V_{\pm}=\Delta^{2}(\hat{x}_{A}\pm\hat{x}_{B})/2=\Delta^{2}(\hat{p}_{A}\mp\hat{p}_{B})/2$,
which are the correlated variances of the quadrature measurement statistics
between two modes of the EPR entangled states kurochkin2014PRL . Thus,
considering the practical transmission efficiency, the CM elements of Eq. (1)
can be expressed as $n=\eta_{A}(V_{+}+V_{-})/2+(1-\eta_{A})$,
$m=\eta_{B}(V_{+}+V_{-})/2+(1-\eta_{B})$,
$c_{1}=-c_{2}=c=-\sqrt{\eta_{A}\eta_{B}}(V_{-}-V_{+})/2$ supp . In our
experiment, the entangled state generated by $30$ mW pump power shows lower
squeezing but higher purity $-1.302/+1.407$ dB (i.e.,
$V_{+}=0.74,\leavevmode\nobreak\ V_{-}=1.38$), while by $50$ mW pump power has
higher squeezing but lower purity $-1.74/+2.08$ dB (i.e.,
$V_{+}=0.67,\leavevmode\nobreak\ V_{-}=1.61$).

To perform the single-photon subtraction on the steered mode $A$, around $4\%$
energy of optical field $A$ is reflected by a beamsplitter and directed to
SNSPD. When the SNSPD clicks at Alice’s station, Bob measures his conditional
state with a homodyne detector and record the output signals by the digital
storage oscilloscope. Note that in case of no click at SNSPD, the non-Gaussian
operation fails and the Bob’s conditional state remains Gaussian. We record
over $30000$ quadrature values of Bob’s mode for each chosen transmission
efficiency, then reconstruct the Wigner functions of mode $B$ by using the
maximum-likelihood algorithm lvovsky2009 .

Fig. 2 shows the reconstructed Wigner functions of mode $B$ conditioned on
single-photon subtraction performed on the distant mode $A$ at different
transmission efficiencies $\eta_{B}$ and fixed $\eta_{A}=0.9$ for two sets of
input squeezing levels: $-$1.74/+2.08 dB (a-c) and $-$1.302/+1.407 dB (d-f).
The corresponding Wigner negativities $\mathcal{N}$ become larger with the
increase of $\eta_{B}$. The fidelity
$F(\rho,\sigma)=\left(\operatorname{Tr}\sqrt{\sqrt{\rho}\sigma\sqrt{\rho}}\right)^{2}$
is a measure which quantifies the overlap between the experimentally
reconstructed reduced quantum state of mode $B$ ($\sigma$) after single-photon
subtraction on mode $A$ and the theoretical result with $\xi$. They are all
above $95\%$ for the presented transmission efficiencies $\eta_{B}$, which
manifests high quality of the RSP process. The results reveal that the case
with lower squeezing $-$1.302/+1.407 dB indicated in Figs. 2(d-f) performs
better than the other case with $-$1.74/+2.08 dB given in Figs. 2(a-c),
showing more significant negative values as the transmission efficiency
increases. Especially, for the transmission efficiency $\eta_{B}=0.7$, Fig.
2(d) already presents nonzero Wigner negativity, while Fig. 2(a) doesn’t.

Since this remote preparation of non-Gaussian state of mode $B$ is based on
Gaussian EPR steering shared between modes $A$ and $B$, to understand the
physics behind the results given by Fig. 2, we investigate the connection
between the Wigner negativity and the Gaussian steerability from a
quantitative perspective, and establish the decisiveness factors imposing
constraints on the degree of Wigner negativity that is remotely created by
single-photon subtraction performed in a distant station. Especially, we take
into account the lossy channels and examine the effects of loss in the
protocol.

