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# Practical distributed quantum information processing with LOCCNet

Xuanqiang Zhao  Benchi Zhao  Zihe Wang  Zhixin Song  Xin Wang  Institute for
Quantum Computing, Baidu Research, Beijing 100193, China

###### Abstract

Distributed quantum information processing is essential for building quantum
networks and enabling more extensive quantum computations. In this regime,
several spatially separated parties share a multipartite quantum system, and
the most natural set of operations is Local Operations and Classical
Communication (LOCC). As a pivotal part in quantum information theory and
practice, LOCC has led to many vital protocols such as quantum teleportation.
However, designing practical LOCC protocols is challenging due to LOCC’s
intractable structure and limitations set by near-term quantum devices. Here
we introduce LOCCNet, a machine learning framework facilitating protocol
design and optimization for distributed quantum information processing tasks.
As applications, we explore various quantum information tasks such as
entanglement distillation, quantum state discrimination, and quantum channel
simulation. We discover protocols with evident improvements, in particular,
for entanglement distillation with quantum states of interest in quantum
information. Our approach opens up new opportunities for exploring
entanglement and its applications with machine learning, which will
potentially sharpen our understanding of the power and limitations of LOCC. An
implementation of LOCCNet is available in Paddle Quantum, a quantum machine
learning Python package based on PaddlePaddle deep learning platform.

##  I Introduction

In the past few decades, quantum technologies have been found to have an
increasing number of powerful applications in areas including optimization
Farhi2014 ; Harrigan2020 , chemistry McArdle2018a ; Arute2020 , security
Bennett1984 ; Ekert1991 , and machine learning Biamonte2017a . To realize
large-scale quantum computers and deliver real-world applications, distributed
quantum information processing will be essential in the technology road map,
where quantum entanglement and its manipulation play a crucial role.

Quantum entanglement is central to quantum information by serving as a
fundamental resource which underlies many important protocols such as
teleportation bennett1993teleporting , superdense coding
bennett1992communication , and quantum cryptography Ekert1991 . To achieve
real-world applications of quantum technologies, protocols for manipulating
quantum entanglement are essential ingredients, and it will be important to
improve existing methods. The study of entanglement manipulation is one of the
most active and important areas in quantum information Plenio2007 ;
horodecki2009quantum .

In entanglement manipulation and distributed quantum information processing,
multiple spatially separated parties are usually involved. As direct transfers
of quantum data between these nodes are not feasible with current technology,
Local Operations and Classical Communication (LOCC) bennett1993teleporting  is
more practical at this stage. Such an LOCC (or distant lab) paradigm plays a
fundamental role in entanglement theory, and many important results have been
obtained within this paradigm horodecki2009quantum . However, how to design
LOCC protocols on near-term quantum devices preskill2018quantum  remains an
important challenge. Such protocols are generally hard to design even with
perfect entanglement due to the complicated and hard-to-characterize structure
of LOCC Chitambar2014 . Moreover, limited capabilities and structure of near-
term quantum devices have to be considered during the design of LOCC
protocols.

Inspired by the breakthroughs of deep learning LeCun2015  in mastering the
game of Go Silver2016  and solving protein folding Jumper2020 , it is
desirable to apply machine learning ideas to explore quantum technologies. For
instance, machine learning has been applied to improve quantum processor
designs Mavadia2017 ; Wan2017 ; Lu2017b ; Niu2019  and quantum communication
Wallnofer2020 ; Bausch2018 . Here, we adopt the ideas from machine learning to
solve the challenges in exploring LOCC protocols. We use parameterized quantum
circuits (PQC) Benedetti2019a  to represent the local operations allowed in
each spatially separated party and then incorporate multiple rounds of
classical communication. Then one can formulate the original task as an
optimization problem and adopt classical optimization methods to search the
optimal LOCC protocol. The PQCs have been regarded as machine learning models
with remarkable expressive power, which leads to applications in quantum
chemistry and optimization Benedetti2019a . Here, we generalize PQC to a
larger deep learning network to deal with distributed quantum information
processing tasks and in particular to explore better entanglement manipulation
protocols.

In this work, we introduce a machine learning framework for designing and
optimizing LOCC protocols that are adaptive to near-term quantum devices,
which consists of a set of PQCs representing local operations. As
applications, we explore central quantum information tasks such as
entanglement distillation, state discrimination, and quantum channel
simulation. We discover protocols with evident improvements via this
framework, sharpening our understanding of the power and limitations of LOCC.
As showcases, we establish hardware-efficient and simple protocols for
entanglement distillation and state discrimination, which outperforms
previously best-known methods. In particular, for distillation of Bell states
with non-orthogonal product noise, the optimized protocol outputs a state
whose distillation fidelity even reaches the theoretical upper bound and hence
is optimal.

##  II Results

###  II.1 The LOCCNet framework

In this section, we introduce LOCCNet, a machine learning framework that
facilitates the design of LOCC protocols for various quantum information
processing tasks, including entanglement distillation Bennett1996 ;
deutsch1996quantum ; Murao1998 ; Dur2007 ; Pan2003 ; Devetak2003a , quantum
state discrimination Bennett1999b ; Walgate2000 ; Fan2004a ; Hayashi2006 ;
Ghosh2004 ; Nathanson2005 ; Duan2007a ; Chitambar2014b ; Duan2009d ;
Childs2013 ; Li2017 ; Bandyopadhyay2014a , and quantum channel simulation
Bennett1996c ; Bennett2014 ; Berta2013 ; Pirandola2015b ; Wilde2018 ; WW18 ;
Fang2018 . An LOCC protocol can be characterized as a sequence of local
quantum operations performed by spatially separated parties with classical
communication of measurement outcomes [see Supplementary Note 1].

According to the number of classical communication rounds, one can divide LOCC
into different classes Chitambar2014 . The one-round protocols correspond to
LOCC operations where one party applies a local operation and sends the
measurement outcome to others, who then apply local operations chosen based on
the outcome they receive. Based on one-round protocols, we are able to
construct an $r$-round protocol recursively. All these protocols belong to the
finite-round LOCC class, and can be visualized as tree graphs. Each node in
the tree represents a local operation and different measurement outcomes
correspond to edges connecting to this node’s children, which represent
different choice of local operations based on the measurement outcomes from
last round.

Although the basic idea of LOCC is relatively easy to grasp, its mathematical
structure is highly complicated Chitambar2014  and hard to characterize. As
indicated by its tree structure, a general $r$-round LOCC protocol could lead
to exponentially many possible results, making LOCC protocol designs for many
essential quantum information processing tasks very challenging. At the same
time, it will be more practical to consider LOCC protocols with hardware-
efficient local operations and a few communication rounds due to the limited
coherence time of local quantum memory. To overcome these challenges, we
propose to find LOCC protocols with the aid of machine learning, inspired by
its recent success in various areas. Specifically, we present the LOCCNet
framework, which incorporates optimization methods from classical machine
learning field into the workflow of designing LOCC protocols and can simulate
any finite round LOCC in principle.

Figure 1: Illustration of the procedure for optimizing an LOCC protocol with
LOCCNet. For simplicity, only two parties are involved in this workflow,
namely Alice and Bob. The tree presented here corresponds to a specific two-
round LOCC protocol. Such a tree can be customized with LOCCNet. With each
node (Local Operation) encoded as a PQC and arrows between nodes referring to
classical communication, one can define a loss function to guide the training
process depending on the task. The tree branch diverges indicating different
possible measurement outcomes. Finally, one can adopt optimization methods to
iteratively update the parameters $\bm{\theta}$ in each local operation and
hence obtain the optimized LOCC protocol.

As illustrated in Fig. 1, each party’s local operations, represented by nodes
in a tree, are described as parameterized quantum circuits (PQC)
Benedetti2019a . Users can measure any chosen qubit and define a customized
loss function from measurement outcomes as well as remaining states. With a
defined loss function for a task of interest, LOCCNet can be optimized to give
a protocol. The effect of classical communication is also well simulated by
LOCCNet in the sense that different PQCs can be built for different
measurement outcomes from previous rounds.

Previously, PQCs have been adapted to many research areas including quantum
simulation peruzzo2014variational , quantum optimization Farhi2014 , and
quantum error correction Johnson2017 . The family of variational quantum
algorithms Cerezo2020 ; Bharti2021 ; Endo2020 , based on PQCs, is one
promising candidate to achieve quantum advantages with near-term devices. In
quantum information, PQCs also help in estimating distance measures for
quantum states Chen2020a ; Cerezo2019  and compressing quantum data Romero2017
; Cao2020 . Here, we take one step further by extending the use of PQCs to the
distributed quantum information processing scenario where LOCC is the most
natural set of operations.

In the next three sections, we will demonstrate the LOCCNet framework in
details with important applications and present some interesting findings,
including protocols that achieve better results than existing ones. We conduct
software implementations of LOCCNet using the Paddle Quantum toolkit
Paddlequantum  on the PaddlePaddle Deep Learning Platform Ma2019 ; Paddle .

###  II.2 Entanglement distillation

Many applications of LOCC involve entanglement manipulation, and the use of
entanglement is generally required to be in its pure and maximal form. Hence,
the efficient conversion of entanglement into such a form, a process known as
entanglement distillation Bennett1996 ; Bennett1996c , is usually a must for
many quantum technologies. The development of entanglement distillation
methods remains at the forefront of quantum information horodecki2009quantum .
For example, the two-qubit maximally entangled state
$|\Phi^{+}\rangle=1/\sqrt{2}(|00\rangle+|11\rangle)$, which is also known as
the entangled bit (ebit), is the fundamental resource unit in entanglement
theory since it is a key ingredient in many quantum information processing
tasks. Thus, an essential goal for entanglement distillation in a two-qubit
setting is to convert a number of copies of some two-qubit state $\rho_{AB}$
shared by two parties, Alice and Bob, into a state as close as possible to the
ebit. Here, closeness between the state $\rho_{AB}$ and the ebit is usually
measured in terms of the fidelity

$\displaystyle F=\langle\Phi^{+}|\rho_{AB}|\Phi^{+}\rangle.$ (1)

Although theory is more concerned with asymptotic distillation with unlimited
copies of $\rho_{AB}$, protocols considering a finite number of copies are
more practical due to the physical limitations of near-term quantum
technologies. Also, practical distillation protocols usually allow for the
possibility of failure as a trade-off for achieving a higher final fidelity.
Furthermore, due to limited coherence time of local quantum memories, schemes
involving only one round of classical communication are preferred in practice.
Under these settings, many practical schemes for entanglement distillation
have been proposed Bennett1996 ; deutsch1996quantum ; fujii2009entanglement ;
kalb2017entanglement ; rozpkedek2018optimizing ; krastanov2019optimized . Not
surprisingly, there is not a single scheme that applies to all kinds of
states. In fact, designing a protocol even for a specific type of states is a
difficult task.

In this section, we apply LOCCNet to entanglement distillation and present
selected results that reinforce the validity and practicality of using this
framework for designing LOCC protocols. To use LOCCNet for finding
distillation protocols for a state $\rho_{AB}$, we build two PQCs, one for
Alice and one for Bob. In the preset event of success, these PQCs output a
state supposed to have a higher fidelity to the ebit. To optimize PQCs, we
define the infidelity of the output state and the ebit, i.e., $1-F$, as the
loss function to be minimized. As soon as the value of the loss function
converges through training, the PQCs along with the optimized parameters form
an LOCC distillation protocol. In principle, this training procedure is
general and can be applied to find distillation protocols for any initial
state $\rho_{AB}$ given its numerical form. Beyond rediscovering existing
protocols, we are also able to find improved protocols with LOCCNet. Below, we
give two distillation protocols for S states and isotropic states,
respectively, as examples of optimized schemes found with LOCCNet.

An S state is a mixture of the ebit $|\Phi^{+}\rangle$ and non-orthogonal
product noise rozpkedek2018optimizing . Here, we define it to be

$\displaystyle\rho_{AB}=p|\Phi^{+}\rangle\langle\Phi^{+}|+(1-p)|00\rangle\langle
00|,$ (2)

where $p\in[0,1]$. A distillation protocol known to perform well on two copies
of some S state is the DEJMPS protocol deutsch1996quantum , which in this case
outputs a state whose fidelity to the ebit is $(1+p)^{2}/(2+2p^{2})$ with a
probability of $(1+p^{2})/2$ [see Supplementary Note 2].

