Neural Network Approach to the Simulation of Entangled States with One
  Bit of Communication
Bell's theorem states that Local Hidden Variables (LHVs) cannot fully explain
the statistics of measurements on some entangled quantum states. It is natural
to ask how much supplementary classical communication would be needed to
simulate them. We study two long-standing open questions in this field with
neural network simulations and other tools. First, we present evidence that all
projective measurements on partially entangled pure two-qubit states require
only one bit of communication. We quantify the statistical distance between the
exact quantum behaviour and the product of the trained network, or of a
semianalytical model inspired by it. Second, while it is known on general
grounds (and obvious) that one bit of communication cannot eventually reproduce
all bipartite quantum correlation, explicit examples have proved evasive. Our
search failed to find one for several bipartite Bell scenarios with up to 5
inputs and 4 outputs, highlighting the power of one bit of communication in
reproducing quantum correlations.
Peter Sidajaya, Aloysius Dewen Lim, Baichu Yu, Valerio Scarani
# Neural Network Approach to the Simulation of Entangled States with One Bit of Communication

###### Abstract

Bell's theorem states that Local Hidden Variables (LHVs) cannot fully explain the statistics of measurements on some entangled quantum states. It is natural to ask how much supplementary classical communication would be needed to simulate them. We study two long-standing open questions in this field with neural network simulations and other tools. First, we present evidence that all projective measurements on partially entangled pure two-qubit states require only one bit of communication. We quantify the statistical distance between the exact quantum behaviour and the product of the trained network, or of a semianalytical model inspired by it. Second, while it is known on general grounds (and obvious) that one bit of communication cannot eventually reproduce all bipartite quantum correlation, explicit examples have proved evasive. Our search failed to find one for several bipartite Bell scenarios with up to 5 inputs and 4 outputs, highlighting the power of one bit of communication in reproducing quantum correlations.

## I Introduction

Quantum Mechanics is famous for having randomness inherent in its prediction. Einstein, Podolski and Rosen argued that this makes quantum mechanics incomplete, and suggested the existence of underlying Local Hidden Variables (LHVs) [1]. While this view was disproved by Bell's theorem [2; 3], it has nevertheless proved fruitful to approach quantum correlations, without committing to an ontology of the quantum world, by asking _which resources would one use to simulate them_. Though insufficient, LHVs provide an intuitive starting point - then, the question becomes: _which additional resources, on top of the LHVs, are needed to simulate quantum correlations?_. Some works have considered nonlocal boxes as supplementary resources [4; 5; 6]: while appealing for their intrinsic no-signaling feature, these hypothetical resources are as counterintuitive as entanglement itself, if not more. Classical communication, on the other hand, is a resource that we use on a daily basis and of which therefore we have developed an intuitive understanding. Because we are thinking in terms of simulations and not of ontology, we are not impaired by the very problematic fact that communication should be instantaneous if taken as the real underlying physical mechanism.

Therefore, we are interested in the question of how much classical communication must supplement LHVs to simulate the behaviour of a quantum state. For the maximally entangled state of two qubits, after some partial results [7; 8], Toner and Bacon provided a definitive solution by describing a protocol that simulates the statistics of all projective measurements using only one bit of communication, which we refer to as _LHV+1_[9]. Subsequently, Degorre and coworkers used a different approach and found another protocol which also requires only one bit of communication [10]. The case of non-maximally entangled pure states proved harder. By invoking the Toner-Bacon model, two bits of communication are certainly sufficient [9]; while Brunner and coworkers proved that one PR-box is not [5]. But the simulation of those states in LHV+1 remained open. Only recently, Renner and coworkers reported an LHV+1 protocol that simulates exactly weakly entangled pure states [11]. Our neural network will provide evidence that projective measurements on all two-qubit states can be very closely approximated in LHV+1.