As shown in Fig. 3(a), Gaussian steerability from Bob to Alice only exists
($\mathcal{G}^{B\rightarrow A}>0$) when $\eta_{B}>0.623$ (the case with lower
squeezing but higher purity described in blue) or $\eta_{B}>0.701$ (the case
with higher squeezing but lower purity indicated in red) for a fixed value of
$\eta_{A}=0.9$. Correspondingly, the Wigner negativity of the reduced state of
mode $B$ appears only when Gaussian steerability is larger than zero, as shown
in Fig. 3(b). The states produced with a higher squeezing level possess
stronger Gaussian steerability as expected, while the created Wigner
negativity in the steering mode $B$ is lower (red lines). This means there is
a one-to-one correspondence of nonzero Gaussian steerability and Wigner
negativity, but lack of a quantitative connection between them. In fact, there
is a tradeoff between the quality of the initial EPR entanglement (the two-
mode squeezing) and the final Wigner negativity. For instance, if the
entanglement (squeezing) is too high the reduced single-mode state (by tracing
over the mode that was subject to photon subtraction) has a “higher
temperature” lvovsky2015squeezed  which prevents (high) Wigner negativity
after photon subtraction. As indicated in supp , the state purity plays main
role instead of the squeezing level in our experiment.

Figure 4: The metrological power remotely created in mode $B$ after single-
photon subtraction on mode $A$, varying with (a) transmission efficiency
$\eta_{B}$ of Bob and (b) transmission efficiency $\eta_{A}$ of Alice. The
squeezing levels are $-$1/+1 dB (blue) and $-$3/+3dB (red)

.

The Gaussian steerability enhances with the increase of transmission
efficiency in two channels, as shown in Figs. 3(a) and (c), while the created
Wigner negativity is only affected by the channel loss existing in the
steering mode $B$ (b), but does not vary with the loss in the channel of the
steered mode (d). This is because that Wigner function of the reduced state
only depends on $\eta_{B}$, as shown in Eq. (4) in the Appendix supp .
However, the transmission efficiency in Alice’s channel will affect the
generation rate of non-Gaussian state at Bob’s node. For instance, for the
case with lower squeezing, the generation rate of creating Wigner negativity
in mode $B$ is decreased from $\sim$3 kHz to $\sim$500 Hz when $\eta_{A}$
decreases from $0.9$ to $0.3$.

As an application of the remotely prepared non-Gaussian state with Wigner
negativity, we examine its metrological power in quantum precision
measurement, as demonstrated in Fig. 4. The metrological power is defined as
$\mathcal{M}(\rho)=1/4\max[F_{x}(\rho)-2,0]$ metrology2019PRL ; metrologyPRR ;
metrologyPRA , where $\mathcal{M}(\rho)$ quantifies the metrological advantage
beyond the standard quantum limit, and $F_{x}(\rho)$ is the optimized quantum
Fisher information over all possible quadratures $\hat{x}$ toth2014quantum ;
demkowicz2015quantum . Similar with the Wigner negativity, the metrological
power of the reduced state becomes stronger with a lower level of input
squeezing (blue lines), and sensitive to the loss existing in Bob’s channel
but robust to the loss in Alice’s channel.

In summary, we experimentally prepare non-Gaussian state with negative Wigner
function in a remote mode (hold by Bob) by performing local single-photon
subtraction on the steered mode (hold by Alice) of the previously shared
Gaussian EPR entangled state. We demonstrate that the appearance of Wigner
negativity at Bob’s node depends on the presence of the steerability from Bob
to Alice, confirming the connection between the remotely created Wigner
negativity and quantum steering. To further examine the quantitative relation
we take the channel loss between two distant nodes, and find that the created
Wigner negativity is sensitive to the loss in channel head to Bob, but robust
to the loss in channel head to Alice who performs photon subtraction. We also
show a potential application of the prepared non-Gaussian state in quantum
phase estimation, where less initial squeezing produces higher Wigner
negativity and thus stronger metrological power in the steering mode. Our
results present a significant advance in a concrete in-depth understanding of
the connection between remotely creating Wigner negativity and the Gaussian
EPR steering, and pave the way for remote preparation of multimode non-
Gaussian states for reaching further quantum advantage.