Here, we present a protocol learned by LOCCNet that can output a state
achieving a fidelity higher than DEJMPS and close to the highest possible
fidelity. Details on this protocol after simplification are given in Fig. 2,
where Alice and Bob apply local operations to their own qubits independently
and then compare their measurement outcomes through classical communication.
The distillation succeeds only when both Alice and Bob get $0$ from
computational basis measurements.

@C=1em @R=1.7em A_0 &
A_1 -1 R_y(θ)
B_0 1
B_1 -1 R_y(θ)

Figure 2: Circuit of a distillation protocol learned by LOCCNet for S states.
This simplified circuit represents local operations in a distillation protocol
learned by LOCCNet for two copies of an S state, $\rho_{A_{0}B_{0}}$ and
$\rho_{A_{1}B_{1}}$. The rotation angles of both $R_{y}$ gates are
$\theta=\arccos(1-p)+\pi$, which depend on the parameter $p$ of the S states
to be distilled.

The final fidelity achieved by this protocol is compared with that achieved by
the DEJMPS protocol in Fig. 3. For the aim of benchmarking, the techniques
based on partial positive transpose (PPT) were introduced to derive
fundamental limits of entanglement distillation Rains2001 ; Matthews2008 ;
Wang2016 ; Fang2017 ; rozpkedek2018optimizing ; Wang2016c . The entanglement
theory under PPT operations has been extensively studied in the literature
(e.g., Audenaert2003 ; Wang2016d ; Regula2019 ; Chitambar2017 ; Wang2017d ;
Wang2020c ) and offers valuable limitations of LOCC. Here, the PPT bound
obtained with semi-definite programming rozpkedek2018optimizing  is an upper
bound to the fidelity achieved by any LOCC protocol [see Supplementary Note
2].

As shown in the figure, the protocol learned by LOCCNet achieves near-optimal
fidelity in the sense that it is close to the PPT bound. Analytically, for two
copies of some S state with a parameter $p$, the post-measurement state in the
event of success is
$\sigma_{AB}=F|\Phi^{+}\rangle\langle\Phi^{+}|+(1-F)|\Phi^{-}\rangle\langle\Phi^{-}|$,
where

$\displaystyle F=\frac{1+\sqrt{2p-p^{2}}}{2}$ (3)

is its fidelity to the ebit and
$|\Phi^{-}\rangle=1/\sqrt{2}(|00\rangle-|11\rangle)$. The probability of
arriving at this state is $p_{\text{succ}}=p^{2}-p^{3}/2$ [see Supplementary
Note 2]. It is noteworthy that the distilled state is a Bell diagonal state of
rank two. For two copies of such a state, the DEJMPS protocol achieves the
optimal fidelity rozpkedek2018optimizing ; Ruan2018 . Thus, combining our
protocol with the DEJMPS protocol offers an efficient and scalable
distillation scheme for more copies of some S state.

Figure 3: Fidelity achieved by distillation protocols for two copies of some S
state. The orange dashed line depicts the performance of the protocol learned
by LOCCNet, which outperforms the DEJMPS protocol (green dotted). Also, the
learned protocol is near optimal in the sense that its line almost aligns with
the PPT bound (blue solid).

Another important family of entangled states is the isotropic state family,
defined as

$\displaystyle\rho_{AB}=p|\Phi^{+}\rangle\langle\Phi^{+}|+(1-p)\frac{I}{4},$
(4)

where $p\in[0,1]$ and $I$ is the identity matrix. Distillation protocols for
two copies of some isotropic state have been well studied, and the DEJMPS
protocol achieves empirically optimal fidelity in this case. Given four copies
of some isotropic state with a parameter $p$, a common way to distill
entanglement is to divide them into two groups of two copies and apply the
DEJMPS protocol to each group. Conditioned on success, we then apply the
DEJMPS protocol again to the two resulting states from the previous round.
Since the DEJMPS protocol was originally designed for two-copy distillation,
such a generalization is probably unable to fully exploit the resources
contained in four copies of the state. Indeed, with the aid of LOCCNet, we
find a protocol optimized specifically for four copies of some isotropic
state. As illustrated in Fig. 4, Alice and Bob first apply similar local
operations with three pairs of qubits being measured and then compare their
measurement outcomes through classical communication. If their measurement
outcomes for each pair of qubits are identical, the distillation procedure
succeeds.

@C=1em @R=1.7em A_0 & 1
A_1 1 R_x(+π2)
A_2 1 R_x(+π2)
A_3 -3 R_x(+π2)

Figure 4: Circuit of a distillation protocol learned by LOCCNet for isotropic
states. This simplified circuit represents Alice’s local operation in a
protocol learned by LOCCNet for entanglement distillation with four copies of
some isotropic state. Bob’s local operation is identical to Alice’s, except
that the rotation angles of Bob’s $R_{x}$ gates are $-\pi/2$.

The fidelity achieved by this protocol for different input isotropic states is
plotted in Fig. 5, along with that of the generalized DEJMPS protocol. For
four copies of some isotropic state with a parameter $p$, our protocol
achieves a final fidelity of

$\displaystyle F=\frac{1-2p+9p^{2}}{4-8p+12p^{2}},$ (5)

which is slightly higher than the DEJMPS protocol, as shown in Fig. 5. Details
are referred to [Supplementary Note 2]. Another advantage of this optimized
protocol is that the output state in the event of success is still an
isotropic state, implying the possibility of a generalized distillation
protocol for $4^{n}$ copies of some isotropic state.

We remark that our protocols are optimized with the goal to achieve the
highest possible fidelity, so their probabilities of success are not high. For
situations where the probability of success is important, one can also design
a customized loss function to optimize a protocol according to their metrics.

Figure 5: Fidelity achieved by distillation protocols for four copies of some
isotropic state. The blue solid line depicts the fidelity achieved by the
protocol learned by LOCCNet, which outperforms the generalized DEJMPS protocol
(orange dashed).

###  II.3 Distributed quantum state discrimination

Another important application of LOCC is quantum state discrimination (QSD).
Distinguishing one physical configuration from another is central to
information theory. When messages are encoded into quantum states for
information transmission, the processing of this information relies on the
distinguishability of quantum states. Hence, QSD has been a central topic in
quantum information Bae2015 ; reviewQSD ; Li2015a , which investigates how
well quantum states can be distinguished and underlies various applications in
quantum information processing tasks, including quantum data hiding
DiVincenzo2002  and dimension witness Gallego2010 .

QSD using global quantum operations is well-understood in the sense that the
optimal strategy maximizing the success probability can be solved efficiently
via semi-definite programming (SDP) Eldar2003 ; Sun2002 ; Jezek2002 . However,
for an important operational setting called distant lab paradigm or
distributed regime, our knowledge of QSD remains limited despite substantial
efforts in the past two decades Bennett1999b ; Walgate2000 ; Fan2004a ;
Hayashi2006 ; Ghosh2004 ; Nathanson2005 ; Duan2007a ; Chitambar2014b ;
Duan2009d ; Childs2013 ; Li2017 ; Bandyopadhyay2014a . In the distributed
regime, multipartite quantum states are distributed to spatially separated
labs, and the goal is to distinguish between these states via LOCC.

For two orthogonal pure states shared between multiple parties, it has been
shown that they can be distinguished via LOCC alone no matter if these states
are entangled or not Walgate2000 . However, it is not easy to design a
concrete LOCC protocol for practical implementation on near-term quantum
devices. Using LOCCNet, one can optimize and obtain practical LOCC protocols
for quantum state discrimination. Furthermore, for non-orthogonal states,
limited aspects have been investigated in terms of the feasibility of LOCC
discrimination. However, LOCCNet can provide an optimized and practical
protocol in this realistic setting.

Here, to explore the power of LOCCNet in state discrimination, we focus on the
optimal success probability of discriminating between noiseless and noisy Bell
states via LOCC. Consider two Bell states, $|\Phi^{+}\rangle$ and
$|\Phi^{-}\rangle$, and an amplitude damping (AD) channel $\mathcal{A}$ with
noise parameter $\gamma$ such that $\mathcal{A}(\rho)=E_{0}\rho
E_{0}^{\dagger}+E_{1}\rho E_{1}^{\dagger}$ with $E_{0}=|0\rangle\langle
0|+\sqrt{1-\gamma}|1\rangle\langle 1|$ and
$E_{1}=\sqrt{\gamma}|0\rangle\langle 1|$. If we send $|\Phi^{-}\rangle$’s two
qubits respectively through this AD channel, then the resulting state is
$\mathcal{A}\otimes\mathcal{A}(|\Phi^{-}\rangle\langle\Phi^{-}|)$. The goal is
now to distinguish between $|\Phi^{+}\rangle\\!\langle\Phi^{+}|$ and
$\mathcal{A}\otimes\mathcal{A}(|\Phi^{-}\rangle\\!\langle\Phi^{-}|)$.

Suppose $\Phi_{0}$ and $\Phi_{1}$ are some pair of two-qubit states. To find a
protocol discriminating between them, we build an ansatz with measurements on
both qubits. As illustrated in Fig. 6, Alice performs a unitary gate on her
qubit followed by a measurement, whose outcome determines Bob’s operation on
his qubit. Given an ideal discrimination protocol, Bob’s measurement outcome
should be $0$ if and only if the input state is $\Phi_{0}$ so that he can tell
which state the input state is for sure. Based on this observation, we define
a loss function

$\displaystyle L=P(1|\Phi_{0})+P(0|\Phi_{1}),$ (6)

where $P(j|\Phi_{k})$ is the probability of Bob’s measurement outcome being
$j$ given the input state being $\Phi_{k}$. By minimizing this loss function,
we are able to obtain a protocol for distinguishing between states $\Phi_{0}$
and $\Phi_{1}$ with an optimized probability of success. Specifically, for
$\Phi_{0}\equiv|\Phi^{+}\rangle\\!\langle\Phi^{+}|$ and
$\Phi_{1}\equiv\mathcal{A}\otimes\mathcal{A}(|\Phi^{-}\rangle\\!\langle\Phi^{-}|)$,
through optimization we find a protocol where Alice’s local unitary operation
is $U=R_{y}(\pi/2)$ and Bob’s local unitary operation is
$V=R_{y}((-1)^{a}\theta)$ where $\theta=\pi-\arctan((2-\gamma)/\gamma)$ and
$a=0\text{ or }1$ is Alice’s measurement outcome. This optimized protocol
achieves an average success probability of

$\displaystyle
p_{\text{succ}}=\frac{1}{2}+\frac{\sqrt{2-2\gamma+\gamma^{2}}}{2\sqrt{2}},$
(7)

as given in [Supplementary Note 3].

@C=0.6em @R=1.7em Alice & U [1]
Bob V 0 or 1

Figure 6: Ansatz used for finding QSD protocols with LOCCNet. Alice performs a
unitary gate on her qubit and measures. Then Bob performs on his qubit a
unitary gate chosen based on Alice’s measurement result. Bob’s measurement
outcome is supposed to tell which state the input state is.

In Fig. 7, we compare the protocol learned by LOCCNet with the optimal
protocol for perfect discrimination between two noiseless and orthogonal Bell
states $|\Phi^{+}\rangle$ and $|\Phi^{-}\rangle$. The PPT bound shown in Fig.
7 is obtained via SDP and serves as an upper bound to the average probability
of any LOCC protocol recognizing the input state correctly Yu2014a , where the
input state is either $\Phi_{0}$ or $\Phi_{1}$ with equal chance. While the
noiseless protocol is consistently better than random guessing as noise in the
AD channel increases, it inevitably suffers from a decrease in its
discrimination ability. The gap between its probability of success and the PPT
bound steadily widens. On the other hand, the protocol optimized with LOCCNet
can achieve a near-optimal probability of success for each noise setting, as
shown in the figure.

Figure 7: Average success probability of distinguishing a Bell state and a
noisy Bell state. The orange dashed line depicts the behavior of the protocol
via LOCCNet, which outperforms the protocol for distinguishing perfect
orthogonal Bell states (green dotted). Moreover, the protocol from LOCCNet is
near optimal since it almost matches the upper bounds obtained via PPT POVMs
(blue solid).

