Machine Learning meets Quantum Foundations: A Brief Survey

Most of the recent advances in machine learning were facilitated by using artificial neurons. The basic structure is a single artificial neuron (AN), which is a real-valued function of the form,

In equation 3, the function $\phi$ is usually known as activation function. Some of the well-known activation functions are

1. 1.

   Threshold function: $\phi\left(a\right)=1$ if $a>0$ and $0$ otherwise,

2. 2.

   Sigmoid function: $\phi\left(a\right)=\sigma(a)=\frac{1}{1+\exp\left(-a\right)}\,,$ and

3. 3.

   Rectified Linear (ReLu) function: $\phi(a)=$ max$\left(0,a\right).$

The $\bm{w}$ vector in equation 3 is known as weight vector.

Many of those artificial neurons can be combined together via communication links to perform complex computation. This can be achieved by feeding the output of neurons (weighted by the weights $w_{i}$) as an input to another neuron, where the activation function is applied again. Such a graph structure $G=\left(V,E\right),$ with the set of nodes $(V)$ as artificial neurons and edges in $E$ as connections, is known as an artificial neural network or neural network in short. The first layer, where the data is fed in, is called input layer. The layers of neurons in-between are the hidden layers, which are defining feature of deep learning. The last layer is called output layer. A neural network with more than one hidden layer is called deep neural network. Machine learning involving deep neural networks is called Deep learning. A feedforward neural network is a directed acyclic graph, which means that the output of the neurons is fed only into forward direction (see Fig.2). Apart from the feedforward neural network, some of the popular neural network architectures are convolutional neural networks (CNN), recurrent neural networks (RNN), generative adversarial network (GAN), Boltzmann machine and restricted Boltzmann machines (RBM). We provide a brief summary of RBM here. For a detailed understanding of various other neural networks, refer to goodfellow2016deep .

An RBM is a bipartite graph with two kinds of binary-valued neuron units, namely visible $\left(\bm{v}=\left\{v_{1},v_{2},\cdots\right\}\right)$ and hidden $\left(\bm{h}=\left\{h_{1},h_{2},\cdots\right\}\right)$ (see Fig.3) The weight matrix $W=\left(w_{ij}\right)$ encodes the weight corresponding to the connection between visible unit $v_{i}$ and hidden unit $h_{j}$ hinton2012practical . Let the bias weight (offset weight) for the visible unit $v_{i}$ be $a_{i}$ and hidden unit $h_{j}$ be $b_{j}$.

For a given configuration of visible and hidden neurons $\left(\bm{v,h}\right)$, one can define an energy function $E\left(\bm{v,h}\right)$ inspired from statistical models for spin systems as

The probability of a configuration $\left(\bm{v,h}\right)$ is given by Boltzmann distribution,

where $Z=\sum_{\bm{v,h}}\exp\left(-E\left(\bm{v,h}\right)\right)$ is a normalization factor, commonly known as partition function. As there are no intra-layer connections, one can sample from this distribution easily. Given a set of visible units $\bm{v}$, the probability of a specific hidden unit being $h_{j}=1$ is

where $\sigma(a)$ is the sigmoid activation function as introduced earlier. A similar relation holds for the reverse direction, e.g. given a set of hidden units, what is the probability of the visible unit.

With these concepts, complicated probability distributions $P\left(\bm{v}\right)$ over some variable $\bm{v}$ can be encoded and trained within the RBM. Given a set of training vectors $\bm{v}\in\mathbb{V}$, the goal is find the weights $W^{\prime}$ of the RBM that fit the training set best

RBMs have been shown to be a good Ansatz to represent many-body wavefunctions, which are difficult to handle with other methods due to the exponential scaling of the dimension of the Hilbert space. This method has been successfully applied to quantum many-body physics and quantum foundations problems deng2018machine ; carleo2017solving .

In context beyond physics, machine learning with deep neural networks has accomplished many significant milestones, such as mastering various games, image recognition, self-driving cars, and so forth. Its impact on the physical sciences is just about to start carleo2019machine ; deng2018machine .

The term “artificial intelligence” was first coined in the famous 1956 Dartmouth conference mccarthy2006proposal . Though the term was invented in 1956, the operational idea can be traced back to Alan Turing’s influential “Computing Machinery and Intelligence” paper in 1950, where Turing asks if a machine can think turing2009computing . The idea of designing machines that can think dates back to ancient Greece. Mythical characters such as Pygmalion, Hephaestus and Daedalus can be interpreted as some of the legendary inventors and Galatea, Pandora and Hephaestus can be thought of as examples of artificial life goodfellow2016deep . The field of artificial intelligence is difficult to define, as can be seen by four different and popular candidate definitions russell2002artificial . The definitions start with “AI is the discipline that aims at building …”

1. 1.