The LHV+1 problem could, in principle, be approached systematically, since the behaviours that can be obtained with those resources are contained in a _polytope_. However, the size of this polytope grows very fast with the number of inputs and outputs: as of today, after some initial works [12; 13], the largest LHV+1 polytope to be completely characterized has three measurements per party and binary outcomes; and no quantum violation is found [14]. Addressing the problem for higher-dimensional systems has been challenging without going to the asymptotic limit. The only work we are aware of is of Vertesi and Bene, who showed that a pair of maximally entangled four-dimensional quantum systems cannot be simulated with only one bit of communication, by presenting a scenario involving an infinite number of measurements [15].

In recent years, there have also been increasing attempts to study quantum correlations with machine learning. Many of them reveal the great potential neural network has in tackling the complexities in detecting nonlocality and entanglement [16; 17; 18; 19; 20; 21]. The choice of tackling the LHV+1 problem with machine learning is prompted
by the fact that there is no compact parametrisation of LHVs, nor of the dependence of the bit of communication from the parameters of the problem. Thus, we are looking for a solution to a problem, whose variables are themselves poorly specified. Moreover, similar to an LHV model, everything inside a neural network has definite values. Thus, it seems natural to devise a machine learning tool, specifically an artificial neural network (ANN), to act as an LHV model.

This work is separated into two sections. In Section II, we study the simulability of the correlations of entangled state with classical resource and one bit of communication using neural network. We also present a semi-analytical protocol which _approximates_ the behaviour of partially entangled two-qubit states with one bit of communication, and we also study the errors of our protocol. In Section III, we also tried to find a quantum behaviour in dimensions higher than two qubits that could not be simulated by a single bit of communication.

## II Two-qubit entangled states using machine learning

### Using Neural Network to generate protocols

Inspired by the use of a neural network as an oracle of locality [19], we approached the problem using an artificial neural network. The network takes in measurement settings \(\hat{a}\) and \(\hat{b}\) as an input and outputs an LHV+1 bit probability distribution, enforced by an architecture that forces the suitable locality constraints, which we will discuss below. The output distribution is then compared against the target distribution using a suitable error function, which is the Kullback-Leibler divergence.

The Local Hidden Variables (LHV) are described by a random variable \(\lambda\) shared among both parties. \(\lambda\) can be of any form; the model of Toner and Bacon uses a pair of uniformly distributed Bloch vectors, the model by Renner [11] uses a biased distribution on the Bloch sphere, and the neural network of [19] only uses a single number, distributed normally or uniformly, as the LHV. In theory, the choice is ultimately redundant because the different LHV models can be made equivalent by some transformation. However, the neural network will perform differently since it can only process a certain amount of complexity in the model. From trial and error, we settled on Toner-Bacon's _uniformly distributed vector pair_ as the LHV model in our neural network.

A probability distribution \(P(A,B)\) is _local_ if it can be written as

\[P_{L}(A,B\mid\hat{a},\hat{b})=\int P(A\mid\hat{a},\lambda)\;P(B\mid\hat{b}, \lambda)\;d\lambda. \tag{1}\]

The network approximates a local distribution by the Monte Carlo method as

\[P_{L}(A,B\mid\hat{a},\hat{b})=\frac{1}{N}\sum_{i=1}^{N}P(A\mid\hat{a},\lambda_ {i})\;P(B\mid\hat{b},\lambda_{i}), \tag{2}\]

where \(N\) is a sufficiently large number (\(\geq 1000\)). In the network, Alice and Bob are represented as a series of hidden layers. Each of the parties takes in their inputs according to the locality constraint and outputs their own local probability distribution. The activation functions used in the hidden layers are the standard functions, such as the rectified linear unit (ReLU) and the softmax function used to normalise the probabilities. The forward propagation is done \(N\) times using varying values of \(\lambda_{i}\) sampled from a probability distribution. Thereafter we take the average of the probabilities over \(N\) to get the probability distribution as expressed in equation (2).