###  II.4 Quantum channel simulation

One central goal of quantum information is to understand the limitations
governing the use of quantum systems to take advantage of quantum physics
laws. Quantum channel lies at the heart of this question since it
characterizes what we can do with the quantum states physically Nielsen2010 ;
Wilde2017book ; Watrous2011b . To fully exploit quantum resources, the ability
to manipulate quantum channels under operational settings is important.
Particularly, in distributed quantum computing, one fundamental primitive,
dubbed quantum channel simulation, is to realize quantum channels from one
party to another using entanglement and LOCC protocols. Quantum channel
simulation, exploiting entanglement to synthesize a target channel through
LOCC protocols Bennett1996c ; Bennett2014 ; Berta2013 ; Pirandola2015b ;
Wilde2018 ; WW18 ; Fang2018 ; Pirandola2018 , servers as the basis of many
problems in quantum information, including quantum communication, quantum
metrology Pirandola2017 , and quantum key distribution Pirandola2020 .

One famous example of quantum channel simulation is quantum teleportation
(i.e., simulation of the identity channel). As one of the most important
quantum information processing protocols bennett1993teleporting ;
Pirandola2015 , quantum teleportation exploits the physical resource of
entanglement to realize noiseless quantum channels between different parties
and it is an important building block for quantum technologies including
distributed quantum computing and quantum networks. Similar to quantum
teleportation, quantum channel simulation is a general technique to send an
unknown quantum state $\psi$ from a sender to a receiver such that the
receiver could obtain ${\cal N}_{A^{\prime}\to B}(\psi_{A^{\prime}})$ with the
help of a pre-shared entangled state $\rho_{AB}$ and an LOCC protocol $\Pi$.
The overall scheme simulates the target channel ${\cal N}$ in the sense that

$\displaystyle\Pi(\psi_{A^{\prime}}\otimes\rho_{AB})={\cal N}_{A^{\prime}\to
B}(\psi_{A^{\prime}}),\forall\psi_{A^{\prime}}.$ (8)

For some classes of channels such as Pauli channels, the LOCC-based simulation
protocols were known Bennett1996c ; Horodecki1999 ; Pirandola2015b . However,
the LOCC protocols for general quantum channel simulation is hard to design
due to the complexity of LOCC. Even for the qubit amplitude damping (AD)
channel, the LOCC protocol for simulating this channel in the non-asymptotic
regime is still unknown, and its solution would provide a better estimate of
its secret key capacity Pirandola2020 . Note that the asymptotic simulation of
this channel involving infinite dimensions was introduced in Pirandola2015b .

Here, we apply our LOCCNet to explore the simulation of an AD channel
$\mathcal{A}$ using its Choi state Choi1975
$\rho_{\mathcal{A}}=(I\otimes\mathcal{A})(\Phi^{+})$ as the pre-shared
entangled state. Note AD channel is one of the realistic sources of noise in
superconducting quantum processor Chirolli2018 .

To train the LOCCNet for simulating ${\cal A}$, we select a set of linearly
independent density matrices $S$ as the training set. The loss function for
this channel simulation task is then defined as

$\displaystyle L=-\sum_{\psi\in S}F({\cal A}(\psi),{\cal B}(\psi)),$ (9)

where ${\cal B}$ is the actual channel simulated by LOCCNet with current
parameters and
$F(\rho,\sigma)=\text{Tr}\left(\sqrt{\rho^{1/2}\sigma\rho^{1/2}}\right)^{2}$
gives the fidelity between states $\rho$ and $\sigma$. With this loss function
to be minimized, the parameters in LOCCNet are optimized to maximize the state
fidelity between $\mathcal{A}(\psi)$ and $\mathcal{B}(\psi)$ for all $\psi\in
S$.

Once the LOCCNet is trained to teleport all the basis states in $S$ with near
perfect fidelity, we obtain a protocol for simulating ${\cal A}$. For
benchmarking, we randomly generate 1000 pure states and teleport them to Bob.
The results are summarized in Fig. 8. Compared with the original teleportation
protocol, we could achieve an equivalent performance at low noise level and a
better performance at noise level $\gamma>0.4$. Note that the numerical
simulations are conducted on Paddle Quantum Paddlequantum .

Figure 8: Average fidelity of simulating AD channel with LOCC protocols. The
blue curve depicts the behavior of the protocol via LOCCNet, which outperforms
the original teleportation (orange) at high noise level (noise parameter
$\gamma>0.4$). Each data point contains the statistical results of $1000$
randomly generated states.

##  III Discussion

We established LOCCNet for exploring LOCC protocols in distributed quantum
information processing. Its overall pipeline is standard for machine learning
algorithms. For a specific task, one firstly designs an appropriate loss
function and then utilizes different LOCCNet structures and optimization
methods to train the model to obtain an optimal or near-optimal protocol.
Depending on the nature of the task, a selected training data set may be
required, as in the case of channel simulation. Based on the current design of
LOCCNet, more machine learning techniques, such as reinforcement learning
could be incorporated into this framework, making it a more powerful tool for
exploring LOCC protocols.

LOCCNet not only unifies and extends the existing LOCC protocols, but also
sheds light on the power and limitation of LOCC in the noisy intermediate-
scale quantum (NISQ) era preskill2018quantum  by providing a plethora of
examples. We developed improved protocols for entanglement distillation, local
state discrimination, and quantum channel simulation as applications. As a
showcase, we applied LOCCNet to establish hardware-efficient and state-of-the-
art protocols for entanglement distillation of noisy entangled states of
interest. In addition to making a significant contribution to entanglement
distillation, LOCCNet finds direct practical use in many settings, as we
exemplified with several explicit applications in distinguishing noisy and
noiseless Bell states as well as simulating amplitude damping channels.

As we have shown the ability of LOCCNet in discovering improved LOCC
protocols, one future direction is to apply LOCCNet to further enhance
practical entanglement manipulation and quantum communication and explore
fundamental problems in quantum information theory. While in this paper we
mainly focus on bipartite cases, LOCCNet also supports multipartite
entanglement manipulation. For example, as an essential part in quantum
repeaters Briegel1998 , entanglement swapping aims to transform two entangled
pairs shared between Alice and Bob and between Bob and Carol into a new
entangled pair shared by Alice and Carol using only LOCC. Indeed, we could use
LOCCNet to design such a protocol. For instance, we can build an LOCCNet where
Bob first operates on and measure his subsystem, and then Alice and Carol
perform local operations according to the measurement results from Bob. The
loss function to minimize can be defined as the infidelity of a target state
and the output state shared between Alice and Carol. Similar procedures can be
followed to apply LOCCNet in optimizing other multipartite protocols as well,
which is worth exploring in future works.

Another important direction is to extend the framework to the continuous-
variable quantum information processing, which may be applied to explore
better LOCC protocols of private communication based on continuous variable
systems Pirandola2020 . As we have seen the potential of advancing distributed
quantum information processing with the aid of machine learning, we expect
more of such cases with classical machine learning being used to improve
quantum technologies, which in turn will enhance quantum machine learning
applications.

## Data Availability

Data that support the plots and other findings of this study are available
from the corresponding authors upon reasonable request.

## Code Availability

Code used in the numerical experiments on quantum channel simulation is
available at https://github.com/vsastray/LOCCNetcodes. Other Code used in this
study is available from the corresponding authors upon reasonable request.

## Acknowledgements

We would like to thank Runyao Duan and Kun Fang for helpful discussions.

## Competing Interests

The authors declare no competing interests.

## Author Contributions

X. W. formulated the initial idea and the framework; X. Z. and B. Z. developed
the theory; X. Z., B. Z., Z. W., and Z. S. performed the experiments. All co-
authors contributed to the preparation of the manuscript.