###### Acknowledgements.

This work is supported by National Natural Science Foundation of China (Grants
No. 12125402, No. 11834010, No. 12004011), National Key R&D Program of China
(Grant No. 2019YFA0308702). X. S. thanks the Fund for Shanxi “1331 Project”
Key Subjects Construction. Q. H. acknowledges the Beijing Natural Science
Foundation (Z190005) and the Key R&D Program of Guangdong Province (Grant No.
2018B030329001).

‡S. H. Liu and D. M. Han contributed equally to this work.

*

## Appendix I: The derivation of remotely created Wigner negativity

In our experiment, an EPR entangled state was directly generated from the
nondegenerate optical parametric amplifier, which is fully described by its
covariance matrix (CM)

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

where $n=\Delta^{2}\hat{x}_{A}=\Delta^{2}\hat{p}_{A}$,
$m=\Delta^{2}\hat{x}_{B}=\Delta^{2}\hat{p}_{B}$ represent the variances of
amplitude and phase quadratures of the output optical modes,
$c_{1}=Cov(\hat{x}_{A},\hat{x}_{B})$ and $c_{2}=Cov(\hat{p}_{A},\hat{p}_{B})$
indicate their cross correlations. The CM elements of Eq. (1) can be retrieved
from single mode measurements as detailed in APPENDIX II-B JOB ; PRA69 ; PRA76
; PRA87 ; OL46 ; EPJ05 ; PRL09 .

We can define the correlated variances of the quadrature measurement
statistics between two modes of the EPR entangled states by
$V_{+}=\Delta^{2}(\hat{x}_{A}+\hat{x}_{B})/2=\Delta^{2}(\hat{p}_{A}-\hat{p}_{B})/2$
and
$V_{-}=\Delta^{2}(\hat{x}_{A}-\hat{x}_{B})/2=\Delta^{2}(\hat{p}_{A}+\hat{p}_{B})/2$.
Considering the practical losses characterized by transmission efficiency
$\eta_{A}$ and $\eta_{B}$, the CM elements become
$n=\eta_{A}(V_{+}+V_{-})/2+(1-\eta_{A})$,
$m=\eta_{B}(V_{+}+V_{-})/2+(1-\eta_{B})$,
$c_{1}=-c_{2}=c=-\sqrt{\eta_{A}\eta_{B}}(V_{-}-V_{+})/2$. The local purities
$\mu_{A,B}$ of modes $A,\leavevmode\nobreak\ B$ and the global purity
$\mu_{AB}$ are then given by

$\displaystyle\mu_{A}=\frac{1}{\sqrt{\text{det}\sigma_{A}}}=\frac{2}{2+(V_{-}+V_{+}-2)\eta_{A}},$
$\displaystyle\mu_{B}=\frac{1}{\sqrt{\text{det}\sigma_{B}}}=\frac{2}{2+(V_{-}+V_{+}-2)\eta_{B}},$
(2) $\displaystyle\mu_{AB}=\frac{1}{\sqrt{\text{det}\sigma_{AB}}}$
$\displaystyle=\frac{2}{2\eta_{A}\eta_{B}(V_{-}-1)(V_{+}-1)+\eta_{A}(V_{-}+V_{+}-2)+\eta_{B}(V_{-}+V_{+}-2)+2}.$

The Gaussian steerability $\mathcal{G}^{B\rightarrow A}$ in such system can be
also quantified from its CM AdessoPRL2015 ,

$\displaystyle\mathcal{G}^{B\rightarrow
A}=\max\left\\{0,\frac{1}{2}\ln\frac{\text{det}\sigma_{B}}{\text{det}\sigma_{AB}}\right\\}=\max\left\\{0,\ln\frac{\mu_{AB}}{\mu_{B}}\right\\}$
(3)
$\displaystyle=\max\left\\{0,-\ln\left|\frac{\left[V_{-}+V_{+}+2(V_{-}-1)(V_{+}-1)\eta_{B}-2\right]\eta_{A}}{(V_{-}+V_{+}-2)\eta_{B}+2}+1\right|\right\\}.$