To move from LHV to LHV+1, we notice that sending one bit of communication is equivalent to giving Alice the power of making the decision to choose between one out of two local strategies. The recipe looks as follows:

* Alice and Bob pre-agreed on _two_ local strategies \(P_{L,1}\) and \(P_{L,2}\), as well as on the \(\lambda\) to be used in each round. It seems to us that all previous works in LHV+1 assumed \(P_{1}(A\mid\hat{a},\lambda)=P_{2}(A\mid\hat{a},\lambda)\), but of course there is no need to impose such a constraint.
* Upon receiving her input \(\hat{a}\), Alice decides which of the two strategies should be used for that round, taking also \(\lambda\) into account. Although all previous LHV+1 models used a deterministic decision, there is no reason to impose that: Alice's decision could be stochastic. She informs Bob of her choice with one bit of communication \(c\), and Bob consequently keeps his outcome for the chosen strategy.

Thus, given a randomly sampled LHV \(\lambda_{i}\), the LHV+1 model is described by

\[\begin{split} P(A,B\mid\hat{a},\hat{b},\lambda_{i})& =P(c=+1\mid\hat{a},\lambda_{i})P_{L,1}(A,B\mid\hat{a},\hat{b}, \lambda_{i})+P(c=-1\mid\hat{a},\lambda_{i})P_{L,2}(A,B\mid\hat{a},\hat{b}, \lambda_{i})\\ &=P(c=+1\mid\hat{a},\lambda_{i})P_{1}(A\mid\hat{a},\lambda_{i})P _{1}(B\mid\hat{b},\lambda_{i})+P(c=-1\mid\hat{a},\lambda_{i})P_{2}(A\mid\hat {a},\lambda_{i})P_{2}(B\mid\hat{b},\lambda_{i}).\end{split} \tag{3}\]

where we labeled \(c=+1\) (respectively \(c=-1\)) the value of the bit of communication when Alice decides for strategy 1 (resp. 2).

Now the complete model consists of two local networks
and one communication network. The communication network consists of a series of layers whose inputs are the same as Alice's and outputs a number between 0 and 1 by using a sigmoid activation function, representing \(P(c\mid\hat{a},\lambda_{i})\), which then is used to make a convex mixture of the two local strategies, for the particular inputs and LHV. The final network architecture, then, can be seen in Fig. 1.

This approach of using a neural network to generate local strategies was originally used in a network setting [19]. In that work, the network was used to verify non-locality by looking for transitions in the behaviours of distributions when mixed with noise. When a state is mixed with noise, it lies within a local set, up to a certain noise threshold; reducing the amount of noise in the state allows for the identification of sharp transitions in the network's error, indicating when the state exits the local set. Here, instead of such an oracle, we will use the network to generate a protocol to simulate the quantum state by analysing its outputs.

### Simulating Two-qubit States

For a two-qubit scenario, the joint measurements can be defined by two vectors in the Bloch sphere, i.e. \(\hat{a},\hat{b}\in S^{2}\). Thus, the behaviour is the set \(\mathcal{P}(\rho)=\{P_{\rho}(A,B\mid\hat{a},\hat{b})\mid\hat{a},\hat{b}\in S ^{2}\}\).

#### ii.2.1 Maximally Entangled State

The maximally entangled state case has been solved analytically by Toner and Bacon [9]. Thus, we used this state as a test bed for our machine learning approach by training the machine to simulate the distribution of the maximally entangled state \(|\Psi^{-}\rangle\). A snapshot of the behaviour of the trained model can be seen in Fig. 5. By scrutinising similar figures for different LHVs, we can infer an analytical model of the machine.

Maximally entangled state protocol:

1. Alice sends to Bob \[c=\text{sgn}(\hat{a}\cdot\hat{\lambda}_{1})\text{sgn}(\hat{a}\cdot\hat{ \lambda}_{2}).\]
2. Alice outputs \[A=-\text{sgn}(\hat{a}\cdot(\hat{\lambda}_{1}+c\hat{\lambda}_{2})).\]
3. Bob outputs \[B=\text{sgn}(\hat{b}\cdot(\hat{\lambda}_{1}+c\hat{\lambda}_{2})).\]

The protocol bears much resemblance to Toner-Bacon's original protocol, with the only difference being the output of Alice, which are simply \(-\text{sgn}(\hat{a}\cdot\hat{\lambda}_{1})\) in the original protocol. However, one can check [22] that it indeed fulfills all the correct expectation values.

