Machine Learning Bell Nonlocality in Quantum Many-body Systems

Dong-Ling DengCondensed Matter Theory Center and Joint Quantum Institute,

Department of Physics, University of Maryland, College Park, MD 20742-4111, USA

Machine learning, the core of artificial intelligence and big data science, is one of today’s most rapidly grow-ing interdisciplinary fields. Recently, its tools and techniques have been adopted to tackle intricate quantummany-body problems. In this work, we introduce machine learning techniques to the detection of quantumnonlocality in many-body systems, with a focus on the restricted-Boltzmann-machine (RBM) architecture. Us-ing reinforcement learning, we demonstrate that RBM is capable of finding the maximum quantum violationsof multipartite Bell inequalities with given measurement settings. Our results build a novel bridge betweencomputer-science-based machine learning and quantum many-body nonlocality, which will benefit future stud-ies in both areas.

Nonlocality is one of the most fascinating and enigmaticfeatures of quantum mechanics that denies any local realisticdescription of our world [1, 2]. It represents the most profounddeparture of quantum from classical physics and has been ex-perimentally confirmed in a number of systems through vi-olations of Bell inequalities [3–18]. In addition to its fun-damental interest, in practice nonlocality is the key resourcefor device-independent quantum technologies, such as securekey distribution [19–21] or certifiable random number gener-ators [22–26]. Thus, characterizing and detecting nonlocalityis one of the central problems in both quantum informationtheory and experiment. Here, we introduce machine learn-ing, a branch of computer science [27–29], to the detection ofquantum nonlocality (see Fig. 1 for a pictorial illustration).

For quantum many-body systems, whereas entanglementhas been extensively studied [30], nonlocality remains rarelyexplored. Mathematically, it has been proved that the com-plete characterization of classical correlations for a genericmany-body system is an NP-hard problem [31]. Neverthe-less, an incomplete list of multipartite Bell inequalities withhigh-order correlation functions has indeed been discoveredfor a long time [2]. More recently, Bell inequalities with onlytwo-body correlators were constructed [32–36] and multipar-tite nonlocality has been demonstrated experimentally in aBose-Einstein condensate by violating one of these inequal-ities [37]. This sparks a new wave of interest in the study ofnonlocality in many-body systems.

A particular question of both theoretical and experimen-tal relevance is that for a given multipartite Bell inequality,how to obtain its quantum violation in a numerical simula-tion? To tackle this problem, one has to face at least two chal-lenges. First of all, the Hilbert space of a quantum many-bodysystem grows exponentially with the system size and a com-plete description of its state requires an exponential amountof information in general, rendering the computation of thequantum expectation value corresponding to the inequality aformidably demanding task. Second, the measurement set-tings for each party involved in a Bell experiment is arbitraryin principle, making the problem even more complicated. Infact, it has been shown that the computation of the maximumviolation of a multipartite Bell inequality is an NP-problem[38]. In this paper, we will not attempt to solve this problem

RBM-based reinforcement learning

Classical correlations

Quantum correlations

Tight Bell inequality

Nonlocal point

VisibleLayer

HiddenLayer

⋯

⋯

"#$

ℎ#

(a)

(b)

FIG. 1. (a) A sketch of the restricted-Boltzmann-machine (RBM)representation of quantum many-body states. For each spin config-uration Ξ = (σz1 , σ

z2 , · · · , σzN ), the artificial neural network out-

puts its corresponding coefficient Φ(Ξ). (b) A pictorial illustrationof the essential idea of machine learning Bell nonlocality in quan-tum many-body systems. The set of all classical correlations forms ahigh-dimensional polytope (yellow region), which is a subset of thequantum-correlation set that consists of all possible correlations al-lowed by quantum mechanics. The black line represents a tight Bellinequality (facet of the polytope). We start with a random RBM,which typically shows only classical correlations (in the sense that itdoes not violate a given Bell inequality). We then optimize its inter-nal parameters, through reinforcement learning, so as to violate theBell inequality maximally.