## References

  * (1) Farhi, E., Goldstone, J. & Gutmann, S.  A Quantum Approximate Optimization Algorithm.  Preprint at http://arxiv.org/abs/1411.4028 (2014).
  * (2) Harrigan, M. P. _et al._ Quantum approximate optimization of non-planar graph problems on a planar superconducting processor.  _Nat. Phys._ 17, 332–336 (2021).
  * (3) McArdle, S., Endo, S., Aspuru-Guzik, A., Benjamin, S. & Yuan, X.  Quantum computational chemistry.  _Rev. Mod. Phys._ 92, 015003 (2020).
  * (4) Arute, F. _et al._ Hartree-Fock on a superconducting qubit quantum computer.  _Science_ 369, 1084–1089 (2020).
  * (5) Bennett, C. H. & Brassard, G.  Quantum cryptography: Public key distribution and coin tossing.  In _International Conference on Computers, Systems & Signal Processing, Bangalore, India, Dec 9-12, 1984_, 175–179 (1984).
  * (6) Ekert, A. K.  Quantum cryptography based on Bell’s theorem.  _Phys. Rev. Lett._ 67, 661–663 (1991).
  * (7) Biamonte, J. _et al._ Quantum machine learning.  _Nature_ 549, 195–202 (2017).
  * (8) Bennett, C. H. _et al._ Teleporting an unknown quantum state via dual classical and einstein-podolsky-rosen channels.  _Phys. Rev. Lett._ 70, 1895 (1993).
  * (9) Bennett, C. H. & Wiesner, S. J.  Communication via one-and two-particle operators on einstein-podolsky-rosen states.  _Phys. Rev. Lett._ 69, 2881 (1992).
  * (10) Plenio, M. B. & Virmani, S. S.  An introduction to entanglement measures.  _Quantum Information and Computation_ 7, 1–51 (2007).
  * (11) Horodecki, R., Horodecki, P., Horodecki, M. & Horodecki, K.  Quantum entanglement.  _Rev. Mod. Phys._ 81, 865 (2009).
  * (12) Preskill, J.  Quantum Computing in the NISQ era and beyond.  _Quantum_ 2, 79 (2018).
  * (13) Chitambar, E., Leung, D., Mančinska, L., Ozols, M. & Winter, A.  Everything You Always Wanted to Know About LOCC (But Were Afraid to Ask).  _Commun. Math. Phys._ 328, 303–326 (2014).
  * (14) LeCun, Y., Bengio, Y. & Hinton, G.  Deep learning.  _Nature_ 521, 436–444 (2015).
  * (15) Silver, D. _et al._ Mastering the game of Go with deep neural networks and tree search.  _Nature_ 529, 484–489 (2016).
  * (16) Jumper, J. _et al._ Highly accurate protein structure prediction with AlphaFold.  _Nature_ 596, 583–-589 (2021).
  * (17) Mavadia, S., Frey, V., Sastrawan, J., Dona, S. & Biercuk, M. J.  Prediction and real-time compensation of qubit decoherence via machine learning.  _Nat. Commun._ 8, 14106 (2017).
  * (18) Wan, K. H., Dahlsten, O., Kristjánsson, H., Gardner, R. & Kim, M. S.  Quantum generalisation of feedforward neural networks.  _npj Quantum Inf._ 3, 36 (2017).
  * (19) Lu, D. _et al._ Enhancing quantum control by bootstrapping a quantum processor of 12 qubits.  _npj Quantum Inf._ 3, 1–7 (2017).
  * (20) Niu, M. Y., Boixo, S., Smelyanskiy, V. N. & Neven, H.  Universal quantum control through deep reinforcement learning.  _npj Quantum Inf._ 5, 33 (2019).
  * (21) Wallnöfer, J., Melnikov, A. A., Dür, W. & Briegel, H. J.  Machine Learning for Long-Distance Quantum Communication.  _PRX Quantum_ 1, 010301 (2020).
  * (22) Bausch, J. & Leditzky, F.  Quantum codes from neural networks.  _New J. Phys._ 22, 023005 (2020).
  * (23) Benedetti, M., Lloyd, E., Sack, S. & Fiorentini, M.  Parameterized quantum circuits as machine learning models.  _Quantum Sci. Technol._ 4, 043001 (2019).
  * (24) Bennett, C. H. _et al._ Purification of Noisy Entanglement and Faithful Teleportation via Noisy Channels.  _Phys. Rev. Lett._ 76, 722–725 (1996).
  * (25) Deutsch, D. _et al._ Quantum privacy amplification and the security of quantum cryptography over noisy channels.  _Phys. Rev. Lett._ 77, 2818 (1996).
  * (26) Murao, M., Plenio, M. B., Popescu, S., Vedral, V. & Knight, P. L.  Multiparticle entanglement purification protocols.  _Phys. Rev. A_ 57, R4075–R4078 (1998).
  * (27) Dür, W. & Briegel, H. J.  Entanglement purification and quantum error correction.  _Rep. Prog. Phys._ 70, 1381–1424 (2007).
  * (28) Pan, J.-W., Gasparoni, S., Ursin, R., Weihs, G. & Zeilinger, A.  Experimental entanglement purification of arbitrary unknown states.  _Nature_ 423, 417–422 (2003).
  * (29) Devetak, I. & Winter, A.  Distillation of secret key and entanglement from quantum states.  _Proc. R. Soc. A._ 461, 207–235 (2005).
  * (30) Bennett, C. H. _et al._ Quantum nonlocality without entanglement.  _Phys. Rev. A_ 59, 1070–1091 (1999).
  * (31) Walgate, J., Short, A. J., Hardy, L. & Vedral, V.  Local distinguishability of multipartite orthogonal quantum states.  _Phys. Rev. Lett._ 85, 4972 (2000).
  * (32) Fan, H.  Distinguishability and indistinguishability by local operations and classical communication.  _Phys. Rev. Lett._ 92, 177905 (2004).
  * (33) Hayashi, M., Markham, D., Murao, M., Owari, M. & Virmani, S.  Bounds on multipartite entangled orthogonal state discrimination using local operations and classical communication.  _Phys. Rev. Lett._ 96, 40501 (2006).
  * (34) Ghosh, S., Kar, G., Roy, A. & Sarkar, D.  Distinguishability of maximally entangled states.  _Phys. Rev. A_ 70, 22304 (2004).
  * (35) Nathanson, M.  Distinguishing bipartitite orthogonal states using LOCC: Best and worst cases.  _J. Math. Phys._ 46, 062103 (2005).
  * (36) Duan, R., Feng, Y., Ji, Z. & Ying, M.  Distinguishing arbitrary multipartite basis unambiguously using local operations and classical communication.  _Phys. Rev. Lett._ 98, 230502 (2007).
  * (37) Chitambar, E., Duan, R. & Hsieh, M.-H.  When Do Local Operations and Classical Communication Suffice for Two-Qubit State Discrimination?  _IEEE Trans. Inf. Theory_ 60, 1549–1561 (2014).
  * (38) Duan, R., Feng, Y., Xin, Y. & Ying, M.  Distinguishability of quantum states by separable operations.  _IEEE Trans. Inf. Theory_ 55, 1320–1330 (2009).
  * (39) Childs, A. M., Leung, D., Mančinska, L. & Ozols, M.  A framework for bounding nonlocality of state discrimination.  _Commun. Math. Phys._ 323, 1121–1153 (2013).
  * (40) Li, Y., Wang, X. & Duan, R.  Indistinguishability of bipartite states by positive-partial-transpose operations in the many-copy scenario.  _Phys. Rev. A_ 95, 052346 (2017).
  * (41) Bandyopadhyay, S. _et al._ Limitations on separable measurements by convex optimization.  _IEEE Trans. Inf. Theory_ 61, 3593–3604 (2014).
  * (42) Bennett, C. H., DiVincenzo, D. P., Smolin, J. A. & Wootters, W. K.  Mixed-state entanglement and quantum error correction.  _Phys. Rev. A_ 54, 3824–3851 (1996).
  * (43) Bennett, C. H., Devetak, I., Harrow, A. W., Shor, P. W. & Winter, A.  The Quantum Reverse Shannon Theorem and Resource Tradeoffs for Simulating Quantum Channels.  _IEEE Trans. Inf. Theory_ 60, 2926–2959 (2014).
  * (44) Berta, M., Brandao, F. G. S. L., Christandl, M. & Wehner, S.  Entanglement Cost of Quantum Channels.  _IEEE Trans. Inf. Theory_ 59, 6779–6795 (2013).
  * (45) Pirandola, S., Laurenza, R., Ottaviani, C. & Banchi, L.  Fundamental limits of repeaterless quantum communications.  _Nat. Commun._ 8, 15043 (2017).
  * (46) Wilde, M. M.  Entanglement cost and quantum channel simulation.  _Phys. Rev. A_ 98, 042338 (2018).
  * (47) Wang, X. & Wilde, M. M.  Exact entanglement cost of quantum states and channels under PPT-preserving operations.  Preprint at http://arxiv.org/abs/1809.09592 (2018).
  * (48) Fang, K., Wang, X., Tomamichel, M. & Berta, M.  Quantum Channel Simulation and the Channel’s Smooth Max-Information.  _IEEE Trans. Inf. Theory_ 66, 2129–2140 (2020).
  * (49) Peruzzo, A. _et al._ A variational eigenvalue solver on a photonic quantum processor.  _Nat. Commun._ 5, 4213 (2014).
  * (50) Johnson, P. D., Romero, J., Olson, J., Cao, Y. & Aspuru-Guzik, A.  QVECTOR: an algorithm for device-tailored quantum error correction.  Preprint at http://arxiv.org/abs/1711.02249 (2017).
  * (51) Cerezo, M. _et al._ Variational quantum algorithms.  _Nat. Rev. Phys._ 3, 625–644 (2021).
  * (52) Bharti, K. _et al._ Noisy intermediate-scale quantum (NISQ) algorithms.  Preprint at http://arxiv.org/abs/2101.08448 (2021).
  * (53) Endo, S., Cai, Z., Benjamin, S. C. & Yuan, X.  Hybrid Quantum-Classical Algorithms and Quantum Error Mitigation.  _J. Phys. Soc. Jpn._ 90, 032001 (2021).
  * (54) Chen, R., Song, Z., Zhao, X. & Wang, X.  Variational Quantum Algorithms for Trace Distance and Fidelity Estimation.  Preprint at http://arxiv.org/abs/2012.05768 (2020).
  * (55) Cerezo, M., Poremba, A., Cincio, L. & Coles, P. J.  Variational Quantum Fidelity Estimation.  _Quantum_ 4, 248 (2020).
  * (56) Romero, J., Olson, J. P. & Aspuru-Guzik, A.  Quantum autoencoders for efficient compression of quantum data.  _Quantum Sci. Technol._ 2, 045001 (2017).
  * (57) Cao, C. & Wang, X.  Noise-Assisted Quantum Autoencoder.  _Phys. Rev. Applied_ 15, 054012 (2021).
  * (58) Paddle Quantum.  https://github.com/PaddlePaddle/Quantum. (2020).
  * (59) Ma, Y., Yu, D., Wu, T. & Wang, H.  PaddlePaddle: An Open-Source Deep Learning Platform from Industrial Practice.  _Frontiers of Data and Computing_ 1, 105–115 (2019).
  * (60) PArallel Distributed Deep LEarning: Machine Learning Framework from Industrial Practice.  https://github.com/paddlepaddle/paddle (2016).
  * (61) Fujii, K. & Yamamoto, K.  Entanglement purification with double selection.  _Phys. Rev. A_ 80, 042308 (2009).
  * (62) Kalb, N. _et al._ Entanglement distillation between solid-state quantum network nodes.  _Science_ 356, 928–932 (2017).
  * (63) Rozpedek, F. _et al._ Optimizing practical entanglement distillation.  _Phys. Rev. A_ 97, 062333 (2018).
  * (64) Krastanov, S., Albert, V. V. & Jiang, L.  Optimized entanglement purification.  _Quantum_ 3, 123 (2019).
  * (65) Rains, E. M.  A semidefinite program for distillable entanglement.  _IEEE Trans. Inf. Theory_ 47, 2921–2933 (2000).
  * (66) Matthews, W. & Winter, A.  Pure-state transformations and catalysis under operations that completely preserve positivity of partial transpose.  _Phys. Rev. A_ 78, 012317 (2008).
  * (67) Wang, X. & Duan, R.  Improved semidefinite programming upper bound on distillable entanglement.  _Phys. Rev. A_ 94, 050301 (2016).
  * (68) Fang, K., Wang, X., Tomamichel, M. & Duan, R.  Non-asymptotic Entanglement Distillation.  _IEEE Trans. Inf. Theory_ 65, 6454–6465 (2019).
  * (69) Wang, X. & Duan, R.  Nonadditivity of Rains’ bound for distillable entanglement.  _Phys. Rev. A_ 95, 062322 (2017).
  * (70) Audenaert, K., Plenio, M. B. & Eisert, J.  Entanglement Cost under Positive-Partial-Transpose-Preserving Operations.  _Phys. Rev. Lett._ 90, 027901 (2003).
  * (71) Wang, X. & Duan, R.  Irreversibility of Asymptotic Entanglement Manipulation Under Quantum Operations Completely Preserving Positivity of Partial Transpose.  _Phys. Rev. Lett._ 119, 180506 (2017).
  * (72) Regula, B., Fang, K., Wang, X. & Gu, M.  One-shot entanglement distillation beyond local operations and classical communication.  _New J. Phys._ 21, 103017 (2019).
  * (73) Chitambar, E., de Vicente, J. I., Girard, M. W. & Gour, G.  Entanglement manipulation beyond local operations and classical communication.  _J. Math. Phys._ 61, 042201 (2020).
  * (74) Wang, X., Fang, K. & Duan, R.  Semidefinite Programming Converse Bounds for Quantum Communication.  _IEEE Trans. Inf. Theory_ 65, 2583–2592 (2019).
  * (75) Wang, X. & Wilde, M. M.  Cost of Quantum Entanglement Simplified.  _Phys. Rev. Lett._ 125, 040502 (2020).
  * (76) Ruan, L., Dai, W. & Win, M. Z.  Adaptive recurrence quantum entanglement distillation for two-Kraus-operator channels.  _Phys. Rev. A_ 97, 052332 (2018).
  * (77) Bae, J. & Kwek, L.-C.  Quantum state discrimination and its applications.  _J. Phys. A: Math. Theor._ 48, 083001 (2015).
  * (78) Barnett, S. M. & Croke, S.  Quantum state discrimination.  _Adv. Opt. Photon._ 1, 238–278 (2009).
  * (79) Li, K.  Discriminating quantum states: the multiple Chernoff distance.  _Ann. Statist._ 44, 1661–1679 (2016).
  * (80) DiVincenzo, D. P., Leung, D. W. & Terhal, B. M.  Quantum data hiding.  _IEEE Trans. Inf. Theory_ 48, 580–598 (2002).
  * (81) Gallego, R., Brunner, N., Hadley, C. & Acín, A.  Device-independent tests of classical and quantum dimensions.  _Phys. Rev. Lett._ 105, 230501 (2010).
  * (82) Eldar, Y.  A semidefinite programming approach to optimal unambiguous discrimination of quantum states.  _IEEE Trans. Inf. Theory_ 49, 446–456 (2003).
  * (83) Sun, X., Zhang, S., Feng, Y. & Ying, M.  Mathematical nature of and a family of lower bounds for the success probability of unambiguous discrimination.  _Phys. Rev. A_ 65, 44306 (2002).
  * (84) Ježek, M., Řeháček, J. & Fiurášek, J.  Finding optimal strategies for minimum-error quantum-state discrimination.  _Phys. Rev. A_ 65, 60301 (2002).
  * (85) Yu, N., Duan, R. & Ying, M.  Distinguishability of Quantum States by Positive Operator-Valued Measures With Positive Partial Transpose.  _IEEE Trans. Inf. Theory_ 60, 2069–2079 (2014).
  * (86) Nielsen, M. A. & Chuang, I. L.  _Quantum Computation and Quantum Information: 10th Anniversary Edition_ (Camb. Univ. Press, Cambridge, 2010).
  * (87) Wilde, M. M.  _Quantum Information Theory_ (Camb. Univ. Press, Cambridge, 2017).
  * (88) Watrous, J.  _The Theory of Quantum Information_ (Camb. Univ. Press, Cambridge, 2018).
  * (89) Pirandola, S. _et al._ Theory of channel simulation and bounds for private communication.  _Quantum Sci. Technol._ 3, 035009 (2018).
  * (90) Pirandola, S. & Lupo, C.  Ultimate Precision of Adaptive Noise Estimation.  _Phys. Rev. Lett._ 118, 100502 (2017).
  * (91) Pirandola, S. _et al._ Advances in quantum cryptography.  _Adv. Opt. Photon._ 12, 1012 (2020).
  * (92) Pirandola, S., Eisert, J., Weedbrook, C., Furusawa, A. & Braunstein, S. L.  Advances in quantum teleportation.  _Nat. Photonics_ 9, 641–652 (2015).
  * (93) Horodecki, M., Horodecki, P. & Horodecki, R.  General teleportation channel, singlet fraction, and quasidistillation.  _Phys. Rev. A_ 60, 1888–1898 (1999).
  * (94) Choi, M.-D.  Completely positive linear maps on complex matrices.  _Linear Algebra Appl._ 10, 285–290 (1975).
  * (95) Chirolli, L. & Burkard, G.  Decoherence in solid-state qubits.  _Adv. Phys._ 57, 225–285 (2008).
  * (96) Briegel, H.-J., Dür, W., Cirac, J. I. & Zoller, P.  Quantum Repeaters: The Role of Imperfect Local Operations in Quantum Communication.  _Phys. Rev. Lett._ 81, 5932–5935 (1998).