#### ii.2.2 Non-maximally Entangled States

We now apply the same method to the non-maximally entangled two-qubit states. Without loss of generality,

Figure 1: The architecture of the Artificial Neural Network (ANN). The model consists of two local distributions and a communication network. In each distribution, the two parties are constrained by locality by routing the input accordingly. The communication network outputs a value between 0 and 1, and represents the probability of Alice sending a certain bit to Bob. The output for a particular round is then simply the convex combination of the two local distributions.
any pure two-qubit states can be written in the form of

\[\left|\psi(\theta)\right\rangle=\cos(\theta)\left|01\right\rangle-\sin(\theta) \left|10\right\rangle,\quad\theta\in\left[0,\frac{\pi}{4}\right]\]

using a suitable choice of bases. The state is maximally entangled when \(\theta=\frac{\pi}{4}\) and separable when \(\theta=0\). We train the network to simulate the distribution of \(\left|\psi(\theta)\right\rangle\) with \(\theta\in\left[0,\frac{\pi}{4}\right]\). A selection of the resulting protocol is shown in Fig. 6.

The error of the models for these states is lower than the one for the maximally entangled state (see Fig. 2). This does not necessarily mean that they successfully simulate the states exactly, instead of simply approximating them. If the behaviour were actually nonlocal, we should expect a transition in the error when we mix the state with noise, signifying the exit of the state from the local+1 bit set. However, we observe no clear transition occurring when noise is added to the state, only a shallow gradient, suggesting that it is still inside the local+1 bit set. While encouraging, this does not constitute a proof, and we would still need to write an analytical protocol. Unlike for the case of maximally entangled states, the models we obtained for the non-maximally entangled states are more complex and our attempt to infer a protocol begins by looking at figures similar to Fig. 6.

We start from the parties' outputs:

The outputs of Alice are of the form of

\[P(A_{1}=+1\mid\hat{a})=\frac{1}{2}(1-\operatorname{sgn}(\hat{a}\cdot\vec{ \lambda}_{a1}+b_{a1})),\]

where \(\hat{\lambda}_{a1}=u_{a1}\vec{\lambda}_{1}+\vec{\lambda_{2}}+v_{a1}\hat{z}\) decides the hemisphere direction and \(b_{a1}=w_{a1}+x_{a1}\vec{\lambda}_{1}\cdot\hat{z}+y_{a1}\vec{\lambda}_{2}\cdot \hat{z}\) decides the size of the hemisphere. Similarly,

\[P(A_{2}=+1\mid\hat{a})=\frac{1}{2}(1+\operatorname{sgn}(\hat{a}\cdot\vec{ \lambda}_{a2}+b_{a2})),\]

\[P(B_{1}=+1\mid\hat{b})=\frac{1}{2}(1+\operatorname{sgn}(\hat{b}\cdot\vec{ \lambda}_{b1}+b_{b1})),\]

\[P(B_{2}=+1\mid\hat{b})=\frac{1}{2}(1-\operatorname{sgn}(\hat{b}\cdot\vec{ \lambda}_{b2}+b_{b2})).\]

Using numerical algorithms, we can approximately obtain the relevant coefficients, laid out in Appendix A for the different states.

The (simplified) bit of communication is given by

\[P(c=+1\mid\hat{a})=\frac{1}{2}(1-\operatorname{clip}(f_{c},-1,1)),\]

where

\[f_{c} =\Theta(\hat{a}\cdot\vec{\lambda}_{1}+b_{c})\Theta(\hat{a}\cdot \vec{\lambda}_{2}+b_{c})\] \[+\Theta(-\hat{a}\cdot\vec{\lambda}_{1}+b_{c})\Theta(-\hat{a} \cdot\vec{\lambda}_{2}+b_{c})\] \[-\Theta(-\hat{a}\cdot\vec{\lambda}_{1}-b_{c})\Theta(\hat{a} \cdot\vec{\lambda}_{2}-b_{c})\] \[-\Theta(\hat{a}\cdot\vec{\lambda}_{1}-b_{c})\Theta(-\hat{a} \cdot\vec{\lambda}_{2}-b_{c}),\]