completely, which is implausible due to the NP complexity.Instead, we study a simplified scenario where the given mul-tipartite Bell inequality only involves a polynomial number ofcorrelation functions and the measurement settings for eachparty are restricted (due to experimental requirements, for in-stance) and preassigned. We show that machine learning mayprovide an unprecedented perspective for solving this simpli-

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fied, but still sufficiently intricate, quantum many-body prob-lem. Within physics, applications of machine-learning tech-niques have recently been invoked in various contexts [39–72], such as black hole detection [56], gravitational lenses[57] and wave analysis [58, 59], material design [60], glassydynamics [61], Monte Carlo simulation [62, 63], topologicalcodes [64], quantum machine learning [72], and topologicalphases and phase transitions [41–50], etc. Here, we apply ma-chine learning to the detection of quantum many-body nonlo-cality, focusing on one of the simplest stochastic neural net-works for unsupervised learning—the restricted Boltzmannmachine (RBM) [73–75] as an example. We demonstrate,through three concrete examples, that RBM-based reinforce-ment learning is capable of finding the maximum quantum vi-olations of multipartite Bell inequalities with given measure-ment settings. Our method works for generic Bell inequalitiesthat involve a polynomial number (in system size) of correla-tion functions, independent of dimensionality, the order of thecorrelations, or whether the correlation functions are short-range or not. Our results showcase the exceptional power ofmachine learning in the detection of quantum nonlocality formany-body systems, thus would provide a valuable guide forboth theory and experiment.

To begin with, let us first briefly introduce the RBM repre-sentation of quantum states [39] and the general recipe for ma-chine learning Bell nonlocality. We consider a quantum sys-tem withN spin- 12 particles (qubits) Ξ = (σ1, σ2, · · ·σN ) anduse a RBM neural network to describe its many-body wave-function Φ(Ξ). A RBM consists of two layers: one calledvisible layer withN nodes (visible neurons), corresponding tothe physical spins; the other called hidden layer withM auxil-iary nodes (hidden neurons). The hidden neurons are coupledto the visible ones, but there is no coupling among neurons inthe same layer, as schematically illustrated in Fig. 1(a). Bytracing out the hidden neurons, we obtain a RBM representa-tion of a quantum many-body state [39]:

ΦM (Ξ,Ω) =∑hk

e∑k akσ

zk+

∑k′ bk′hk′+

∑kk′ Wk′khk′σ

zk , (1)

where Ω ≡ (a, b,W ) are internal parameters that fully spec-ify the RBM neural network and hk = −1, 1M denotesthe possible hidden neuron configurations. We mention thatany quantum state can be approximated to arbitrary accuracyby the above RBM representation, as long as the number ofhidden neurons is large enough [76–78]. It is shown in Ref.[44] that RBM can represent topological states, either sym-metry protected or with intrinsic topological order, in an exactand efficient fashion, and the entanglement properties of RBMstates are extensively studied in Ref [79].

We consider a standard Bell experiment in which N partieseach can freely choose to perform one of K possible mea-surements M(i)

k (i = 1, · · · , N and k = 0, · · · ,K − 1)with binary outcomes ±1. We describe the observed corre-lations by using a collection of expectation values of corre-lators 〈M(i1)

k1· · ·M(iα)

kα〉 and we say that the correlations are

classical when they can be simulated with only shared clas-sical information between parties (or in other words, can bedescribed by a local hidden variable theory [80]). Classicalcorrelations form a high-dimensional (exponential inN ) poly-tope P, which is a bounded convex set with a finite number ofextreme points. Each facet of P corresponds to a tight Bellinequality and correlations that fall outside of P will violatea Bell inequality and thus manifest nonlocality. We write theBell inequalities in a generic form: I ≥ B(c), where I is afunction of the expectation values of the correlators and B(c)is the classical bound. Within this framework, our generalrecipe for machine learning of nonlocality through violationof a given Bell inequality is as follows: we begin with a ran-dom RBM state, whose observed correlations may or may notfall inside P, but typically do not violate the given inequality;we then use a reinforcement learning scheme recently intro-duced by Carleo and Troyer [39] to iteratively optimize the in-ternal parameters, such that the minimal expectation value ofI within quantum mechanics will be achieved. If the minimalvalue is smaller than B(c), the Bell inequality is maximallyviolated with a given measurement setting and nonlocality isdetected. A pictorial illustration of the classical polytope, atight Bell inequality, and the essential idea of machine learn-ing Bell nonlocality is shown in Fig. 1(b).