Supplemental Information: Practical distributed quantum information processing
with LOCCNet

## SUPPLEMENTARY NOTE 1: Details of LOCC

Preliminaries. We begin with the preliminaries on quantum information. We will
frequently use symbols such as $A$ (or $A^{\prime}$) and $B$ (or $B^{\prime}$)
to denote finite-dimensional Hilbert spaces associated with Alice and Bob,
respectively. We use $d_{A}$ to denote the dimension of system $A$. The set of
linear operators acting on $A$ is denoted by ${\cal L}(A)$. We usually write
an operator with a subscript indicating the system that the operator acts on,
such as $M_{AB}$, and write $M_{A}\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Tr}_{B}M_{AB}$.

A quantum state on system $A$ is a positive operator $\rho_{A}$ with unit
trace. The set of quantum states is denoted as $S(A)\mathrel{\mathop{\mathchar
58\relax}}=\\{\,\rho_{A}\geq 0\,|\,\operatorname{Tr}\rho_{A}=1\,\\}$. We call
a positive operator separable if it can be written as a convex combination of
tensor product positive operators. A bipartite positive semidefinite operator
$E_{AB}\in{\cal L}(A\otimes B)$ is said to be Positive-Partial-Transpose (PPT)
if $E_{AB}^{T_{B}}$ is positive semidefinite. Note that the action of partial
transpose (with respect to $B$) is defined as $(|i_{A}\rangle\langle
k_{A}|\otimes|j_{B}\rangle\langle l_{B}|)^{T_{B}}=|i_{A}\rangle\langle
k_{A}|\otimes|l_{B}\rangle\langle j_{B}|$.

LOCC. When a quantum system is distributed to spatially separated parties, it
is natural to consider how the system evolves when the parties perform local
quantum operations with classical communication. A systematic definition of
LOCC can be found in Chitambar2014 . Here, for self-consistency, we give a
detailed description of LOCC as follows.

Consider a setting involving multiple spatially separated parties sharing a
multipartite quantum system. The set $\text{LOCC}_{1}$ consists of the most
elementary LOCC operations corresponding to LOCC protocols with one classical
communication round, where one party performs a local operation and sends the
measurement outcome to others, who then perform corresponding local operations
on their local systems upon receiving the outcome. A local operation can be
described as a set of completely positive (CP) maps $\\{\mathcal{E}_{m}\\}$
such that $\sum_{m}\mathcal{E}_{m}$ is trace-preserving. The subscript $m$
corresponds to an operation’s measurement outcome, which could affect each
party’s choices of subsequent local operations. A more complicated LOCC
operation can be seen as a sequence of $\text{LOCC}_{1}$ operations.
Specifically, for any $r\geq 2$, $\text{LOCC}_{r}$ is defined to be a set of
LOCC operations, in which each operation is constructed from an
$\text{LOCC}_{r-1}$ operation followed by an $\text{LOCC}_{1}$ operation. A
common characteristic of these LOCC operations is that they can implemented
with finite rounds of classical communication. Thus, we define a set
$\text{LOCC}_{\mathbb{N}}$, corresponding to finite round protocols, such that
an LOCC operation is in this set if it belongs to $\text{LOCC}_{r}$ for some
$r$ in $\mathbb{N}=\\{1,2,\dots\\}$. As there are finite round protocols,
there also exist infinite round protocols in theory. These infinite round
protocols, together with operations in $\text{LOCC}_{\mathbb{N}}$, form the
set known as LOCC.

LOCCNet is a machine learning framework developed for designing and exploring
LOCC protocols for various quantum information processing tasks. In the main
text , we give a brief introduction to this framework. Here, we give some
common types of LOCC protocols involving two parties, Alice and Bob, as
examples to explain how a protocol can be constructed and optimized using the
LOCCNet.

Optimizing one-round LOCC protocols. One-round LOCC protocols are protocols
having only one round of classical communication. An example is shown in Fig.
S1. An application of such a protocol is quantum state teleportation. To
optimize a one-round protocol with LOCCNet, we need to build and train three
PQCs, shown as a tree in Fig. S2. The PQC $U(\theta_{0})$ is used to optimize
Alice’s local operation $U$, and PQCs $V_{0}(\theta_{1})$ and
$V_{0}(\theta_{2})$ are for Bob’s local operation in the case of Alice
measuring $0$ and $1$, respectively.

@C=1em @R=1.7em A & / U [1]
B / V

Figure S1: A circuit illustration of one-round LOCC. Alice first performs a
local operation and sends the measurement outcome to Bob. Bob then perform a
local operation accordingly. Figure S2: Tree structure of the LOCCNet used for
optimizing a one-round protocol.

Optimizing two-round LOCC protocols. A general two-round LOCC protocol
includes Alice performing a local operation and telling Bob her measurement
outcome, then Bob performing a corresponding local operation and telling Alice
his measurement outcome, and finally Alice performing another local operation.
Such a protocol is already a little complicated and optimizing such a protocol
requires seven PQCs. Here, we give two special types of two-round protocols
that are easier to train and has practical applications.

The first type of protocols is shown in Fig. S3 and are widely used for
entanglement distillation. In such a protocol, Alice and Bob first perform
local operations independently and then exchange their measurement outcomes
through classical communication to check whether the expected task is
completed. To optimize such a protocol, we only need to build two PQCs, one
for Alice’s local operation and one for Bob’s local operation.

@C=1em @R=1.7em A & / U 0 or 1
B / V 0 or 1

Figure S3: A circuit illustration of a type of two-round LOCC where Alice and
BOB perform local operations independently before exchanging measurement
outcomes.

Another type of protocols is given in Fig. S4. In such a protocol, after Bob
obtains his measurement outcome and tells it to Alice, Alice does not need to
perform a local operation. An application of such a protocol is state
discrimination, as we show in the main text. Like training a one-round
protocol, optimizing a protocol of this type only requires three PQCs.

@C=1em @R=1.7em A & / U [1]
B / V 0 or 1

Figure S4: A circuit illustration of another type of two-round LOCC. In such a
protocol, Bob sending his measurement outcome to Alice is the last step.

## SUPPLEMENTARY NOTE 2: Analysis of entanglement distillation

The aim of entanglement distillation is to compensate for the impurity caused
by noise and restore a maximally entangled state at the cost of many noisy
entangled states. In this sense, one could also refer an entanglement
distillation protocol as a purification or error-correction protocol. The Bell
states are four two-qubit maximally entangled states defined as

$\displaystyle|\Phi^{\pm}\rangle=\frac{1}{\sqrt{2}}(|00\rangle\pm|11\rangle),\quad|\Psi^{\pm}\rangle=\frac{1}{\sqrt{2}}(|01\rangle\pm|10\rangle).$
(S1)

The state $|\Phi\rangle$ is also known as the entangled bit (ebit), and
entanglement distillation in two-qubit settings usually means to convert
copies of a state $\rho_{AB}$ shared by two parties, Alice and Bob, into a
state closer to the ebit. Here, closeness between the state $\rho_{AB}$ and
the ebit is usually measured in terms of the fidelity

$\displaystyle F=\langle\Phi^{+}|\rho_{AB}|\Phi^{+}\rangle.$ (S2)

A well known protocol for two-copy entanglement distillation is the DEJMPS
protocol, which is illustrated in Fig. S5. Sharing two copies of an initial
state, $\rho_{A_{0}B_{0}}$ and $\rho_{A_{1}B_{1}}$, both Alice and Bob first
apply $R_{x}$ gates and CNOT gates to their local qubits and then measure a
pair of qubits from the same copy. Finally, they exchange measurement outcomes
and output the unmeasured copy when their outcomes agree. Otherwise, the
distillation procedure fails.

@C=1em @R=1.7em A_0 &R_x(+π2) 1
A_1 R_x(+π2)
B_0 R_x(-π2) 1
B_1 R_x(-π2)

Figure S5: The DEJMPS protocol for two-copy entanglement distillation.

The DEJMPS protocol has been shown to be optimal in purifying two copies of
any Bell diagonal state with of rank at most three rozpkedek2018optimizing ,
where a Bell diagonal state is a state of the form

$\displaystyle\rho_{AB}=p_{0}|\Phi^{+}\rangle\\!\langle\Phi^{+}|+p_{1}|\Psi^{+}\rangle\\!\langle\Psi^{+}|+p_{2}|\Phi^{-}\rangle\\!\langle\Phi^{-}|+p_{3}|\Psi^{-}\rangle\\!\langle\Psi^{-}|,$
(S3)

which is a convex combination of the four Bell states. For conciseness, we can
write such a Bell diagonal state as a $4$-tuple,

$\displaystyle\rho_{AB}=(p_{0},p_{1},p_{2},p_{3}).$ (S4)

The DEJMPS protocol can also distill some states besides Bell diagonal states,
like S states. In the following, we will analyze the performance of the DEJMPS
protocol on two copies of an S state and compare it with a protocol learned by
LOCCNet. After that, we will compare the DEJMPS protocol with another protocol
learned by LOCCNet for distilling four copies of an isotropic state, which is
a special Bell diagonal state.

S state. The S state is defined as the Bell state with a non-orthogonal
product noise,

$\displaystyle\rho_{AB}(p)=p|\Phi^{+}\rangle\langle\Phi^{+}|+(1-p)|00\rangle\langle
00|,$ (S5)

where $p\in[0,1]$. In the main text, we give expressions of fidelity achieved
by the DEJMPS protocol and the protocol learned by LOCCNet for two copies of
an S state. Here, we give a detailed derivation of these two expressions.

###### Proposition S1

For two copies of an S state with parameter $p$, the DEJMPS protocol outputs a
state whose fidelity to the ebit is

$\displaystyle F=\frac{(1+p)^{2}}{2+2p^{2}}$ (S6)

with a probability of success

$\displaystyle p_{\text{succ}}=\frac{1+p^{2}}{2}.$ (S7)

###### Proof.

By its definition in Equation (S5), an S state $\rho$ with parameter $p$ can
be written in the matrix form as

$\displaystyle\rho=\begin{pmatrix}1-\frac{p}{2}&0&0&\frac{p}{2}\\\ 0&0&0&0\\\
0&0&0&0\\\ \frac{p}{2}&0&0&\frac{p}{2}\\\ \end{pmatrix}.$ (S8)

Applying the circuit in Fig. S5 to two copies of such an state, Alice and Bob
both get $0$ for measurement outcomes with a probability of
$p_{00}=(1+p^{2})/4$. By matrix calculation, we obtain the post-measurement
state of the unmeasured copy in this case as

$\displaystyle\sigma_{-}=\begin{pmatrix}\alpha&-\beta&-\beta&\alpha\\\
-\beta&\beta&\beta&-\beta\\\ -\beta&\beta&\beta&-\beta\\\
\alpha&-\beta&-\beta&\alpha\\\ \end{pmatrix},$ (S9)

where $\alpha=(1+p)^{2}/(4+4p^{2})$ and $\beta=(1-p)^{2}/(4+4p^{2})$. The
probability that Alice and Bob both get $1$ for measurement outcomes is
$p_{11}=(1+p^{2})/4$, and the post-measurement state in this case is

$\displaystyle\sigma_{+}=\begin{pmatrix}\alpha&\beta&\beta&\alpha\\\
\beta&\beta&\beta&\beta\\\ \beta&\beta&\beta&\beta\\\
\alpha&\beta&\beta&\alpha\\\ \end{pmatrix}.$ (S10)

According to the definition of fidelity, the fidelity of state $\sigma_{\pm}$
to the ebit is

$\displaystyle F$
$\displaystyle=\text{Tr}(\sigma_{\pm}|\Phi^{+}\rangle\langle\Phi^{+}|)=\frac{(1+p)^{2}}{2+2p^{2}}.$
(S11)

The probability of Alice and Bob arriving at state $\sigma_{\pm}$ is

$\displaystyle
p_{\text{succ}}=p_{00}+p_{11}=\frac{1+p^{2}}{4}+\frac{1+p^{2}}{4}=\frac{1+p^{2}}{2}.$
(S12)

$\sqcap$$\sqcup$

With LOCCNet, we are able to find a new protocol that achieves a higher
fidelity than the DEJMPS protocol when distilling two copies of an S state.
Indeed, we show in the main text that this protocol is optimal in the sense
that it achieves the highest possible fidelity. With some simplification, we
obtain a circuit shown in Fig. S6. Below, we offer analysis on the performance
of this optimized protocol in Proposition S2.

