Certifying Unknown Genuine Multipartite Entanglement by Neural Networks Zhenyu Chen1 Xiaodie Lin1 and Zhaohui Wei23 1Institute for Interdisciplinary Information Sciences Tsinghua University Beijing 100084 China

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Certifying Unknown Genuine Multipartite Entanglement by Neural Networks

Zhenyu Chen1, Xiaodie Lin1, and Zhaohui Wei2,3,∗

1Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing 100084, China

2Yau Mathematical Sciences Center, Tsinghua University, Beijing 100084, China

3Yanqi Lake Beijing Institute of Mathematical Sciences and Applications, 101407, China

Suppose we have an unknown multipartite quantum state, how can we experimentally ﬁnd out

whether it is genuine multipartite entangled or not? Recall that even for a bipartite quantum

state whose density matrix is known, it is already NP-Hard to determine whether it is entangled or

not. Therefore, it is hard to eﬃciently solve the above problem generally. However, since genuine

multipartite entanglement is such a fundamental concept that plays a crucial role in many-body

physics and quantum information processing tasks, ﬁnding realistic approaches to certify genuine

multipartite entanglement is undoubtedly necessary. In this work, we show that neural networks can

provide a nice solution to this problem, where measurement statistics data produced by measuring

involved quantum states with local measurement devices serve as input features of neural networks.

By testing our models on many speciﬁc multipartite quantum states, we show that they can certify

genuine multipartite entanglement very accurately, which even include some new results unknown

before. We also exhibit a possible way to improve the eﬃciency of our models by reducing the size of

features. Lastly, we show that our models enjoy remarkable robustness against ﬂaws in measurement

devices, implying that they are very experiment-friendly.

I. INTRODUCTION

Quantum entanglement plays a central role in many

quantum information processing tasks, including quan-

tum communication [1], quantum cryptography [2] and

quantum key distribution [3]. As a consequence, certi-

fying the existence of quantum entanglement is a funda-

mental problem.

However, it has been proved that even for a bipar-

tite quantum state that the density matrix is completely

known, to determine whether it is entangled or not is al-

ready NP-Hard [4], implying that it is hard to generally

solve this problem eﬃciently. Nevertheless, due to its im-

portance, numerous approaches have been put forward to

certify bipartite entanglement [5–9].

Meanwhile, we often face the situation that the target

bipartite quantum state we would like to characterize is

unknown to us, i.e., its density matrix is not available,

making the task even tougher. For this, one may ﬁrst

reconstruct the density matrix by quantum state tomog-

raphy [10–12], and then try to solve the problem accord-

ingly. However, it is well-known that this procedure is ex-

tremely expensive and can only be implementable when

the size of the target quantum state is small. To overcome

this diﬃculty, some realistic alternative approaches can

be utilized, which include entanglement witness [13–15]

and device-independent schemes [16–19]. But they also

suﬀer from other drawbacks, say being sensitive to op-

eration errors or being easy to fail in providing valuable

outcomes.

When it comes to multipartite quantum states, the

problem becomes even more complicated, as the mathe-

matical structures of multipartite quantum entanglement

∗Email: weizhaohui@gmail.com

are much richer than the bipartite case. Particularly,

as a speical form of multipartite entanglement that is

highly valuable, genuine multipartite entanglement has

signiﬁcant applications in quantum teleportation [20, 21],

quantum state sharing [22], quantum metrology [23, 24],

and even chemical and biological processes [25]. To cer-

tify it, many methods have been proposed. For example,

when the full information of density matrices are known,

quite a lot of mathematical criteria that can detect mul-

tipartite genuine entanglement have been proposed [26–

32].

Similar to the bipartite case, we also have to han-

dle the case that full information of target multipartite

quantum states is not available, which is actually an ex-

tremely common and realistic problem from the view-

point of engineering. For this, known methods such as

quantum state tomography, entanglement witness, and

device-independent schemes have been applied to certify

genuine multipartite entanglement [17, 33–37]. However,

facing similar diﬃculties as in the bipartite case, it is not

hard to understand that these approaches cannot provide

satisfying solutions for this task. As a result, due to its

central role in quantum computing and quantum engi-

neerings, ﬁnding realistic approaches to certify unknown

genuine multipartite entanglement is a challenging yet

urgent task, which is also the main motivation of the

current paper.

Recently, machine learning approaches have been em-

ployed to characterize quantum properties [38–45]. Dif-

ferent from analytic methods, machine learning is a data-

driven approach which aims at making predictions on

unseen data by learning from training data. In these

works, following standard procedures of machine learn-

ing tasks, certain features are extracted from training

quantum states, which are then fed into machine learn-

ing models. Then we train these models by adjusting the

parameters they contain such that the models can make

arXiv:2210.13837v2 [quant-ph] 15 Nov 2022

predictions on training quantum states with high accu-

racy, and if the model details are chosen properly, they

can also make accurate predictions on target quantum

states unseen before. Here, based on the chosen forms

of features, diﬀerent machine learning models can be de-

signed to detect quantum properties.