One may also choose another measurement setting and runthe same process to obtain the maximal violation for this set-ting. In order to obtain the maximal violation of the Bell in-equality for all measurement settings, one can just scan allpossible settings and do the same process repeatedly. We men-tion that an alternative and more efficient way is to regard allthe parameters that specify the measurements as variationalparameters as well (on an equal footing as the RBM parame-ters Ω) and optimize them together with the RBM parametersusing a similar reinforcement learning procedure. But this ismore technically involved. Here, we will only focus on theformer cases with fixed measurement settings (parameters formeasurements are preassigned) for simplicity and leave thelater approach for future studies.

To show more precisely how this RBM-based reinforce-ment learning protocol works, we give three concrete exam-ples. The first one concerns Bell inequalities with only short-range two-body correlators in one dimension (1D). This is acase where traditional methods, such as density-matrix renor-malization group (DMRG), also work remarkably well [81–83]. We compare our RBM results with that from exact di-agonalization (ED) for small system size N and DMRG forlarger N , and find that they agree excellently. This validatesthe effectiveness of our RBM approach. The second and thirdexamples are about Bell inequalities with, either all-to-all buttwo-body or multipartite, correlators. These examples are be-yond the capacity of the DMRG or ED methods for large sys-tem sizes and show a striking advantage of RBMs in detectingmany-body nonlocality.

Bell inequalities with short-range two-body correla-tors.—Now, let us consider a 1D system with N (an eveninteger) qubits. A Bell inequality involving only two-body

3

(a) (b) (c)(1

) (1) (1

)

FIG. 2. RBM-based reinforcement learning of many-body Bell nonlocality through quantum violations of the Ineq. (2). The red dashedlines represent the classical bounds B(c)

1 , the regions below which show quantum nonlocality and thus are not attainable by any local hiddenvariable models. We denote the quantum expectation value corresponding to I1 asQ(1)

v , and without loss of generality, we have fixed δ = 0.9throughout this figure for simplicity. (a) A comparison between results from RBM and exact diagonalization (ED) for a small system sizeN = 20. The results match each other very well. (b) The obtained quantum expectation value as a function of the iteration number of thelearning process, for a larger system size N = 100. For this particular learning process,Q(1)

v begins to cross the classical bound B(c)1 after 65

iterations, and all the RBM states thereafter violate Ineq. (2) and thus show many-body nonlocality. As the iteration number increases, Q(1)v

converges quickly to the value computed from DMRG [84]. (c) RBM learned Q(1)v as a function of N for ∆ = 2. The inset shows the result

for ∆ = 3, where no quantum violation is observed.

correlators with nearest-neighbor couplings has recently beenobtained through a dynamic programming procedure [34]:

I1 =

N/2−1∑k=0

(1 + δ)I(k)even + (1− δ)I(k)odd ≥ B(c)1 , (2)

where I(k)even =∑4a=0

∑3b=0 Λa,b(∆)〈M(2k)

a M(2k+1)b 〉 and

I(k)odd = I(k)even(2k → 2k + 1) with Λ(∆) a four-by-three ma-trix [85]; B(c)1 is the classical bound depending on the realparameters δ and ∆: B(c)1 = −(4 + 2|∆|)N for |∆| ≤ 2 and|δ| ≤ 1, and B(c)1 = −4|∆|N for 2 ≤ |∆| ≤ 3 and |δ| ≤ 1.By choosing the measurement settings properly [86], the Belloperator corresponding to the above inequality reduces to thefollowing XXZ-type Hamiltonian:

H =

N−1∑k=0

gk(δ)[σxk σxk+1 + σyk σ

yk+1 + ∆σzkσ

zk+1],

where gk(δ) = 4[1 + (−1)k]/√

3, and σx, σy and σz are theusual Pauli matrices. For this particular measurement settings,the maximal quantum violation of Ineq. (2) corresponds to theground state energy ofH and can be calculated using DMRG,as already discussed in Ref. [34]. Here, we use the aboveintroduced reinforcement learning method to obtain the samequantum violation.