@C=1em @R=1.7em A_0 &
A_1 -1 R_y(θ)
B_0 1
B_1 -1 R_y(θ)

Figure S6: The simplified circuit of a distillation protocol learned by
LOCCNet for two copies of some S state, $\rho_{A_{0}B_{0}}$ and
$\rho_{A_{1}B_{1}}$. The rotation angles of both $R_{y}$ gates are
$\theta=\arccos(1-p)+\pi$, which depends on the parameter $p$ of the S states
to be distilled.

###### Proposition S2

For two copies of an S state with parameter $p$, the protocol illustrated in
Fig. S6 outputs a state whose fidelity to the ebit is

$\displaystyle F=\frac{1+\sqrt{2p-p^{2}}}{2}$ (S13)

with probability $p_{\text{succ}}=p^{2}-\frac{p^{3}}{2}$ of success.

###### Proof.

The matrix form of an S state $\rho$ with parameter $p$ is given in Equation
(S8). Applying the circuit in Fig. S6 to two copies of such an state, Alice
and Bob both get $0$ for measurement outcomes with a probability of
$p_{00}=p^{2}-p^{3}/2$. By matrix calculation, we obtain the post-measurement
state of the unmeasured copy as

$\displaystyle\sigma=\begin{pmatrix}\frac{1}{2}&0&0&\frac{\sqrt{2p-p^{2}}}{2}\\\
0&0&0&0\\\ 0&0&0&0\\\ \frac{\sqrt{2p-p^{2}}}{2}&0&0&\frac{1}{2}\\\
\end{pmatrix}.$ (S14)

Note that the state $\sigma$ can be written as

$\displaystyle\sigma=\begin{pmatrix}\alpha+\beta&0&0&\alpha-\beta\\\
0&0&0&0\\\ 0&0&0&0\\\ \alpha-\beta&0&0&\alpha+\beta\\\
\end{pmatrix}=\alpha|\Phi^{+}\rangle\\!\langle\Phi^{+}|+\beta|\Phi^{-}\rangle\\!\langle\Phi^{-}|,$
(S15)

where $\alpha=(1+\sqrt{2p-p^{2}})/2$, $\beta=(1-\sqrt{2p-p^{2}})/2$, and
$|\Phi^{-}\rangle=1/\sqrt{2}(|00\rangle-|11\rangle)$. By the definition of
fidelity, we have

$\displaystyle F$
$\displaystyle=\text{Tr}(\sigma|\Phi^{+}\rangle\\!\langle\Phi^{+}|)=\text{Tr}((\alpha|\Phi^{+}\rangle\\!\langle\Phi^{+}|+\beta|\Phi^{-}\rangle\\!\langle\Phi^{-}|)|\Phi^{+}\rangle\\!\langle\Phi^{+}|)$
(S16) $\displaystyle=\alpha=\frac{1+\sqrt{2p-p^{2}}}{2}$ (S17)

since $\langle\Phi^{+}|\Phi^{+}\rangle=1$ and
$\langle\Phi^{-}|\Phi^{+}\rangle=0$ for $|\Phi^{-}\rangle$ is orthogonal to
$|\Phi^{+}\rangle$. The probability of Alice and Bob arriving at state
$\sigma$ is

$\displaystyle p_{\text{succ}}=p_{00}=p^{2}-\frac{p^{3}}{2}.$ (S18)

$\sqcap$$\sqcup$

Isotropic state. A two-qubit isotropic state is of the form

$\displaystyle\rho_{AB}=p|\Phi^{+}\rangle\langle\Phi^{+}|+(1-p)\frac{I}{4},$
(S19)

where $p\in[0,1]$. Alternatively, one can write an isotropic state as a Bell
diagonal state

$\displaystyle\rho_{AB}=\left(\frac{1+3p}{4},\frac{1-p}{4},\frac{1-p}{4},\frac{1-p}{4}\right).$
(S20)

Distillation with the DEJMPS protocol. The DEJMPS protocol is known to achieve
a high fidelity when distilling two copies of an Bell diagonal state, and the
resulting state in the event of success is still a Bell diagonal state.
Specifically, the DEJMPS protocol’s circuit, excluding the measurements, acts
on an Bell diagonal state as a permutation of the Bell states’ coefficients.
For a Bell diagonal state

$\displaystyle\rho=p_{0}|\Phi^{+}\rangle\\!\langle\Phi^{+}|+p_{1}|\Psi^{+}\rangle\\!\langle\Psi^{+}|+p_{2}|\Phi^{-}\rangle\\!\langle\Phi^{-}|+p_{3}|\Psi^{-}\rangle\\!\langle\Psi^{-}|,$
(S21)

the operator $R_{x}(\pi/2)\otimes R_{x}(-\pi/2)$ maps it to another Bell
diagonal state

$\displaystyle\sigma=p_{0}|\Phi^{+}\rangle\\!\langle\Phi^{+}|+p_{1}|\Psi^{+}\rangle\\!\langle\Psi^{+}|+p_{3}|\Phi^{-}\rangle\\!\langle\Phi^{-}|+p_{2}|\Psi^{-}\rangle\\!\langle\Psi^{-}|.$
(S22)

As stated in Eq. (S22), a pair of $R_{x}(\pm\pi/2)$ gates transforms a Bell
diagonal state $(p_{0},p_{1},p_{2},p_{3})$ to another Bell diagonal state
$(p_{0},p_{1},p_{3},p_{2})$. Similarly, a pair of bilateral CNOT gates shown
in Fig. S5 acts on the tensor product of two Bell diagonal states as a
permutation of coefficients. The effect of the bilateral CNOT gates is
summarized as a Table in Bennett1996 . Specifically, for a pair of Bell
diagonal states $(a_{0},a_{1},a_{2},a_{3})$ and $(b_{0},b_{1},b_{2},b_{3})$,
applying the bilateral CNOT gates on the state

$\displaystyle(p_{0},p_{1},\dots,p_{14},p_{15})$
$\displaystyle=(a_{0},a_{1},a_{2},a_{3})\otimes(b_{0},b_{1},b_{2},b_{3})$
(S23) $\displaystyle\equiv(a_{0}b_{0},a_{0}b_{1},\dots,a_{3}b_{2},a_{3}b_{3})$
(S24)

results in a state

$\displaystyle\text{CNOT}(p_{0},p_{1},\dots,p_{14},p_{15})=(p_{0},p_{1},p_{10},p_{11},p_{5},p_{4},p_{15},p_{14},p_{8},p_{9},p_{2},p_{3},p_{13},p_{12},p_{7},p_{6}).$
(S25)

Although the coincidence measurement, referring to Alice and Bob getting
identical measurement outcomes, on a Bell diagonal state is not a Bell basis
permutation, the post-measurement state is still a Bell diagonal state. To be
specific, note that since only $00$ and $11$ are counted as valid results,
Bell states $|\Psi^{\pm}\rangle\\!\langle\Psi^{\pm}|$ are filtered out and
thus a Bell diagonal state $(p_{0},p_{1},p_{2},p_{3})$ collapses to
$(p_{0},0,p_{2},0)$ up to a normalization factor after the coincidence
measurement.

The final fidelity and the probability of success achieved by the DEJMPS
protocol can be derived by permuting coefficients in the Bell basis, and the
results is given in deutsch1996quantum . For self-consistency, we give a
derivation as below.

######  Proposition S3 (deutsch1996quantum )

For two copies of a Bell diagonal state $(a_{0},a_{1},a_{2},a_{3})$, the
DEJMPS protocol outputs a state whose fidelity to the ebit is

$\displaystyle F=\frac{a_{0}^{2}+a_{3}^{2}}{p_{\text{succ}}},$ (S26)

where
$p_{\text{succ}}=a_{0}^{2}+a_{3}^{2}+a_{1}^{2}+a_{2}^{2}+2a_{0}a_{3}+2a_{1}a_{2}$
is the probability of success.

###### Proof.

After the first layer of $R_{x}$ gates, the input state
$(a_{0},a_{1},a_{2},a_{3})^{\otimes 2}$ becomes
$(a_{0},a_{1},a_{3},a_{2})^{\otimes 2}$ according to Eq. (S22). Then,
transformed by the layer of bilateral CNOT gates, the state
$(p_{0},p_{1},\dots,p_{14},p_{15})=(a_{0},a_{1},a_{3},a_{2})^{\otimes 2}$
becomes
$(p_{0},p_{1},p_{10},p_{11},p_{5},p_{4},p_{15},p_{14},p_{8},p_{9},p_{2},p_{3},p_{13},p_{12},p_{7},p_{6})$.
The coincidence measurement in the computational basis on the second copy
filters out $|\Psi^{\pm}\rangle\\!\langle\Psi^{\pm}|$ and the remaining state
is either $|00\rangle\\!\langle 00|$ or $|11\rangle\\!\langle 11|$. In either
case, the first copy becomes

$\displaystyle\sigma$
$\displaystyle=(p_{0}+p_{10},p_{5}+p_{15},p_{8}+p_{2},p_{13}+p_{7})$ (S27)
$\displaystyle=(a_{0}^{2}+a_{3}^{2},a_{1}^{2}+a_{2}^{2},a_{3}a_{0}+a_{0}a_{3},a_{2}a_{1}+a_{1}a_{2})$
(S28)
$\displaystyle=(a_{0}^{2}+a_{3}^{2},a_{1}^{2}+a_{2}^{2},2a_{0}a_{3},2a_{1}a_{2})$
(S29)

up to a normalization factor. The sum of all the unnormalized coefficients is
the probability of measuring $00$ or $11$, i.e.,

$\displaystyle
p_{\text{succ}}=a_{0}^{2}+a_{3}^{2}+a_{1}^{2}+a_{2}^{2}+2a_{0}a_{3}+2a_{1}a_{2}.$
(S30)

The the normalized output state is $\sigma/p_{\text{succ}}$, and its fidelity
to the ebit, which is the coefficient before
$|\Phi^{+}\rangle\\!\langle\Phi^{+}|$, is

$\displaystyle F=\frac{a_{0}^{2}+a_{3}^{2}}{p_{\text{succ}}}.$ (S31)

$\sqcap$$\sqcup$

Since the DEJMPS protocol is for distilling two copies of a Bell diagonal
state, to distill four copies of an isotropic state, we can follow these
steps. First, we divide them into two groups where each group consists of two
copies. Then we apply the DEJMPS protocol to both groups independently. In the
event of success, we will get two copies of a Bell diagonal state, to which we
apply the DEJMPS protocol again.

###### Proposition S4

For four copies of an isotropic state $((1+3p)/4,(1-p)/4,(1-p)/4,(1-p)/4)$,
the generalized DEJMPS protocol given above outputs a state whose fidelity to
the ebit is

$\displaystyle F=\frac{1+10p^{2}+8p^{3}+13p^{4}}{4+8p^{2}+20p^{4}}$ (S32)

with $p_{\text{succ}}=\frac{1}{8}\left(1+2p^{2}+5p^{4}\right)$ probability of
success.