   (Reasoning-based and human-based): agents that can reason like humans.

2. 2.

   (Reasoning-based and ideal rationality): agents that think rationally.

3. 3.

   (behavior-based and human-based): agents that behave like humans.

4. 4.

   (behavior-based and ideal rationality): agents that behave rationally.

Apart from its foundational impact in attempting to understand “intelligence,” AI has reaped practical impacts such as automated routine labour and automated medical diagnosis, to name a few among many. The real challenge of AI is to execute tasks which are easy for people to perform, but hard to express formally. An approach to solve this problem is by allowing machines to learn from experience, i.e. via machine learning. From a foundational point of judgment, the study of machine learning is vital as it helps us understand the meaning of “intelligence”. It is worthwhile mentioning that deep learning is a subset of machine learning which can be thought of as a subset of AI (see Fig.9).

Quantum interaction inevitably leads to quantum entanglement. The individual states of two classical systems after an interaction are independent of each other. Yet, this is not the case for two quantum systems raimond2001manipulating ; horodecki2009quantum . Quantum states can be pure or mixed. For a bipartite pure quantum state $|\psi\rangle\in\cal{H}^{A}\otimes\cal{H}^{B}$ is entangled if it cannot be written as a product state, i.e. $|\psi\rangle=|\psi^{A}\rangle\otimes|\psi^{B}\rangle$ for some $|\psi^{A}\rangle\in\cal{H}^{A}$ and $|\psi^{B}\rangle\in\cal{H}^{B}$. A mixed bipartite state is however expressed in terms of a density matrix, $\rho$. Like for pure state, a density matrix $\rho$ is entangled if it cannot be expressed in the form $\displaystyle\rho=\sum_{i}p_{i}|\psi_{i}^{A}\rangle$ $\langle\psi_{i}^{A}|\otimes|\psi_{i}^{\rangle}$ $\langle\psi_{i}^{B}|$. An $n-$partite state $\rho_{sep}$ is called separable if it can be represented as a convex combination of product states i.e.

where $0\leq p_{i}\leq 1$ and $\sum_{i}p_{i}=1.$ A quantum state is called separable if it is not entangled.

Computationally it is not easy to check if a mixed state, especially in higher dimensions and for more parties, is separable or entangled. Numerous measures of quantum entanglement for both pure and mixed states are proposed horodecki2009quantum : for bipartite systems where the dimension of $\mathcal{H}^{i}(i=A,B)$ is 2, a good measure is concurrence wootters2001entanglement . Other measures for quantifying entanglement are entanglement of formation, robustness of entanglement, Schmidt rank, squashed entanglement and so forth horodecki2009quantum . It turns out that for any entangled state $\rho$, there exists a Hermitian matrix $W$ such that $\text{Tr}(\rho W)<0$ and for all separable states $\rho_{\text{sep}}$, $\text{Tr}(\rho W)\geq 0$  horodecki1996necessary ; terhal2002detecting ; krammer2009multipartite ; ekert2002direct .

Quantum entanglement was mooted a long time ago by Erwin Schrödinger, but it took another thirty years or so for John Bell to show that quantum theory imposes strong constraints on statistical correlations in experiments. Yet, correlations are not tantamount to causation and one wonders if machine learning could do better pearle2000causality . In the sixties and seventies, this Gedanken experiment was given further impetus with some ingenious experimental designs freedman1972experimental ; kocher1967polarization ; aspect1981experiences ; aspect1982experimental . The advent of quantum information in the nineties gave a further push: quantum entanglement became a useful resource for many quantum applications, ranging from teleportation bennett1993teleporting , communication ekert1991quantum , purification bennett1996purification , dense coding bennett1992communication and computation deutsch1985quantum ; feynman2018feynman . Interestingly, quantum entanglement is a fascinating area that emerges almost serendipitously from the foundation of quantum mechanics into real practical applications in the laboratories.