Our results are plotted in Fig. 2. In Fig. 2(a), we compareour results with that from ED for N = 20. As shown in thisfigure, the RBM result matches the ED result very well [84].We find that the quantum expectation value of I1 (denotedby Q(1)

v ) decreases approximately linearly as we increase ∆.

There is a critical value ∆ ≈ 2.4, after which no quantum vio-lation will be observed. In Fig. 2(b), we show the convergenceof the RBM learning and compare the obtained results withthat of DMRG. We find that the initial random RBM statestypically do not violate the Ineq. 2, but as the learning processgoes on,Q(1)

v will decrease and begin to violate the inequalityafter a certain critical iteration number. As the iteration num-ber increases further, Q(1)

v quickly converges to the DMRGvalue, validating the effectiveness of the RBM method. Fig.2(c) shows the converged Q(1)

v as a function of N . We findthat Q(1)

v decreases linearly with increasing N for the chosenparameters (δ,∆) = (0.9, 2). For ∆ = 2, the slope forQ(1)

v issmaller than that of B(c)1 , thus the larger N the stronger quan-tum violations. For ∆ = 3, no violation is observed for all N ,which is consistent with the results in Ref. [34].

Bell inequalities with all-to-all two-body correlators.—Asthe second example to show the power of RBM in detectingnonlocality, we consider the following Bell inequality intro-duced by Tura et al [32], which involves all-to-all two-bodycorrelators and thus are beyond the scope of DMRG:

I2 = −2S0 − S01 +1

2(S00 + S11) ≥ B(c)2 , (3)

where the one- and two- body correlators are defined as: Sa =∑Nk=1〈M

(k)a 〉 and Sab =

∑Nk 6=l〈M

(k)a M(l)

b 〉 (a, b = 0, 1),

and the classical bound B(c)2 = −2N . This inequality hasbeen used in a recent experiment to demonstrate many-bodynonlocality of about 480 atoms in a Bose-Einstein condensate[37]. For permutationally-symmetric states, its quantum vio-lations were numerically studied in Ref. [32]. Here, we find

4

(b)(a)(2) (3)

FIG. 3. (a) RBM-learned quantum expectation value (Q(2)v ) as a

function of the iteration number, for a typical random sample of mea-surement angles (see [84]). (b) RBM-learned quantum violations(Q(3)

v ) of Ineq. (4) as a function of measurement angle θ. In both (a)and (b), the system size is fixed to be N = 20.

that, using the RBM approach, one can obtain the same max-imal violations readily if one chooses a permutation-invariantneural network. More interestingly, we find that the RBM ap-proach also works for the cases where the permutation sym-metry is released. To this end, we consider a scenario wherethe measurement settings are chosen as: M(k)

0 = σz andM(k)

1 = cos θkσz + sin θkσ

x (k = 1, 2, · · · , N ), where θkare random rotation angles drawn from some uniform distri-butions [84]. We mention that in a real experiment, the mea-surement angles will never be exact due to various control im-perfections or system noises. For instance, in quantum dotspin-qubit experiments, the precision of single qubit rotationsis typically limited due to charge fluctuations [87] and Over-hauser noise [88, 89]. Thus our consideration of random mea-surement settings is of both theoretical and experimental rel-evance. In Fig. 3(a), we show the quantum expectation valuecorresponding to I2 (denoted as Q(2)

v ) as a function of the it-eration number of the learning process for a typical randomsample of θks. It is clear that Q(2)

v decreases as the learn-ing process continues and becomes smaller than the classicalbound at a critical iteration number. It converges to the exactminimal value as we increase the iteration number further.