After a single-photon subtraction applied on the steered mode $A$, the Wigner
function of the steering mode $B$ becomes mattiaPRL

$\displaystyle
W_{B}^{A-}\left(x_{B},p_{B}\right)=\frac{\exp[{-\frac{x_{B}^{2}+p_{B}^{2}}{2m}}]\left[-2c^{2}m+c^{2}\left(x_{B}^{2}+p_{B}^{2}\right)+2m^{2}(n-1)\right]}{4\pi
m^{3}(n-1)}$ (4)
$\displaystyle=\frac{\exp[-\frac{x_{B}^{2}+p_{B}^{2}}{\eta_{B}(V_{-}+V_{+}-2)+2}]}{\pi(V_{-}+V_{+}-2)\left[\eta_{B}(V_{-}+V_{+}-2)+2\right]^{3}}\bigg{\\{}\eta_{B}(x_{B}^{2}+p_{B}^{2})(V_{-}-V_{+})^{2}$
$\displaystyle\quad+2\left[2\eta_{B}(V_{-}-1)(V_{+}-1)+V_{-}+V_{+}-2\right]\left[\eta_{B}(V_{-}+V_{+}-2)+2\right]\bigg{\\}}.$

It is clearly seen that the above Wigner function only depends on the channel
loss existing in the steering mode $B$, i.e., $\eta_{B}$, but does not vary
with the loss in the channel of the steered mode $A$. It is straightforward to
calculate the Wigner negativity possessed in mode $B$ which is defined as the
doubled volume of the integrated negative part of the above Wigner function
kenfack2004negativity

$\mathcal{N}_{B}^{A-}=\frac{2c^{2}e^{\frac{m(n-1)}{c^{2}}-1}}{m(n-1)}-2.$ (5)

Since the Wigner negativity is calculated based on Eq. (4), it should be
irrelevant to $\eta_{A}$.

When taking dark counts of the single-photon detector into account, we have
$W_{B,\text{s}}^{A-}=\xi W_{B}^{A-}+(1-\xi)W_{B}$, where $\xi$ is the ratio of
the true counts from single-photon subtraction on $A$ to the total detector
clicks, and $W_{B}$ is the initial Gaussian Wigner function of mode $B$
corresponding to the failure event of single-photon subtraction darkcount .
Thus, the negativity of the Wigner function $W_{B,\text{s}}^{A-}$ accounting
for dark counts is expressed as

$\mathcal{N}_{B,\text{s}}^{A-}=\frac{2c^{2}\xi
e^{\frac{m(n-1)}{c^{2}\xi}-1}}{m(n-1)}-2.$ (6)

We can also express it by the local and global purities of the initial
Gaussian EPR entangled state,

$\mathcal{N}_{B,\text{s}}^{A-}=\frac{2\xi(\mu_{A}\mu_{B}-\mu_{AB})e^{\frac{\mu_{AB}-\mu_{A}\mu_{AB}}{\mu_{AB}\xi-\mu_{A}\mu_{B}\xi}-1}}{(\mu_{A}-1)\mu_{AB}}-2.$
(7)

It is clearly that the Wigner negativity $\mathcal{N}_{B,\mathrm{s}}^{A-}$ is
determined by the purities of initial Gaussian state.

## Appendix II: Details of experiment

### A. Generation of EPR entangled state

Figure 5: Experimental setup. MC 1: mode cleaner for the $1080$ nm light, MC
2: mode cleaner for the $540$ nm light, EOM: electro-optic modulator, AOM:
acousto-optic modulator, ID: iris diaphragm, Det: detector, NOPA: non-
degenerate optical parametric amplifier, PBS: polarization beam splitter, VBS:
variable beam splitter which is composed of a half wave plate (HWP) and a PBS,
OI: optical isolator, IF: interference filter, FPC: Fabry-Perot cavity, LS:
laser shutters, LPF: low-pass filter, SNSPD: superconducting nanowire single-
photon detector, LO: local oscillator, HD 1: homodyne detector at Alice’s
station, HD 2: homodyne detector at Bob’s station, OSC: oscilloscope.