According to John Bell Bell64 , any theory based on the collective premises of locality and realism must be at variance with experiments conducted by spatially separated parties involving shared entanglement, if the underlying measurement events are space-like separated. The phenomenon, as discussed before, is known as Bell nonlocality brunner2014bell . Apart from its significance in understanding foundations of quantum theory, Bell nonlocality is a valuable resource for many emerging device-independent quantum technologies like quantum key distribution (QKD), distributed computing, randomness certification and self-testing ekert1991quantum ; pironio2010random ; Yao_self . The experiments which can potentially manifest Bell nonlocality are known as Bell experiments. A Bell experiment involves $N$ spatially separated parties $A_{1},A_{2},\cdots,A_{N}$. Each party receives an input $x_{1},x_{2},x_{3,}\cdots,x_{N}\in\mathcal{X}$ and gives an output $a_{1},a_{2},a_{3},\cdots,a_{N}\in\mathcal{A}.$ For various input-output combinations one gets the statistics of the following form:

We will refer to $\mathcal{P}$ as behavior. A Bell experiment involving $N$ space-like separated parties, each party having access to $m$ inputs and each input corresponding to $k$ outputs is referred to as $\left(N,m,k\right)$ scenario. The famous Clauser-Horn-Shimony-Holt (CHSH) experiment is a $(2,2,2)$ scenario CHSH , and it is the simplest scenario in which Bell nonlocality can be demonstrated (see Fig. 5).

A behavior $\mathcal{P}$ admits local hidden variable description if and only if

This is known as local behavior. The set of local behaviors $(\mathcal{L})$ forms a convex polytope, and the facets of this polytope are Bell inequalities (see Fig. 6). In quantum theory, Born rule governs probability according to

where $\left\{M_{a_{i}|x_{i}}\right\}_{i}$ are positive-operator valued measures (POVMs) and $\rho$ is a shared density matrix. If a behavior satisfying Eq.11 falls outside $\mathcal{L}$, it then violates at least one Bell inequality, and such behavior is said to manifest Bell nonlocality. The condition that parties do not communicate during the course of the Bell experiment is known as no-signalling condition. Intuitively speaking, it means that the choice of input of one party can not be used for signalling among parties. Mathematically it means,

The set of behaviours which satisfy the no signalling condition are known as no-signalling behaviours. We will denote the aforementioned set by $\mathcal{NS}$. The no-signalling set also forms a polytope. Furthermore, $\mathcal{L}\subseteq\mathcal{Q}\subseteq\mathcal{NS}$ (see Fig. 6). The no-signalling behaviours which do not lie in $\mathcal{Q}$ are also known as post-quantum behaviours. In the (2,2,2) scenario i.e. CHSH Scenario, there is a unique Bell inequality, namely CHSH inequality, upto relabelling of inputs and outputs. The CHSH inequality is given by

where $E_{x_{1},x_{2}}=P\left(a_{1}=a_{2}|x_{1}=x_{2}\right)-P\left(a_{1}\neq a_{2}|x_{1}=x_{2}\right).$ All local hidden variable theories satisfy CHSH inequality. In quantum theory, suitably chosen measurement settings and state can lead to violation of CHSH inequality and thus the CHSH inequality can be used to witness Bell nonlocal nature of quantum theory. Quantum behaviours achieve upto $2\sqrt{2},$ known as the Tsirelson bound. The upper bound for no-signalling behaviours (no-signalling bound) on the CHSH inequality is $4$.

An intuitive feature of classical models is non-contextuality which means that any measurement has a value independent of other compatible measurements it is performed together with. A set of compatible measurements is called context. It was shown by Simon Kochen and Ernst Specker (as well as John Bell) kochen1967problem that non-contextuality conflicts with quantum theory.

The contextual nature of quantum theory can be established via simple constructive proofs. In Fig. 7 (A), one considers an array of operators on a two-qubit system in any quantum state. There are nine operators, and each of them has eigenvalue $\pm 1$. The three operators in each row or column commute and so it is easy to check that each operator is the product of the other two operators on a particular row or column, with a single exception, the third operator in the third column equals the minus of the product of the other two. Suppose there exist pre-assigned values ($-1$ or $+1$) for the outcomes of the nine operators, then we can replace the none operators by the pre-assigned values. However, there is no consistent way to assign such values to the nine operators so that the product of the numbers in every row or column (except for the operators along the bold line) yield 1 (the product yields -1). Notice that each operator (node) appears in exactly two lines or context. Kochen and Specker provided the first proof of quantum contextuality with a complicated construct involving 117 operators on a 3-dimensional space kochen1967problem . Another example is the pentagram mermin1993hidden in Fig. 7.