Bell inequalities with multipartite correlators.—To showthat RBM is also capable of dealing with Bell inequalitieswith multipartite correlators, we consider the following Bellinequality introduced in Ref. [90]:

I3 = −〈M(1)0 M

(2)0 · · ·M

(N)0 〉 − 〈M(1)

1 M(2)0 · · ·M

(N)0 〉

+1

N − 1

N∑k=2

[〈M(1)0 M

(k)1 〉 − 〈M

(1)1 M

(k)1 〉] ≥ −2. (4)

This inequality contains only two dichotomic measurementsper party, hence in order to find its maximal quantum viola-tion it is sufficient to consider only traceless real observables[91, 92]. As a result, we consider the following choice of

measurements: M(1)0 = σz, M(1)

1 = cos θσx + sin θσz ,M(k)

0 = σz and M(k)1 = σx for all k = 2, · · ·N . By us-

ing RBM-based reinforcement learning, we have computedthe quantum violations of Ineq. (4) and part of our resultsare plotted in Fig. 3(b) [84]. From this figure, we find thatthe Ineq. (4) is always violated when θ 6= π/2 and the max-imal violation is achieved at θ = 0 or π. When θ = π/2,M(1)

0 =M(1)1 = σz and the first party actually has only one

measurement hence no quantum violation can be obtained. Inaddition, from our numerical results we also find that the max-imal violation of Ineq. (4) is always−2

√2, independent of the

system size [84]. This can be understood from the observationthat the Ineq. (4) is in fact reminiscent of the Clauser-Horne-Shimony-Holt inequality [93], whose maximal quantum vio-lation has proved to be bounded by 2

√2 [94].

We emphasize that in the last two examples, we did notspecify the spatial dimensionality of the systems. UnlikeDMRG, our RBM approach works for any dimension. In ad-dition, as shown in Ref. [79], entanglement is not a limitingfactor for the efficiency of the neural-network representationof quantum many-body states. Thus, we expect that RBM canbe used to detect many-body nonlocality for quantum stateswith massive (e.g., volume-law) entanglement as well. Thisimplies another unparalleled advantage of the RBM approach,when compared with traditional methods, such as DMRG,PEPS [95] (projected entangled pair states), or MERA [96](multiscale entanglement renormalization ansatz). We alsonote that one may use other type of neural networks (e.g., deepBoltzmann machine [65] or feedforward neural networks [97],etc.) with different learning algorithms to detect many-bodynonlocality. A complete study on detecting nonlocality withdifferent neural network would not only bring new powerfultools for solving intricate problems in the quantum informa-tion area, but also provide helpful insight on understanding theinternal data structures of the networks themselves. We leavethis interesting and important topic for future investigation.

Discussion and conclusion.—Finding out experimentally-friendly Bell inequalities for a given many-body system isa challenging problem, since in general the complexity ofcharacterizing the set of classical correlations scales exponen-tially with the system size. In the future, it would also beinteresting to study how machine learning can provide valu-able ideas in designing optimal Bell inequalities for many-body systems. Particularly, recent experiments in cold atomic[98] and trapped ion [99] systems have realized programmablequantum simulators with more than fifty qubits and observedexotic quantum dynamics and phases transitions. It is highlydesirable to find appropriate Bell inequalities that can be usedin these experiments to demonstrate many-body nonlocality.We believe that machine learning will provide valuable wis-dom in tackling this problem as well.

In summary, we have introduced machine learning to thedetection of quantum nonlocality in many-body systems. Ourdiscussion is mainly based on the RBM architecture, but itsgeneralizations to other artificial neural networks are possible

5

and straightforward. Through three concrete examples, wehave demonstrated that RBM-based reinforcement learningshows remarkable power in computing quantum violations ofcertain multipartite Bell inequalities. Our results open a doorfor machine learning Bell nonlocality, which would benefitfuture studies across quantum information, machine learning,and artificial intelligence.