The detailed experimental setup is shown in Fig. 5. The laser used in our
experiment is a continuous wave intracavity frequency-doubled and frequency-
stabilized Nd:YAP-LBO (Nd-doped YAlO${}_{\text{3}}$ perovskite-lithium
triborate) laser which generates $1080$ nm and $540$ nm light simultaneously.
The $1080$ nm and $540$ nm lasers are filtered by mode cleaners (MC 1 and MC
2) respectively and then used as the seed and pump beams of the non-degenerate
optical parametric amplifier (NOPA). Two acousto-optic modulators (AOM) are
used to chop the seed beam into a cyclic form: $50$ ms for each locking and
hold period. During the locking period, the seed beam is injected into the
NOPA for the cavity locking. When the seed beam is chopped off, the cavity
length of NOPA is holding and the EPR entangled state is generated. The
measurement is performed during the hold period.

The NOPA is composed of an $\alpha$-cut type-II KTiOPO4 (KTP) crystal
($3\times 3\times 10\leavevmode\nobreak\ mm^{3}$) and a concave mirror with
curvature $50$ mm. The front face of the KTP crystal is used as the input
coupler, of which the transmittances at $540$ nm and $1080$ nm are $40\%$ and
$0.04\%$, respectively. The end face of the KTP crystal is antireflection
coated for both $1080$ nm and $540$ nm. The transmittances of the output
coupler are $12.5\%$ and $0.5\%$ at $1080$ nm and $540$ nm, respectively. Two
modes of the EPR entangled state are separated by a polarization beam splitter
(PBS) placed behind the NOPA.

### B. Measurement of Gaussian covariance matrix

As shown in Fig. 5, mode $A$ is totally reflected by a variable beam splitter
(VBS) and directed to homodyne detector 1 (HD 1), and mode $B$ is transmitted
through a lossy channel simulated by a half wave plate (HWP) and a PBS. The
optical isolators (OI) placed before homodyne detectors are used to avoid the
back-scattered light from the local oscillators. The output signals of HDs are
filtered by two $60$ MHz low-pass filters (LPFs) and recorded by a digital
storage oscilloscope (OSC, TELEDNE LECROY, HDO8108A). Quadrature values of two
modes in the time domain are sampled with the sampling rate of $500$ MS/s. The
relative phase of HD is locked to $0(90)$ degrees to measure the
amplitude(phase) quadrature.

The OI together with mirrors in the signal path to each HD lead to around
$10\%$ transmission loss. In addition to this, the detection efficiency of
each HD is around $90\%$ which includes the quantum efficiency of the photo
diode ($98\%$), the mode matching efficiency ($98\%$), and the clearance of HD
($96\%$). So, our reconstructed CMs are all corrected with $90\%$ detection
efficiency.

All Gaussian properties can be determined from the CM defined in Eq. (1). The
variances $\Delta^{2}\hat{x}_{A(B)}$ and $\Delta^{2}\hat{p}_{A(B)}$ are
obtained by calculating the variances of the measured quadrature values
$x_{A(B)}$ and $p_{A(B)}$. The variances of the cross correlations are
obtained by
$\Delta^{2}(\hat{x}_{A}\hat{x}_{B})=[\Delta^{2}\hat{x}_{A}+\Delta^{2}\hat{x}_{B}-\Delta^{2}(\hat{x}_{A}-\hat{x}_{B})]/2$
and
$\Delta^{2}(\hat{p}_{A}\hat{p}_{B})=[\Delta^{2}\hat{p}_{A}+\Delta^{2}\hat{p}_{B}-\Delta^{2}(\hat{p}_{A}-\hat{p}_{B})]/2$,
where $\Delta^{2}(\hat{x}_{A}-\hat{x}_{B})/2$ and
$\Delta^{2}(\hat{p}_{A}-\hat{p}_{B})/2$ are obtained from the simultaneously
measured amplitude and phase quadratures of two modes.