###### Proof.

For two copies of the isotropic state to be distilled, the DEJMPS protocol
outputs a state

$\displaystyle\rho$
$\displaystyle=\left(\frac{(1+3p)^{2}+(1-p)^{2}}{16p^{\prime}_{\text{succ}}},\frac{(1-p)^{2}+(1-p)^{2}}{16p^{\prime}_{\text{succ}}},\frac{2(1+3p)(1-p)}{16p^{\prime}_{\text{succ}}},\frac{2(1-p)(1-p)}{16p^{\prime}_{\text{succ}}}\right)$
(S33)
$\displaystyle=\left(\frac{1+2p+5p^{2}}{8p^{\prime}_{\text{succ}}},\frac{1-2p+p^{2}}{8p^{\prime}_{\text{succ}}},\frac{1+2p-3p^{2}}{8p^{\prime}_{\text{succ}}},\frac{1-2p+p^{2}}{8p^{\prime}_{\text{succ}}}\right)$
(S34)

up to a normalization factor with a probability of success

$\displaystyle p^{\prime}_{\text{succ}}$
$\displaystyle=\frac{1}{16}\left((1+3p)^{2}+3(1-p)^{2}+2(1+3p)(1-p)+2(1-p)^{2}\right)=\frac{1+p^{2}}{2}.$
(S35)

Then, the probability of successful distillation for both groups is
$p_{\text{succ}}^{{}^{\prime}2}$. In that case, applying the DEJMPS protocol
to the resulting two copies of state $\rho$ gives a state whose fidelity to
the ebit is

$\displaystyle F$
$\displaystyle=\frac{(1+2p+5p^{2})^{2}+(1-2p+p^{2})^{2}}{64p_{\text{succ}}^{{}^{\prime}2}p^{\prime\prime}_{\text{succ}}}=\frac{1+10p^{2}+8p^{3}+13p^{4}}{8(1+p^{2})^{2}p^{\prime\prime}_{\text{succ}}}$
(S36)

with a probability of success

$\displaystyle
p^{\prime\prime}_{\text{succ}}=\frac{1+2p^{2}+5p^{4}}{2(1+p^{2})^{2}}.$ (S37)

Substituting $p^{\prime\prime}_{\text{succ}}$ into $F$, we have

$\displaystyle F=\frac{1+10p^{2}+8p^{3}+13p^{4}}{4+8p^{2}+20p^{4}}.$ (S38)

The sucess probability of the whole process is

$\displaystyle
p_{\text{succ}}=p_{\text{succ}}^{{}^{\prime}2}p^{\prime\prime}_{\text{succ}}=\frac{1}{8}\left(1+2p^{2}+5p^{4}\right).$
(S39)

$\sqcap$$\sqcup$

The protocol found with LOCCNet. As we show in the main text, the DEJMPS
protocol does not fully exploit the resources encoded in four copies of an
isotropic states, and there is a protocol learned by LOCCNet that achieves a
higher fidelity.

@C=1em @R=1.7em A_0 & 1
A_1 1 R_x(+π2)
A_2 1 R_x(+π2)
A_3 -3 R_x(+π2)

Figure S7: The simplified circuit of a protocol learned by LOCCNet for
entanglement distillation with four copies of some isotropic state. This
circuit only includes Alice’s operation, while Bob’s operation is identical to
Alice’s, except that the rotation angles of Bob’s $R_{x}$ gates are $-\pi/2$.

###### Proposition S5

For four copies of an isotropic state $\rho$ with parameter $p$, the protocol
illustrated in Fig. S7 outputs a state whose fidelity to the ebit is

$\displaystyle F=\frac{1-2p+9p^{2}}{4-8p+12p^{2}}$ (S40)

with a probability of success

$\displaystyle p_{\text{succ}}=\frac{1+4p^{3}+3p^{4}}{8}.$ (S41)

###### Proof.

Similar to the DEJMPS protocol, this optimized protocol consists of
$Ry(\pm\pi/2)$ gates, bilateral CNOT gates, and coincidence measurements in
the computational basis. Thus, the claimed fidelity and probability of success
can be derived by simulating the circuit shown in Fig. S7 as permutation on
Bell basis. Using similar techniques from the proof of Proposition S3, we
obtain the unnormalized state after three coincidence measurements in the
event of success, which is

$\displaystyle\sigma=\left(\frac{1}{32}(1+p)^{2}(1-2p+9p^{2}),\frac{1}{32}(1-p^{2})^{2},\frac{1}{32}(1-p^{2})^{2},\frac{1}{32}(1-p^{2})^{2}\right).$
(S42)

Then, adding up all the coefficients in $\sigma$, we obtain the probability of
success

$\displaystyle p_{\text{succ}}$
$\displaystyle=\frac{1}{32}(1+p)^{2}(1-2p+9p^{2})+\frac{3}{32}(1-p^{2})^{2}=\frac{1+4p^{3}+3p^{4}}{8}.$
(S43)

Meanwhile, the normalized $\sigma$’s fidelity to the ebit is

$\displaystyle F$
$\displaystyle=\frac{1}{32p_{\text{succ}}}(1+p)^{2}(1-2p+9p^{2})=\frac{1-2p+9p^{2}}{4-8p+12p^{2}}.$
(S44)

$\sqcap$$\sqcup$

PPT bound. As the mathematical structure of LOCC is complex and difficult to
characterize Chitambar2014 , we may consider larger but mathematically more
tractable classes of operations. The operations most frequently employed
beyond LOCC are the PPT operations, which completely preserve the positivity
of the partial transpose Rains2001 . A bipartite quantum operation $\Pi_{AB\to
A^{\prime}B^{\prime}}$ is called a PPT operation if its Choi-Jamiołkowski
matrix $J_{\Pi}=\sum_{i,j,m,k}|i_{A}j_{B}\rangle\langle
m_{A}k_{B}|\otimes\Pi(|i_{A}j_{B}\rangle\langle m_{A}k_{B}|)$ is positive
under partial transpose across the bipartition of
$AA^{\prime}\mathrel{\mathop{\mathchar 58\relax}}BB^{\prime}$, where
$\\{|i_{A}\rangle\\}$ and $\\{|j_{B}\rangle\\}$ are orthonormal bases on
Hilbert spaces $A$ and $B$, respectively.

The entanglement theory under PPT operations has been extensively studied in
the literature (e.g., Audenaert2003 ; Wang2016d ; Matthews2008 ; Wang2020c ;
Regula2019 ; Chitambar2017 ) and offers the limitations of LOCC. In
particular, the limit of finite-copy entanglement distillation was recently
explored in Fang2017 ; rozpkedek2018optimizing ; Regula2019 . In the
following, we compare our results with the PPT bound from
rozpkedek2018optimizing , which gives the fundamental limits on the fidelity
of distillation with given success probability. To be specific, the maximal
fidelity of distilling $D$-dimensional Bell state from $\rho$ with fixed
success probability $\delta$ using PPT operations rozpkedek2018optimizing  is
given by

$\begin{array}[]{ll}\operatorname{maximize}&\frac{d_{A}d_{B}}{\delta}\operatorname{Tr}\rho_{AB}^{T}M_{AB}\\\
\text{ subject to }&M_{AB}\geqslant 0,\quad E_{AB}\geqslant 0,\\\
&M_{AB}+E_{AB}\leqslant\frac{\mathbb{I}_{AB}}{d_{A}d_{B}},\\\
&M_{AB}^{T_{B}}+E_{AB}^{T_{B}}\leqslant\frac{\mathbb{I}_{AB}}{d_{A}d_{B}},d_{A}d_{B}\operatorname{Tr}\left[\rho_{AB}^{T}\left(M_{AB}+E_{AB}\right)\right]=\delta,\\\
&M_{AB}^{T_{B}}+\frac{1}{D+1}E_{AB}^{T_{B}}\geqslant
0,-M_{AB}^{T_{B}}+\frac{1}{D-1}E_{AB}^{T_{B}}\geqslant 0,\end{array}$ (S45)

where $d_{A},d_{B}$ are the dimensions of systems $A$ and $B$, respectively.
Recall that $\rho_{AB}$ is the initial input state that Alice and Bob are
attempting to distill and in most examples considered here, it will consist of
two copies of some two-qubit state.

## SUPPLEMENTARY NOTE 3: Analysis of LOCC state discrimination

To explore the power of LOCCNet in state discrimination, we focus on the
optimal success probability of discriminating noiseless and noisy Bell states
via LOCC. In the following, we present the LOCC protocol from Walgate2000  for
discriminating two Bells states. After that, we show how to distinguish one
Bell state from one noisy Bell state using the protocol learned via LOCCNet
and compare it with the protocol of the noiseless case.

Noiseless case. Consider two Bell states, $|\Phi^{+}\rangle$ and
$|\Phi^{-}\rangle$. Since these two states are pure and orthogonal to each
other, there exist an LOCC protocol that can perfectly distinguish between
them Walgate2000 . Here, we give a specific discrimination protocol for these
two Bell states. Suppose Alice and Bob share a two-qubit state $\rho_{AB}$,
which could be either $|\Phi^{+}\rangle\\!\langle\Phi^{+}|$ or
$|\Phi^{-}\rangle\\!\langle\Phi^{-}|$. To find out which state it is through
LOCC, they can follow the steps below. First, Alice applies a $R_{y}(\pi/2)$
gate on her qubit followed by a measurement. Then, Alice tell Bob her
measurement outcome through classical communication. Receiving the measurement
outcome from Alice, Bob applies on his qubit a $R_{y}$ gate with the rotation
angle $\theta$ being $\pi/2$ or $-\pi/2$, corresponding to the case where the
communicated measurement outcome is $0$ or $1$, respectively. Finally, Bob
measures his qubit. If he gets $0$, then he can be sure that the state
$\rho_{AB}$ is $|\Phi^{+}\rangle\\!\langle\Phi^{+}|$. Otherwise,
$\rho_{AB}=|\Phi^{-}\rangle\\!\langle\Phi^{-}|$. The whole process is also
illustrated with a circuit shown in Fig. S8, which can perfectly discriminate
$|\Phi^{+}\rangle$ and $|\Phi^{-}\rangle$.

@C=1em @R=1.7em A & R_y(π2) [1]
B R_y(θ)

Figure S8: The LOCC protocol distinguishing between the pair of noiseless Bell
states $|\Phi^{+}\rangle$ and $|\Phi^{-}\rangle$. The rotation angle $\theta$
of Bob’s $R_{y}$ gate is either $\pi/2$ or $-\pi/2$, depending on Alice’s
measurement outcome.

Noisy case. Quantum noises unavoidably may occur in quantum information
processing. One common noise of theoretical and experimental interest is the
amplitude damping channel Nielsen2010 , which is one of the realistic sources
of noise in superconducting quantum processor Chirolli2018 . To be specific,
an amplitude damping (AD) channel $\mathcal{A}$ with noise parameter $\gamma$
such that $\mathcal{A}(\rho)=E_{0}\rho E_{0}^{\dagger}+E_{1}\rho
E_{1}^{\dagger}$ with $E_{0}=|0\rangle\langle
0|+\sqrt{1-\gamma}|1\rangle\langle 1|$ and
$E_{1}=\sqrt{\gamma}|0\rangle\langle 1|$. If $|\Phi^{-}\rangle$ is affected by
the amplitude damping noise on each qubit, then the resulting state is
$\mathcal{A}\otimes\mathcal{A}(|\Phi^{-}\rangle\langle\Phi^{-}|)$. The goal is
now to distinguish between $\Phi_{0}\equiv|\Phi^{+}\rangle\langle\Phi^{+}|$
and
$\Phi_{1}\equiv\mathcal{A}\otimes\mathcal{A}(|\Phi^{-}\rangle\langle\Phi^{-}|)$.

PPT bound. The distinguishability of quantum states under PPT (positive
partial transpose) POVMs was introduced in Yu2014a  to better understand the
fundamental limits of the local distinguishability of quantum states. To be
specific, the PPT POVM used for distinguishing a set of $n$ orthogonal quantum
states $\\{\rho_{1},\dots,\rho_{n}\\}$ can be defined as an $n$-tuple of
operators, $(M_{k})_{k=1,\dots,n}$, where $M_{k}$ is PPT for $k=1,\dots,n$ and
$\sum_{k=1}^{n}M_{k}=I_{AB}$. The set of PPT POVMs enjoys a more tractable
mathematical structure than the LOCC POVMs due to the SDP characterization of
PPT condition.