D. L. D acknowledges S. Das Sarma, Lu-Ming Duan, Xi-aopeng Li, Jing-Ling Chen and C. H. Oh for previous collab-orations on related works. D. L. D. thanks Shengtao Wangfor helpful discussions. This work is supported by Labo-ratory for Physical Sciences and Microsoft. We acknowl-edge the University of Maryland supercomputing resources(http://www.it.umd.edu/hpcc) made available in conductingthe research reported in this paper.

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0 =

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whereas on the odd sites, the measurement areM(2k+1)0 = σx,

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8

Supplementary Material for: MachineLearning Bell Nonlocality in Quantum

Many-body Systems

I. BELL INEQUALITIES WITH SHORT-RANGETWO-BODY CORRELATORS

In the main text, we have shown that one can use the RBM-based reinforcement learning to obtain the quantum violationsof Ineq. (2). Here, we give more details on the structure of theneural networks, the DMRG calculations, and the comparisonbetween obtained results from different methods.

As discussed in the main text, with properly chosen mea-surement settings, the Bell operator corresponding to I1 re-duces to the XXZ-type Hamiltonian H [34]. With the openboundary condition, H does not have other obvious symme-try, except that the total Σz =

∑σzk is conserved. Thus,

it is straightforward to choose a RBM without any symme-try. In this case, the number of variational parameters isN + M + N ×M , which is large when N ≈ 100 (∼ 104).Training this RBM is both time and memory consuming. Forthis particular example, we find that one can instead use ashort-range RBM to reduce the number of parameters, andthe accuracy of the final results will not be affected to much.To be more concrete, we consider a RBM with M = αN (αdenotes the hidden unit density and we choose it to be an inte-ger number for simplicity) and we rearrange the positions ofthe hidden neurons, such that at each site there are α hiddenneurons coupling only locally (within range R) to the visibleneurons. This significantly reduces the number of parameters,from O(N2) to O(N).

We begin with a random short-range RBM (i.e., all the in-ternal parameters are chosen randomly and independently),which typically does not violate the Ineq. (1) in the main text.We then use a reinforcement learning algorithm introduced inRef. [39] to optimize the internal parameters of the RBM.The details of this algorithm can be found in the Supplemen-tary Materials of Ref. [39]. In Fig. S1, we plot partial of theweight parameters for the final trained RBM in Fig. 2(b) inthe main text. Here, only the wight parameters correspondingto the neurons in the first hidden layer are plotted. The param-eters associated to other hidden layers looks similar and thusare omitted for the sake of conciseness.

In order to characterize the accuracy of the trained RBM,one can introduced a quantity called the relative error definedas εrel = |E(RBM)

0 − E0|/E0, where E0 is the true ground-state energy of H and E(RBM)

0 denotes the value calculatedvia the RBM approach. For small system sizes (N ≤ 20), wefind that εrel ∼ 10−4 in our calculations. For larger systemsizes, we compare our RBM results with that from DMRG.In our DMRG calculations, we use a MPS representation ofthe quantum many-body states and variationally optimizes theMPS to minimize the ground state energy (see [82] for de-tails). The maximal bond dimension (where we truncate theMPS) is chosen to be χmax = 100 and we have checked

0 20 40 60 80 100k

-0.2

0

0.2

0.4

0.6

Im(W

(1)

k',k)

0 20 40 60 80 100k

-0.2

0

0.2

Re(W

k'k

(1) )

Wk-2,k

Wk-1,k

Wkk

Wk+1,k

Wk+2,k

(a)

(b)

FIG. S1. The learned weight parameters for representing the groundstate ofH with a short-range restricted Boltzmann machine (here, wehave fixed α = 4 andR = 2). We denote byW (η)

k′k (η = 1, 2, · · · , α,k′, k = 1, 2, · · · , N ) the coupling strength between the η-th hiddenneuron at site k′ and the visible neuron at sit k. (a) and (b) show thereal and imaginary parts of W (1)

k′k, respectively. They share the samelegend. The parameters specifying H are chosen the same as in Fig.2(b) in the main text.