with \(b_{c}=u_{c}+v_{c}(\vec{\lambda}_{2}\cdot\hat{z})(1-\vec{\lambda}_{1}\cdot \hat{z})\) and the clip function is defined as

\[\operatorname{clip}(x,a,b)=\begin{cases}a&\text{if }x<a\\ b&\text{if }x>b\\ x&\text{otherwise}\end{cases}.\]

Again, the relevant coefficients obtained using numerical methods are listed in Appendix A.

Figure 2: The relative error between the neural network models’ behaviours and the quantum behaviours. The blue dots are for the original model described, while the red crosses are for the simplified model described in the text. The grey shaded region is the region in which an LHV+1 model is known [11].
### Statistical analysis of the simulations

After presenting our protocols, we can now consider the performance of our protocols, both the neural network protocol itself and the semianalytical protocol we distilled from it. These LHV+1 protocols are not exact protocols, but approximations, and we can describe their closeness to the quantum behaviour by providing statistical error values. To get a better intuition on the error values, let us consider a hypothesis testing scenario [23]. Suppose that we have an unknown sample of length \(n\) generated by the same measurement done to \(n\) identical systems. Suppose also that we know that the systems are all actual quantum systems (\(P_{Q}\)), or our LHV+1 models (\(P_{LHV+1}\)), but we do not know which. Let us take \(P_{LHV+1}\) as the null hypothesis. Let \(a\) be the Type I error (mistakenly rejecting a true null hypothesis). In our case, a Type I error would correspond to our machine learning model successfully spoofing as a quantum system. For any decision-making procedure, the probability of a Type I error is lower bounded by

\[a\geq e^{-nD_{KL}(P_{Q}||P_{LHV+1})}.\]

Thus, in order to have 95% confidence in rejecting a sample from the LHV+1 model, we would need a sample size of

\[n_{95\%}\geq-\frac{\ln 0.05}{D_{KL}(P_{Q}||P_{LHV+1})}.\]

The sample size \(n\) needed to distinguish the probability distributions differs with the measurement settings, with some measurement settings being more difficult to distinguish. The performance of our LHV+1 models (both the machine learning and our semi-analytical approximations) over the measurement settings are given in Fig. 3. It can be seen that from the neural network's protocols to our semianalytical approximations, we have gained about two orders of magnitude in Kullback-Leibler divergence. This is due to the limitations of our numerical methods used to obtain the optimum parameters, and the fact that we were bound to have missed some details from the behaviour of the network when we translated it into analytical expressions.

Our semianalytical protocols require, on average, hundreds of measurements before they can be distinguished from real quantum behaviours, disregarding other noises present in an actual quantum system. Even better, when considering the neural network themselves, it would take upwards of \(10^{4}\) samples to distinguish them from an actual quantum system. Ideally, one might try to see whether the semianalytical protocols, when integrated analytically to give the full behaviour, can be made into an exact protocol with the correct parameters. However, the communication function is very tricky to analytically integrate, and thus this approach might not work. On the other hand, considering that an exact protocol can already simulate some two-qubit states, these pieces of evidence suggest that all two-qubit states can be simulated with just a single bit of communication. However, ultimately, the question of _exactly_ simulating partially entangled states with one bit of communication remains open.