Bell nonlocality is regarded as a particular case of contextuality where the space-like separation of parties involved creates ”context” CSW ; amaral2018graph . The study of contextuality has led not only to insights into the foundations of quantum mechanics, but it also offers practical implications as well Cabello_QKD ; bharti2019non ; raussendorf2013contextuality ; bharti2019robust ; mansfield2018quantum ; saha2019state ; bharti2019local ; delfosse2015wigner ; singh2017quantum ; pashayan2015estimating ; bharti2018simple ; bermejo2017contextuality ; catani2018state ; arora2019revisiting ; Howard2014 . There are several frameworks for contextuality including sheaf-theoretic framework abramsky2011sheaf , graph and hypergraph framework CSW ; acin2015combinatorial , contextuality-by-default framework dzhafarov2017contextuality ; dzhafarov2019joint ; kujala2019measures and operational framework spekkens2005contextuality . A scenario exhibits contextuality if it does not admit the non-contextual hidden variable (NCHV) model KS67 .

Correlations produced by steering lie between the Bell nonlocal correlations and those generated from entangled states quintino2015inequivalence ; wiseman2007steering . A state that manifests Bell nonlocality for some suitably chosen measurement settings also exhibits steering schrodinger1935discussion . Furthermore, a state which exhibits steering must be entangled. A state demonstrates steering if it does not admit “local hidden state (LHS)” model wiseman2007steering . We discuss this formally here.

Alice and Bob share some unknown quantum state $\rho^{AB}.$ Alice can perform a set of POVM measurements $\left\{M_{a}^{x}\right\}_{a}.$ The probability of her getting outcome $a$ after choosing measurement $x$ is given by

where $\rho_{a|x}^{B}=\text{Tr}_{A}\left[\left(M_{a}^{x}\otimes I\right)\rho^{AB}\right]$ is Bob’s residual state upon normalization. A set of operators $\left\{\rho_{a|x}^{B}\right\}_{a,x}$ acting on Bob’s space is called an assemblage if

and

Condition 15 is the analogue of the no-signalling condition. An assemblage $\left\{\rho_{a|x}^{B}\right\}_{a,x}$ is said to admit LHS model if there exists some hidden variable $\lambda$ and some quantum state $\rho_{\lambda}$ acting on Bob’s space such that

A bipartite state $\rho^{AB}$ is said to be steerable from Alice to Bob if there exist measurements for Alice such that the corresponding assemblage does not satisfy equation 17. Determining whether an assemblage admits LHS model is a semidefinite program (SDP). The concept of steering is asymmetric by definition, i.e. even if Alice could steer Bob’s state, Bob may not be able to steer Alice’s state.

Given a behavior, deciding whether it is classical or non-classical is an extremely challenging task since the underlying scenario grows in complexity very quickly. Canabarro et al. canabarro2019machine use supervised machine learning with an ensemble of neural networks to tackle the approximate version of the problem (i.e. with a small margin of error) via regression. The authors ask “How far is a given correlation from the local set.” The input feature vector to the neural network is a random correlation vector. For a given behavior, the output (label) is the distance of the feature vector from the classical, i.e. local set. The nonlocality quantifier of a behavior $q$ is the minimum trace distance, denoted by $NL(q)$ brito2018quantifying . For the two-party scenario, the nonlocality quantifier is given by

where $\mathcal{L}$ is the local set and $|x|=|y|=m$ is the input size for the parties. The training points are generated by sampling the set of non-signalling behaviors randomly and then calculating its distance from the local set via equation 18. Given a behavior $q$, the distance predicted by the neural network is never equal to the actual distance, i.e. there is always a small error $(\epsilon\neq$0). Let us represent the learned hypothesis as $f:q\rightarrow\mathbb{R}.$ The performance metric of the learned hypothesis $f$ is given by

In experiments such as entanglement swapping, which comprises three separated parties sharing two independent sources of quantum states, the local set admits the following form and the set is non-convex,

Here, non-convexity emerges from the independence of the sources i.e. $P\left(\lambda_{1},\lambda_{2}\right)=P\left(\lambda_{1}\right)P\left(\lambda_{2}\right).$

In this case, the Bell inequalities are no longer linear. Characterizing the set of classical as well as quantum behaviors gets complicated for such scenarios tavakoli2014nonlocal ; branciard2010characterizing .

Canabarro et al. train the model for convex as well as non-convex scenarios. They also train the model to learn post-quantum correlations. The techniques studied in the paper are valuable for understanding Bell nonlocality for large quantum networks, for example those in quantum internet.