In this work, we exploit the possibility that utilizes

neural networks to certify genuine multipartite entangle-

ment for general quantum states, where the features we

choose is measurement statistics data produced by mea-

suring involved quantum states with local measurement

devices. Here each subsystem of involved multipartite

quantum states is measured by at least two devices. In-

spired by Bell experiments, we believe that this kind of

measurement statistics data can reveal nontrivial quan-

tum properties for target quantum states.

It turns out that our idea works very well. Particularly,

we successfully train a series of neural network models

that can certify unknown genuine multipartite entangle-

ment very accurately, where the target quantum states

are quite diverse. Taking 4-qubit quantum states for ex-

ample, we ﬁrst train a proper model, and then we run the

same trained model on four diﬀerent classes of 4-qubit

quantum states, on each of which our model successfully

certify genuine 4-qubit entanglement with accuracy over

99%. Interestingly, our model even reports some new re-

sults that are unknown before, indicating that machine

learning models can be highly valuable in such a chal-

lenging task. We conﬁrm the high performance of neu-

ral networks on many other test quantum states, which

include quantum state sets that are sampled randomly

without any assumptions.

Meanwhile, we also proposed a modiﬁed scheme called

k-correlation to reduce the cost of our approach, and

we show that in some cases where certain speciﬁc prior

knowledge is available, the cost of our approach can be

sharply reduced while the prediction accuracy is still

comparable.

Lastly, we provide evidence showing that our approach

enjoys remarkable robustness against ﬂaws in measure-

ment devices, which implies that our approach is very

experiment-friendly.

II. DEEP LEARNING AND ENTANGLEMENT

STRUCTURE

In this work, the certiﬁcation of genuine multipartite

entanglement is formulated as a supervised binary classi-

ﬁcation task, where the deep learning method is applied.

Deep learning is a powerful machine learning model based

on artiﬁcial neural networks. It has great representative

ability and has been widely utilized in a variety of ﬁelds

such as image recognition [46], natural language process-

ing [47], recommendation systems [48] and so on.

For our task we apply fully connected neural net-

works (FNNs) to ﬁt the training set, for more details

one can see Refs.[49, 50]. Following the standard pro-

cedure of machine learning, we need to gather a train-

ing dataset and a test dataset, which have the form of

{(x1, y1),...,(xN, yN)}, where Nis the size of the set,

xiis the feature of the i-th sample, and yiis the label.

The labels of training dataset are known to us, and the

mission of the neural network is to predict the labels for

the test dataset. When training the model, we input the

features of training dataset into the model and adjusting

its parameters such that it can produce correct labels for

the training dataset. For this, a proper loss function, a

reasonable conﬁguration for the nerual network, and an

eﬃcient optimization method such as gradient-descent

have to be chosen. If the model is trained properly, it

can predict accurately the labels of test dataset unseen

before.

In this work we apply the deep learning method to

certify genuine multipartite entanglement, which is also

a typical binary-classiﬁcation task. In general, a multi-

partite quantum state can involves many subsystems and

thus its entanglement structure can be very complicated.

An n-partite pure quantum state |Ψk−sepiis called k-

separable, where 1 ≤k≤n, if and only if it can be

written as a tensor product of ksubstates:

|Ψk−sepi=|Ψ1i⊗|Ψ2i⊗···⊗|Ψki.(1)

Correspondingly, a mixed state is called k-separable, if

and only if it has a decomposition into k-separable pure

states. A multipartite quantum state is called genuinely

multipartite entangled if it can not be written as k-

separable for k= 2, otherwise we call it biseparable state.

In addition, a quantum state is said to be of entanglement

intactness k, if it is k-separable but not (k+1)-separable.

III. DETECTING GENUINE MULTIPARTITE

ENTANGLEMENT FOR QUBIT SYSTEMS

A. 3-qubit case

1. The setup

As the simplest case, we ﬁrst try to detect genuine

multipartite entanglement for 3-qubit quantum states.

As mentioned above, since our approach is based on a

neural network, we need to prepare a large number of

quantum states to train (and test) the neural network.

In this work, we always sample random d-dimensional

quantum state ρ∈ Hdaccording to spectral decomposi-

tion

ρ=

d−1

i=0

λi|uiihui|.(2)

Here we randomly choose nonnegative numbers λi’s such

that they satisfy Piλi= 1. Then we generate a d×

dHaar random unitary U, and set |uiito be the i-th

column of U, which means that {|uii} forms a set of

orthonormal basis for Hd. Particularly, if λi= 1 for

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CertifyingUnknownGenuineMultipartiteEntanglementbyNeuralNetworksZhenyuChen1,XiaodieLin1,andZhaohuiWei2;3;1InstituteforInterdisciplinaryInformationSciences,TsinghuaUniversity,Beijing100084,China2YauMathematicalSciencesCenter,TsinghuaUniversity,Beijing100084,China3YanqiLakeBeijingInstituteofMathemati...

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Certifying Unknown Genuine Multipartite Entanglement by Neural Networks Zhenyu Chen1 Xiaodie Lin1 and Zhaohui Wei23 1Institute for Interdisciplinary Information Sciences Tsinghua University Beijing 100084 China

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