## III Searching for Bell violation of the one-bit of communication polytope

Since two-qubit states are simulatable up to very good precision, we now consider a different question: can we find an explicit quantum behaviour that is unsimulatable

Figure 3: Violin plots for the neural network (blue) and the semianalytical protocol we presented (red) describing the following values: **(a)** The Kullback-Leibler divergence between our protocols and the quantum behaviours. **(b)** The Total Variational Distance between our protocols and the quantum behaviours. **(c)** The minimum sample size needed to have at least 95% confidence in distinguishing the two behaviours as described in the hypothesis testing scenario. In all three, the violin shapes illustrate the distributions of the values over the different projective measurements on the two-qubit state.
with one bit of communication? We try to go to higher dimensional systems and try to find a Bell-like inequality for the communication polytope. As far as we know, no violation of a Bell-like inequality for the one-bit of communication polytope has ever been described. For the rest of the section, let \(\mathcal{L}\) be the local set, \(\mathcal{Q}\) be the quantum set, and \(\mathcal{C}\) be the one-bit of communication set. We are interested in points inside of \(\mathcal{Q}\) that lie outside \(\mathcal{C}\).

### Description of the polytope

Similar to \(\mathcal{L}\), \(\mathcal{C}\) is also a convex polytope. However, unlike it, it does not lie inside the no-signalling \(\mathcal{NS}\) space. Let \(\mathcal{A}\) (\(\mathcal{B}\)) be the output set of Alice (Bob) and \(\mathcal{X}\) (\(\mathcal{Y}\)) her (his) input set. The number of deterministic strategies that can be performed with a single bit is \(|\mathcal{A}|^{|\mathcal{X}|}|\mathcal{B}|^{2|\mathcal{Y}|}2^{|\mathcal{X}|}\). However, due to duplicates, the number reduces to [14]

\[|\mathcal{A}|\left(|\mathcal{B}|^{|\mathcal{Y}|}+(2^{|\mathcal{X}|-1}-1)(| \mathcal{B}|^{2|\mathcal{Y}|}-|\mathcal{B}|^{|\mathcal{Y}|})\right).\]

\(\mathcal{C}\) is the convex polytope formed by these vertices. In practice, we can only generate polytopes of up to around \(2\times 10^{7}\) points due to memory limitations. Since the number of extremum points for \(\mathcal{C}\) is much larger than for \(\mathcal{L}\), we can quickly discard the possibility of performing full facet enumeration. Hence, we would have to resort to other methods for our search.

### Random sampling quantum behaviours in higher dimensions

We first tried to sample points from \(\mathcal{Q}\) by measuring the maximally entangled two-qutrit and two-ququart state with measurements sampled uniformly in the Haar measure, before using linear programming to solve the membership problem for \(\mathcal{C}\). However, this method proved ineffective as we did not manage to find any behaviour which lies inside \(\mathcal{C}\), and even a significant amount still lies inside \(\mathcal{L}\). The statistics of this method can be seen in Table 1.

### Using non-signalling points

The next method we used was to use points in \(\mathcal{NS}\) and mix them with noise in order to find out the threshold noise levels at which they exit the sets \(\mathcal{Q}\) and \(\mathcal{C}\). If we find a point \(P_{\mathcal{NS}}\) which exits \(\mathcal{Q}\) at a lower noise level \(w_{\mathcal{Q}}\) than the corresponding one for \(\mathcal{C}\), \(w_{\mathcal{C}}\), all behaviours

\[wP_{\mathcal{NS}}+(1-w)P_{noise}\]

with \(w_{\mathcal{C}}<w<w_{\mathcal{Q}}\) would be behaviours in \(\mathcal{Q}\) that are unsimulatable by one-bit of communication. The membership problem for \(\mathcal{Q}\) is solved using the NPA hierarchy method [24] with level 2 hierarchy. A graphical illustration can be seen in Fig. 4.

Choosing a suitable \(P_{\mathcal{NS}}\), however, proved to be a challenge. The extremum points of the \(\mathcal{NS}\) space have only been characterised for binary inputs or binary outputs [25; 26]. Here, we mostly used nonlocal points which are locally unbiased, i.e. for all inputs, all the local outputs are of equal probability, and maximally correlated, i.e. for all input combinations, there is a perfect correlation between Alice and Bob's outputs, for a particular output of Alice the output of Bob is guaranteed and vice versa. While we tried other non-signalling points, this particular class of points gave us the closest gap between \(w_{\mathcal{Q}}\) and \(w_{\mathcal{C}}\). Similarly, there are also numerous choices for \(P_{noise}\), but we find that white noise gives the closest gap in most scenarios.