To generate two sets of entangled Gaussian states with different squeezing
levels, we inject $30$ mW and $50$ mW pump beams into the NOPA respectively.
In the case of $30$ mW pump power, the entangled state with lower squeezing
($-1.302/+1.407$ dB) but higher purity is prepared. This can be derived by
measuring CM of the prepared entangled state. For instance, for the
transmission efficiencies $\eta_{A}=\eta_{B}=0.9$, the experimentally measured
CM is

$\left(\begin{array}[]{cccc}1.056\pm 0.004&0&-0.287\pm 0.004&0\\\ 0&1.055\pm
0.003&0&0.287\pm 0.002\\\ -0.287\pm 0.004&0&1.056\pm 0.002&0\\\ 0&0.287\pm
0.002&0&1.056\pm 0.004\end{array}\right).$ (8)

Based on this CM and the values of efficiencies, we can derive the
corresponding variances of sum and difference in Eq. (1) $V_{+}=0.74$ and
$V_{-}=1.38$ ($-1.302/+1.407$ dB).

In the case of $50$ mW pump power, the entangled Gaussian state with higher
squeezing ($-1.74/+2.08$ dB) but lower purity is prepared. The CM at the
transmission efficiencies $\eta_{A}=\eta_{B}=0.9$ is

$\left(\begin{array}[]{cccc}1.130\pm 0.002&0&-0.421\pm 0.004&0\\\ 0&1.127\pm
0.004&0&0.420\pm 0.004\\\ -0.421\pm 0.004&0&1.127\pm 0.004&0\\\ 0&0.420\pm
0.004&0&1.128\pm 0.004\end{array}\right).$ (9)

Similarly, we have $V_{+}=0.67$ and $V_{-}=1.61$ ($-1.74/+2.08$ dB).

### C. Remote preparation of the Wigner-negative state at Bob’s node

To generate the Wigner-negative state at Bob’s node, we need to subtract a
single photon from mode $A$. As shown in Fig. 5, around $4\%$ of the energy
from mode $A$ is tapped by VBS and directed to the superconducting nanowire
single-photon detector (SNSPD). An OI with transmission efficiency $\sim 95\%$
is used to avoid the reflection from the Fabry-Perot cavity (FPC). An
interference filter (IF) of $0.6$ nm bandwidth together with a FPC of around
$400$ MHz bandwidth are used to filter out the non-degenerate modes. A laser
shutter (SR475) is used to match the detection period of the SNSPD with the
hold time period of NOPA. Then, the subtracted photon is coupled to the
optical fiber by a five-axis fiber aligner and detected by the SNSPD with
$\sim 70\%$ detection efficiency whose temperature is cooled to $3$ Kelvin.
The output signal of the SNSPD driver (Single quantum, Argos410) is used to
trigger the OSC. The OSC with sampling rate of $2.5$ GS/s is used to record
the electrical signal of HD 2.

The dark count in our experiment represents the click of the SNSPD when no
down-converted light is incident which comes from the insufficient chopping of
the locking beam and the back-scattered light from the LO. When the
transmission efficiency of mode $A$ is $\eta_{A}=0.9$, the dark count rate
$R_{d}$ is around 60 Hz, considering the total counting rate $R_{t}$ of around
$3$ kHz when the pump power is $30$ mW, $\xi\approx 98\%$ (calculated by
$1-R_{d}/R_{t}$). When the pump power is $50$ mW, the total counting rate
$R_{t}$ is around $7$ kHz, leading to $\xi\approx 99\%$. In our experiment,
the total photon counting rate will decrease with the decrease of transmission
efficiency $\eta_{A}$, and so will the dark count rate, thus the value of
$\xi$ is almost unchanged as it depends on the ratio between them. However,
the generation rate of non-Gaussian state at Bob’s node is decreased, e.g.,
for the case with $30$ mW pump power, the generation rate is decreased from
$\sim$3 kHz to $\sim$500 Hz when $\eta_{A}$ decreases from $0.9$ to $0.3$.