Given an observed probability distribution corresponding to scenarios where several independent sources are distributed over a network, deciding whether it is classical or non-classical is an important question, both from practical as well as foundational viewpoint. The boundary separating the classical and non-classical correlations is extremely non-convex and thus a rigorous analysis is exceptionally challenging.

In reference krivachy2019neural , the authors encode the causal structure into the topology of a neural network and numerically determine if the target distribution is “learnable”. A behavior belongs to the local set if it is learnable. The authors harness the fact that the information flow in feedforward neural networks and causal structures are both determined by a directed acyclic graph. The topology of the neural network is chosen such that it respects the causality structure. The local set corresponding to even elementary causal networks such as triangle network is profoundly non-convex, and thus analytical characterization of the same is a notoriously tricky task. Using the neural network as an oracle, Krivachy et al. krivachy2019neural convert the membership in a local set problem to a learnability problem. For a neural network with adequate model capacity, a target distribution can be approximated if it is local. The authors examine the triangle network with quaternary outcomes as a proof-of-principle example. In such a scenario, there are three independent sources, say $\alpha,\beta$ and $\gamma$. Each of the three parties receives input from two of the three sources and process the inputs to provide outputs via fixed response functions. The outputs for Alice, Bob and Charlie will be indicated by $a,b,c\in\left\{0,1,2,3\right\}.$ The scenario as discussed here can be characterized by the probability distribution $P\left(a,b,c\right)$ over the random variables $a,b$ and $c$. If the network is classical, then the distribution can be represented by a directed acyclic graph known as a Bayesian network (BN).

Assuming the distribution $P\left(a,b,c\right)$ over the random variables $a,b$ and $c$ to be classical, it is assumed without loss of generality that the sources send a random variable drawn uniformly from the interval $0$ to $1.$ A classical distribution for such a case admits the following form:

The neural network is constructed such that it can approximate the distribution of type Eq.21. The inputs to the neural network are $\alpha,\beta$ and $\gamma$ drawn uniformly at random and the outputs are the conditional probabilities i.e. $P_{A}\left(a|\beta,\gamma\right),P_{B}\left(b|\gamma,\alpha\right)$ and $P_{C}\left(c|\alpha,\beta\right)$.

The cost function is chosen to be any measure of the distance between the target distribution $P_{\text{t}}$ and the network’s output $P_{\text{M}}$. The authors employ the techniques to a few other cases, such as the elegant distribution and a distribution proposed by Renou et al. renou2019genuine . The application of the technique to the elegant distribution suggests that the distribution is indeed nonlocal as conjectured in gisin2019entanglement . Furthermore, the distribution proposed by Renou et al. appears to have nonlocal features for some parameter regime. For the sake of completeness, now we discuss the elegant distribution and the distribution proposed by Renou et al.

Elegant distribution – The distribution is generated by three parties performing entangled measurements on entangles systems. The three parties share singlets i.e. $|\psi^{-}\rangle=\frac{1}{\sqrt{2}}\left(|01\rangle-|10\rangle\right)$ Every party performs entangled measurements on their two qubits. The eigenstates of the entangled measurements are given by

where $|\alpha_{j}\rangle$ are vectors symmetrically distributed on the Bloch sphere i.e. point to the vertices of a tetrahedron, for $j\in\{1,2,3,4\}.$

Renou et al. distribution – The distribution is generated by three parties sharing the entangled state $|\phi^{+}\rangle=\frac{1}{\sqrt{2}}\left(|00\rangle+|11\rangle\right)$ and performing the same measurement on each of their two qubits. The measurements are characterized by a single parameter $\kappa\in\left[\frac{1}{\sqrt{2}},1\right]$ with eigenstates $|01\rangle,|10\rangle,u|00\rangle+\sqrt{1-u^{2}}|11\rangle$ and $\sqrt{1-u^{2}}|00\rangle-u|11\rangle$.

Several methods from machine learning have been adopted to tackle intricate quantum many-body problems carleo2017solving ; saito2018machine ; deng2018machine ; gao2017efficient . Dong-Ling Deng deng2018machine employs machine learning techniques to detect quantum nonlocality in many-body systems using the restricted Boltzmann machine (RBM) architecture. The key idea can be split into two parts:

• •

  After choosing appropriate measurement settings, Bell inequalities for convex scenarios can be expressed as an operator. This operator can be thought of as an Hamiltonian. The eigenstate with the maximal eigenvalue is the state that gives the maximum violation for the underlying Bell inequality. This state can be found by calculating the ground state of the negative of the Hamiltonian.