The optimal success probability of discriminating a collection of quantum
states $\\{\rho_{1},\cdots,\rho_{K}\\}$ using PPT POVMs is given by

$\begin{split}p_{s}(\rho_{1},\cdots,\rho_{K})=\max\
&\frac{1}{K}\operatorname{Tr}\sum_{j=1}^{K}\rho_{j}\\\ \text{s.t.}\
&\sum_{j=1}^{K}M_{j}=I,0\leq M_{j}\leq I,\forall j=1,2,\cdots,K,\\\
&M_{j}^{T_{B}}\geq 0,\forall j=1,2,\cdots,K.\end{split}$ (S46)

where we assume that each state in this collection has a equal probability of
appearance. As LOCC POVMs is a proper subset of PPT POVMs, the above SDP gives
the upper bound to the optimal success probability of discriminating a
collection of quantum states.

Optimized LOCC protocol. While the PPT bound serves as an upper bound to the
success probability of LOCC discrimination, an optimal LOCC protocol may not
necessarily reach the bound. Here, we present an LOCC protocol optimized by
LOCCNet that achieves a success probability close to the PPT bound.

The only difference between this optimized protocol and the protocol for
noiseless discrimination is that the rotation angle $\theta$ of Bob’s $R_{y}$
gate is not fixed in the noisy case, as shown in Fig. S9. Specifically, for
Alice’s measurement outcome being $0$,

$\displaystyle\theta=\pi-\arctan\left(\frac{2-\gamma}{\gamma}\right),$ (S47)

where $\gamma$ is the noise parameter of the AD channel $\mathcal{A}$ that
$|\Phi^{-}\rangle\\!\langle\Phi^{-}|$ goes through. For Alice’s measurement
outcome being $1$,

$\displaystyle\theta=-\pi+\arctan\left(\frac{2-\gamma}{\gamma}\right).$ (S48)

@C=1em @R=1.7em A & R_y(π2) [1]
B R_y(θ)

Figure S9: The optimized LOCC protocol for distinguishing between states
$\Phi_{0}$ and $\Phi_{1}$. Depending on Alice’s measurement outcome, the
rotation angle $\theta$ of Bob’s $R_{y}$ gate is either $\pi-\arctan(\alpha)$
or $\arctan(\alpha)-\pi$, where $\alpha=(2-\gamma)/\gamma$ and $\gamma$ is the
noise parameter of the AD channel.

###### Proposition S6

For states $\Phi_{0}=|\Phi^{+}\rangle\\!\langle\Phi^{+}|$ and
$\Phi_{1}=\mathcal{A}\otimes\mathcal{A}(|\Phi^{-}\rangle\\!\langle\Phi^{-}|)$,
the optimized protocol illustrated in Fig. S9 discriminates between them with
an average probability of

$\displaystyle
p_{\text{succ}}=\frac{1}{2}+\frac{\sqrt{2-2\gamma+\gamma^{2}}}{2\sqrt{2}}.$
(S49)

###### Proof.

We consider this protocol in two cases.

Case 1. The input state is $\Phi_{0}=|\Phi^{+}\rangle\\!\langle\Phi^{+}|$. For
this case, after Alice applies $R_{y}(\pi/2)$ to her qubit, state
$|\Phi^{+}\rangle$ becomes

$\displaystyle R_{y}\left(\frac{\pi}{2}\right)\otimes
I|\Phi^{+}\rangle=\frac{1}{2}|0\rangle\otimes(|0\rangle-|1\rangle)+\frac{1}{2}|1\rangle\otimes(|0\rangle+|1\rangle).$
(S50)

Measuring the first qubit of this state, Alice has $50\%$ chance of getting
$0$ with the normalized post-measurement state being
$|\psi_{0}\rangle=1/\sqrt{2}|0\rangle\otimes(|0\rangle-|1\rangle)$ and $50\%$
chance of getting $1$ with state
$|\psi_{1}\rangle=1/\sqrt{2}|1\rangle\otimes(|0\rangle+|1\rangle)$. The
resulting states of the second qubit after Bob’s operation are given below,
where $\theta_{0}=\pi-\arctan((2-\gamma)/\gamma)$ and
$\theta_{1}=-\pi+\arctan((2-\gamma)/\gamma)$.

$\displaystyle R_{y}(\theta_{0})\frac{1}{\sqrt{2}}(|0\rangle-|1\rangle)$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\left(\cos\frac{\theta_{0}}{2}+\sin\frac{\theta_{0}}{2}\right)|0\rangle-\left(\cos\frac{\theta_{0}}{2}-\sin\frac{\theta_{0}}{2}\right)|1\rangle\right);$
(S51) $\displaystyle R_{y}(\theta_{1})\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\left(\cos\frac{\theta_{1}}{2}-\sin\frac{\theta_{1}}{2}\right)|0\rangle+\left(\cos\frac{\theta_{1}}{2}+\sin\frac{\theta_{1}}{2}\right)|1\rangle\right).$
(S52)

Then, the probability of Bob’s measurement outcome being $0$ given $\Phi_{0}$
as the input state is

$\displaystyle P(0|\Phi_{0})$
$\displaystyle=\frac{1}{2}\left|\frac{1}{\sqrt{2}}\left(\cos\frac{\theta_{0}}{2}+\sin\frac{\theta_{0}}{2}\right)\right|^{2}+\frac{1}{2}\left|\frac{1}{\sqrt{2}}\left(\cos\frac{\theta_{1}}{2}-\sin\frac{\theta_{1}}{2}\right)\right|^{2}=\frac{1}{2}+\frac{\sin\theta_{0}-\sin\theta_{1}}{4}.$
(S53)

Since $\theta_{1}=-\theta_{0}$, we have $\sin\theta_{1}=-\sin\theta_{0}$ and

$\displaystyle P(0|\Phi_{0})$
$\displaystyle=\frac{1}{2}+\frac{\sin\theta_{0}+\sin\theta_{0}}{4}=\frac{1+\sin\theta_{0}}{2}.$
(S54)

Let $\alpha\equiv(2-\gamma)/\gamma$. Then, $\theta_{0}=\pi-\arctan\alpha$ and

$\displaystyle P(0|\Phi_{0})$
$\displaystyle=\frac{1+\sin(\pi-\arctan\alpha)}{2}=\frac{1+\sin(\arctan\alpha)}{2}=\frac{1}{2}+\frac{\alpha}{2\sqrt{1+\alpha^{2}}}.$
(S55)

Case 2. The input state is
$\Phi_{1}=\mathcal{A}\otimes\mathcal{A}(|\Phi^{-}\rangle\\!\langle\Phi^{-}|)$.
If the noise parameter of the AD channel $\mathcal{A}$ is $\gamma$, then the
input state in matrix form is

$\displaystyle\Phi_{0}=\frac{1}{2}\begin{pmatrix}1+\gamma^{2}&0&0&\gamma-1\\\
0&\gamma-\gamma^{2}&0&0\\\ 0&0&\gamma-\gamma^{2}&0\\\
\gamma-1&0&0&(1-\gamma)^{2}\end{pmatrix}.$ (S56)

After Alice applies $R_{y}(\pi/2)$ to her qubit, the input state becomes

$\displaystyle R_{y}\left(\frac{\pi}{2}\right)\otimes
I\Phi_{0}R_{y}^{\dagger}\left(\frac{\pi}{2}\right)\otimes
I=\frac{1}{4}\begin{pmatrix}1+\gamma&1-\gamma&1-\gamma+2\gamma^{2}&\gamma-1\\\
1-\gamma&1-\gamma&1-\gamma&-1+3\gamma-2\gamma^{2}\\\
1-\gamma+2\gamma^{2}&1-\gamma&1+\gamma&\gamma-1\\\
\gamma-1&-1+3\gamma-2\gamma^{2}&\gamma-1&1-\gamma\end{pmatrix}.$ (S57)

From this state, we can see that when Alice measures her qubit, she has $50\%$
chance of getting $0$ with this state collapsing to

$\displaystyle\rho_{0}=|0\rangle\\!\langle
0|\otimes\frac{1}{2}\begin{pmatrix}1+\gamma&1-\gamma\\\
1-\gamma&1-\gamma\end{pmatrix}.$ (S58)

If Alice gets $1$ instead, this state collapses to

$\displaystyle\rho_{1}=|1\rangle\\!\langle
1|\otimes\frac{1}{2}\begin{pmatrix}1+\gamma&\gamma-1\\\
\gamma-1&1-\gamma\end{pmatrix}.$ (S59)

The resulting states of the second qubit after Bob’s operation are given
below, where $\theta_{0}=\pi-\arctan((2-\gamma)/\gamma)$ and
$\theta_{1}=-\pi+\arctan((2-\gamma)/\gamma)$.

$\displaystyle R_{y}(\theta_{0})\frac{1}{2}\begin{pmatrix}1+\gamma&1-\gamma\\\
1-\gamma&1-\gamma\end{pmatrix}R_{y}^{\dagger}(\theta_{0})$
$\displaystyle=\frac{1}{2}\begin{pmatrix}1+\gamma\cos\theta_{0}+(\gamma-1)\sin\theta_{0}&(1-\gamma)\cos\theta_{0}+\gamma\sin\theta_{0}\\\
(1-\gamma)\cos\theta_{0}+\gamma\sin\theta_{0}&1-\gamma\cos\theta_{0}+(1-\gamma)\sin\theta_{0}\end{pmatrix};$
(S60) $\displaystyle
R_{y}(\theta_{1})\frac{1}{2}\begin{pmatrix}1+\gamma&\gamma-1\\\
\gamma-1&1-\gamma\end{pmatrix}R_{y}^{\dagger}(\theta_{1})$
$\displaystyle=\frac{1}{2}\begin{pmatrix}1+\gamma\cos\theta_{1}+(1-\gamma)\sin\theta_{1}&(\gamma-1)\cos\theta_{1}+\gamma\sin\theta_{1}\\\
(\gamma-1)\cos\theta_{1}+\gamma\sin\theta_{1}&1-\gamma\cos\theta_{1}+(\gamma-1)\sin\theta_{1}\end{pmatrix}.$
(S61)

Then, the probability of Bob’s measurement outcome being $1$ given $\Phi_{1}$
as the input state is

$\displaystyle P(1|\Phi_{1})$
$\displaystyle=\frac{1}{2}\cdot\frac{1-\gamma\cos\theta_{0}+(1-\gamma)\sin\theta_{0}}{2}+\frac{1}{2}\cdot\frac{1-\gamma\cos\theta_{1}+(\gamma-1)\sin\theta_{1}}{2}$
(S62)
$\displaystyle=\frac{2-\gamma(\cos\theta_{0}+\cos\theta_{1})+(1-\gamma)(\sin\theta_{0}-\sin\theta_{1})}{4}.$
(S63)

Since $\theta_{0}=\theta_{1}$, we have $\sin\theta_{1}=-\sin\theta_{0}$,
$\cos\theta_{1}=\cos\theta_{0}$, and thus

$\displaystyle P(1|\Phi_{1})$
$\displaystyle=\frac{2-\gamma(\cos\theta_{0}+\cos\theta_{0})+(1-\gamma)(\sin\theta_{0}+\sin\theta_{0})}{4}$
(S64)
$\displaystyle=\frac{1-\gamma\cos\theta_{0}+(1-\gamma)\sin\theta_{0}}{2}.$
(S65)

Let $\alpha\equiv(2-\gamma)/\gamma$. Then $\theta_{0}=\pi-\arctan\alpha$ and

$\displaystyle
P(1|\Phi_{1})=\frac{1}{2}+\frac{\gamma+(1-\gamma)\alpha}{2\sqrt{1+\alpha^{2}}}.$
(S66)

Combining these two cases, we can obtain this optimized protocol’s average
probability of success as

$\displaystyle p_{\text{succ}}$
$\displaystyle=\frac{1}{2}P(0|\Phi_{0})+\frac{1}{2}P(1|\Phi_{1})=\frac{1}{2}\left(\frac{1}{2}+\frac{\alpha}{2\sqrt{1+\alpha^{2}}}+\frac{1}{2}+\frac{\gamma+(1-\gamma)\alpha}{2\sqrt{1+\alpha^{2}}}\right)$
(S67)
$\displaystyle=\frac{1}{2}\left(1+\frac{\gamma+(2-\gamma)\alpha}{2\sqrt{1+\alpha^{2}}}\right)=\frac{1}{2}+\frac{\sqrt{2-2\gamma+\gamma^{2}}}{2\sqrt{2}}.$
(S68)

$\sqcap$$\sqcup$