Figure 6: Local purities $\mu_{A}$ (short-dashed lines) and $\mu_{B}$ (solid
lines) and global purity $\mu_{AB}$ (long-dashed lines) of the initial
Gaussian EPR state as functions of transmission efficiency of mode $B$ for two
sets of squeezing levels (a) $-1.302/+1.407\text{dB}$ and (b)
$-1.74/+2.08\text{dB}$. The parameters are the same as those in Fig. 3 in the
main text.

## Appendix III: Relation between Wigner negativity and purities of initial
EPR state

Figure 7: (a) Steerability $\mathcal{G}^{B\rightarrow A}$ of the initial EPR
state as a function of transmission efficiency $\eta_{A}=\eta_{B}=\eta$. (b)
The Wigner negativity remotely created in steering mode $B$ by a single-photon
subtraction on the steered mode $A$. (c) Local purities $\mu_{A}$ (short-
dashed lines), $\mu_{B}$ (solid lines), and global purity $\mu_{AB}$ (long-
dashed lines) of the initial EPR state. In order to facilitate comparison, the
same two sets of squeezing levels are used: $-1.74/+2.08\text{dB}$ (red lines)
and $-1.302/+1.407\text{dB}$ (blue lines).

In the main text Fig. 3, we have shown that the generated Wigner negativity in
mode $B$ only exists when mode $B$ can steer mode $A$, while the amount of
negativity is not determined by the Gaussian steerability. As shown in Eq.
(7), the negativity is quantified by the purities of the prepared EPR state
XiangarXiv . To complement the results, Fig. 6 shows the purities of the
prepared EPR state versus the transmission efficiency $\eta_{B}$ corresponding
to Fig. 3.

In this case, the transmission efficiency of Alice’s channel is fixed at
$\eta_{A}=0.9$. As transmission efficiency $\eta_{B}$ increases, the local
purity of mode $A$ ($\mu_{A}$) remains invariant because the loss in Bob’s
channel does not affect Alice’s state. While both the local purity of mode $B$
($\mu_{B}$) and the global purity ($\mu_{AB}$) change with Bob’s transmission
efficiency, and they become more sensitive to $\eta_{B}$ with higher squeezing
levels. Since the remotely created Wigner negativity is determined by the
purities, it explains why the negativity created by higher squeezing level
(red) is less robust to channel loss, indicated by Fig. 3(b) in the main text.

Then we consider another scheme where both Alice and Bob’s channels suffer
from transmission losses. For simplicity, we suppose the degrees of two
transmission efficiencies are the same $\eta_{A}=\eta_{B}=\eta$. As shown in
Fig. 7(a), Gaussian steerability $\mathcal{G}^{B\rightarrow A}>0$ requires the
transmission efficiency $\eta>0.701$ in the case with higher squeezing but
lower purity (red line), and requires $\eta>0.623$ in the case with lower
squeezing but higher purity (blue line). The Wigner negativity shown in Fig.
7(b) appears when $\eta>0.709$ and $\eta>0.637$ for two cases, respectively,
which are slightly higher due to the imperfect photon subtraction. Similar to
the case shown in Fig. 3 in main text, the EPR state produced by a higher
level of squeezing possesses stronger Gaussian steerability as expected, but
the Wigner negativity created in the steering mode $B$ is lower instead. This
means that the Gaussian steerability is not directly related to the amount of
Wigner negativity. As indicated in Eq. (7), the amount of Wigner negativity
depends on the purity of the initially shared EPR state between two modes.
Comparing Fig. 7(b) and Fig. 7(c), we find that the case with lower squeezing
level but higher purities (blue) remotely generates larger amount of Wigner
negativity.

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