that the neglected weight for all the truncations are smallerthan 10−6. We have also examined that the typical variancesσ2 = 〈H2〉−〈H〉2 is smaller than 10−8, verifying that the ob-tained MPS is indeed an eigenstate ofH (up to a negligible er-ror rate). In comparing our RBM and DMRG results, we findthat the relative error εrel = |E(RBM)

0 − E(DMRG)0 |/E(DMRG)

0 ∼10−3. We mention that the accuracy of the RBM results can besystematically improved by increasingR and α, or the numberof iterations in the training process. In this work of detectingmany-body nonlocality via RBM, high accuracy is not a majorconcern, hence ε ∼ 10−3 is already sufficient for our purpose.

II. BELL INEQUALITIES WITH ALL-TO-ALLTWO-BODY CORRELATORS

The Bell inequality in Ineq. (3) in the main text has apermutation symmetry by construction [32]. Thus, its cor-responding Bell operator will also has a permutation sym-metry if we choose the same measurement settings for eachparty. In this case, it is natural to use a permutation-invariantRBM to calculate the quantum violations. This greatly re-duces the number of the variational parameters and the samequantum violations as given in Ref. [32] can be readily ob-tained. What might be more interesting is the case in which

9

0 200 400 600 800 1000

Iteration number

-3

-2

-1

0

Q(3)

v

θ=0

θ=π/12

θ=π/6

θ=π/4

θ=π/3

θ=5π/12

θ=π/2

FIG. S2. RBM-learned quantum expectation valueQ(3)v as a function

of the iteration number, for different measurement angle θ.

2 4 6 8 10 12

θ

-2.8

-2.6

-2.4

-2.2

-2

Q(3)

v

N=10

N=15

N=20

FIG. S3. RBM-learned quantum violations Q(3)v as a function of θ,

for different system sizes.

different party choose different measurement settings. In thiscase, the permutation symmetry is violated and it is very chal-lenging to compute the quantum violations of Ineq. (3). Inthe main text, we have considered a scenario where the mea-

surement setting for each party is random. More specifically,we have chosen the measurement settings to be: M(k)

0 = σz

andM(k)1 = cos θkσ

z + sin θkσx with θk being random rota-

tion angles drawn independently from a uniform distribution[θ − ε, θ + ε]. Since there is no obvious symmetry for thecorresponding Bell operator and the correlators are all-to-all,we choose the most general RBM with each hidden neuronconnected to all the visible ones. In plotting Fig. 3(a) in themain text, we have fixed θ = 2π/3 and ε = 0.1.

III. BELL INEQUALITIES WITH MULTIPARTITECORRELATORS

For the Bell inequality (4) in the main text, it is easy toobserve that there is a permutation symmetry between par-ties indexed from 2 to N . In addition, the considered mea-surement settings also have this symmetry, and thus so doesthe corresponding Bell operator. Taking this into considera-tion, we choose a RBM with the same symmetry: the RBMcontains M hidden neurons and each of them connects to allvisible neurons, but the baise and weight parameters satisfya2 = a3 = · · · = aN and Wk′,1 = Wk′,2 = · · · = Wk′,N

(for k′ = 2, 3, · · · ,M ), respectively. This reduces the num-ber of parameters from O(MN) to O(M) and significantlysimplified the calculations.

In Fig. S2, we show the RBM-learnedQ(3)v as a function of

the iteration number for different measurement angle θ. It isclear from this figure that Q(3)

v converges rapidly to the cor-responding exact values for different θ. We note that in ourcalculations we have chosen the learning rate to be an expo-nential decaying function of the iteration number, followingRef. [39]. Thus, at the beginning of the learning process, thelearning rate is large. This explains the large fluctuations atthe beginning of the learning process. As the iteration num-ber increases, the learning rate becomes small and the curvesbecome smooth.

In Fig. S3, we plot the RBM-learned quantum violationsQ(3)v as a function of the measurement angle θ for different

system sizes. We find that the quantum violations are inde-pendent of N (up to negligible numerical errors).