The results of the closest gap \((w_{\mathcal{C}}-w_{\mathcal{Q}})\) found in each scenario are listed in Table 3. In the case of \(|\mathcal{A}|=|\mathcal{B}|=3\), we did not find any violation. The closest gap was observed in the \((|\mathcal{X}|,|\mathcal{Y}|,|\mathcal{A}|,|\mathcal{B}|)=(4,4,3,3)\) scenario, where a there exist a point with \(w_{\mathcal{C}}=0.6612\) and \(w_{\mathcal{Q}}=0.6289\). However, in \(|\mathcal{A}|=|\mathcal{B}|=4\), specifically in the \((4,2,4,4)\) setting, we find a \(P_{\mathcal{NS}}\) which have \(w_{\mathcal{Q}}=w_{\mathcal{C}}=\frac{2}{3}\), described in in Table 2. The table itself can also be interpreted as a Bell inequality by taking the terms in

\begin{table}
\begin{tabular}{c|c c} \((|\mathcal{X}|,|\mathcal{Y}|,|\mathcal{A}|,|\mathcal{B}|)\) & Points sampled & Proportion in \(\mathcal{L}\) \\ \hline \hline (3,3,3,3) & 10000 & 25.6\% \\ (3,4,3,3) & 300 & 10.0\% \\ (4,3,3,3) & 300 & 10.3\% \\ (4,4,3,3) & 100 & 1.0\% \\ (3,3,4,4) & 500 & 56.6\% \\ \hline \end{tabular}
\end{table}
Table 1: The statistics for the sampling approach. None of the points sampled fall outside \(\mathcal{C}\).

Figure 4: \(w_{\mathcal{Q}}\) is the threshold weight for the quantum set \(\mathcal{Q}\), while \(w_{\mathcal{C}}\) is the threshold weight for the one-bit communication set \(\mathcal{C}\). Thus, \(w_{\mathcal{C}}<w_{\mathcal{Q}}\) would imply a violation and would give a quantum behaviour that could not be simulated by a single bit of communication.
the table as the coefficients of the correlation terms and adding all of them. When normalised into a Bell game, the value of the game is both \(\frac{3}{4}\) for \(\mathcal{Q}\) and \(\mathcal{C}\). This Bell facet is the hyperplane which has the line connecting \(P_{\mathcal{N}\mathcal{S}}\) and \(P_{noise}\) as its normal.

This point represents our closest attempt at finding a violation with this method. The number of extremal points for \(\mathcal{C}\) in the \((4,2,4,4)\) scenario is around \(1\times 10^{6}\), and it is possible to still go one input higher to \((4,3,4,4)\) or \((5,2,4,4)\). A violation might exist there, but our heuristic search proved unfruitful. In the end, contrary to the prepare-and-measure scenario [27], it still remains an open problem to find a bipartite quantum behaviour that is provably unsimulatable with one bit of communication [28].

## IV Conclusion

In this work, we tried to further the works that have been done on characterising the communication complexity cost of quantum behaviours. We tried to obtain a protocol to simulate partially entangled two-qubit states using a neural network, and we presented a semianalytical LHV+1 protocol based on the protocol of the neural networks. While these protocols could only approximate the quantum behaviours, on average one needs hundreds of measurement data, for the semianalytical protocols, and tens of thousands of measurement data, for the neural network protocols, in order to be distinguished from the quantum behaviour. We also tried to find quantum behaviours in higher dimensions that could not be simulated with one bit of communication. While we were able to find a Bell-like inequality that has the same maximum value in \(\mathcal{Q}\) and \(\mathcal{C}\), we were unable to find a violation.

From this work and all the previous works done on the topic, it can be seen that evaluating the capabilities of entangled quantum states in terms of communication complexity is very difficult. While we are confident that a behaviour that cannot be simulated with a single bit could probably be found, extending the work to more bits and states would probably be too difficult, barring any new revolutionary techniques. On the other hand, from our result that numerical protocols that closely approximate the two-qubit entangled states can be found, the task of simulating partially entangled two-qubit states using one bit of communication _exactly_ is probably possible and a fully analytical protocol could probably be found in the near future.

## V Code availability

The code is available at [https://github.com/PeterSidajava/neural-network-fp/](https://github.com/PeterSidajava/neural-network-fp/).

\begin{table}
\begin{tabular}{c|c|c c c|c c c c|}  & & \multicolumn{3}{c|}{\(Y=1\)} & \multicolumn{3}{c|}{\(Y=2\)} \\ \cline{3-8}  & & \(P(B=1)\) & \(P(B=2)\) & \(P(B=3)\) & \(P(B=4)\) & \(P(B=1)\) & \(P(B=2)\) & \(P(B=3)\) & \(P(B=4)\) \\ \hline \multirow{3}{*}{\(X=1\)} & \(P(A=1)\) & 1 & 0 & 0 & 0 & 1 & 0 & 0 \\  & \(P(A=2)\) & 0 & 1 & 0 & 0 & 0 & 1 & 0 \\  & \(P(A=3)\) & 0 & 0 & 1 & 0 & 0 & 0 & 1 \\  & \(P(A=4)\) & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 \\ \hline \multirow{3}{*}{\(X=2\)} & \(P(A=1)\) & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\  & \(P(A=2)\) & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 \\  & \(P(A=3)\) & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 \\  & \(P(A=4)\) & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 \\ \hline \multirow{3}{*}{\(X=3\)} & \(P(A=1)\) & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\  & \(P(A=2)\) & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 \\  & \(P(A=3)\) & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 \\  & \(P(A=4)\) & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 \\ \hline \multirow{3}{*}{\(X=4\)} & \(P(A=1)\) & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\  & \(P(A=2)\) & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 \\  & \(P(A=3)\) & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 \\  & \(P(A=4)\) & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 \\ \hline \end{tabular}
\end{table}
Table 2: The point in \((4,2,4,4)\) which have \(w_{\mathcal{Q}}=w_{\mathcal{C}}=\frac{2}{3}\). Each of the smaller eight boxes corresponds to the output table for a particular combination of inputs, which Alice’s input indexing the vertical dimension and Bob’s the horizontal. In each of the boxes, the \(4\times 4\) table corresponds to the outputs of Alice and Bob, with Alice’s in the vertical and Bob’s in the horizontal. Note that the 1 here means \(\frac{1}{4}\), which means that each of the boxes sum up to 1, as required.

\begin{table}
\begin{tabular}{c|c c|c} \((|\mathcal{X}|,|\mathcal{Y}|,|\mathcal{A}|,|\mathcal{B}|)\) & \(w_{\mathcal{Q}}\) & \(w_{\mathcal{C}}\) & \(w_{\mathcal{C}}-w_{\mathcal{Q}}\) \\ \hline \hline (3,3,3,3) & 0.5995 & 0.7000 & 0.1005 \\ (3,4,3,3) & 0.6022 & 0.7000 & 0.0978 \\ (4,3,3,3) & 0.5856 & 0.6766 & 0.0910 \\ (4,4,3,3,3) & 0.6289 & 0.6612 & 0.0323 \\ (5,3,3,3) & 0.5768 & 0.6610 & 0.0842 \\ (3,3,4,4) & 0.6159 & 0.7143 & 0.0984 \\ (4,2,4,4) & 0.6666 & 0.6666 & 0.0000 \\ \hline \end{tabular}
\end{table}
Table 3: The closest gap \((w_{\mathcal{C}}-w_{\mathcal{Q}})\) in each scenario we studied.
Acknowledgments

This research is supported by the National Research Foundation, Singapore and A*STAR under its CQT Bridging Grant. We thank Maria Balanzo-Juando, Martin J. Renner, Marco Tulio Quintino, and Marco Tomamichel for discussions. We are also grateful to the authors of [19] for making their codes public.

We also thank the National University of Singapore Information Technology for the use of their high performance computing resources.