• •

  Finding the ground state of a quantum Hamiltonian is QMA-hard and thus in general difficult. However, using heuristic techniques involving the RBM architecture, the problem is recast into the task of finding the approximate answer in some cases.

Techniques like density matrix renormalization group (DMRG) schollwock2005density , projected entangled pair states (PEPS) verstraete2008matrix and multiscale entanglement renormalization ansatz (MERA) vidal2008class are traditionally used to find (or approximate) ground states of many-body Hamiltonians. But these techniques only work reliably for optimization problems involving low-entanglement states. Moreover, DMRG works well only for systems with short-range interactions in the one-dimensional case. As evident from references carleo2017solving , RBM can represent quantum many-body problems beyond 1-D and low-entanglement.

Prediction of winning strategies for (classical) games and decision-making processes with reinforcement learning (RL) has made significant progress in game theory in recent years. Motivated partly by these results, the authors in Bharti et al. bharti2019teach have looked at a game-theoretic formulation of Bell inequalities (known as Bell games) and applied machine learning techniques to it. To violate a Bell inequality, both the quantum state as well as the measurements performed on the quantum states have to be chosen in a specific manner. The authors transform this problem into a decision making process. This is achieved by choosing the parameters in a Bell game in a sequential manner, e.g. the angles of the measurement operators, the angles parameterizing the quantum states, or both. Using RL, these sequential actions are optimized for the best configuration corresponding to the optimal/near-optimal quantum violation (see Fig.10). The authors train the RL agent with a cost function that encourages high quantum violations via proximal policy optimization - a state-of-the-art RL algorithm that combines two neural networks. The approach succeeds for well known convex Bell inequalities, but it can also solve Bell inequalities corresponding to non-convex optimization problems, such as in larger quantum networks. So far, the field has struggled solving these inequalities; thus, this approach offers a novel possibility towards finding optimal (or near-optimal) configurations.

Furthermore, the authors present an approach to find settings corresponding to maximum quantum violation of Bell inequalities on near-term quantum computers. The quantum state is parameterized by a circuit consisting of single-qubit rotations and CNOT gates acting on neighbouring qubits arranged in a linear fashion. Both the gates and the measurement angles are optimized using a variational hybrid classic-quantum algorithm, where the classical optimization is performed by RL. The RL agent learns by first randomly trying various measurement angles and quantum states. Over the course of the training, the agent improves itself by learning from experience, and is capable of reaching the maximal quantum violation.

In reference ma2018transforming , the authors blend Bell inequalities with a feed-forward neural network to use them as state classifiers. The goal is to classify states as entangled or separable. If a state violates a Bell inequality, it must be entangled. However, Bell inequalities cannot be used as a reliable tool for entanglement classification i.e. even if a state is entangled, it might not violate an entanglement witness based on the Bell inequality. For example, the density matrix $\rho=p|\psi_{-}\rangle\langle\psi_{-}|+\left(1-p\right)\frac{I}{4}$ violates the CHSH inequality only for $p>\frac{1}{\sqrt{2}}$, but is entangled for $p>\frac{1}{3}$ werner1989quantum . Moreover, given a Bell inequality, the measurement settings that witness entanglement of a quantum state (if possible) depend on the quantum state. Prompted by these issues, the authors ask if they can transform Bell inequalities into a reliable separable-entangled states classifier. The coefficients of the terms in the CHSH inequalities are $(1,1,1,-1).$ The local hidden variable bound on the inequality is $2$ (see equation 13). Assuming fixed measurement settings corresponding to CHSH inequality, the authors examine whether it is possible to get better performance in terms of entanglement classification compared with the values $\left(1,1,1,-1,2\right)$. The main challenge to answer such a question in a supervised learning setting is to get labelled data that is verified to be either separable or entangled.

To train the neural network, the correlation vector corresponding to the appropriate Bell inequality was chosen as input with the state being entangled or separable $\left(1\text{ versus }0\right)$ as corresponding output. The correlation vector contains the expectation of the product of Alice’s and Bob’s measurement operators. The performance of the network improved as the model capacity was increased, which hints that the hypothesis which separates entangled states from separable ones must be sufficiently “complex.” The authors also trained a neural network to distinguish bi-separable and bound-entangled states.

Harney et al. harney2019entanglement use reinforcement learning with RBMs to detect entangled states. RBMs have demonstrated being capable of learning complex quantum states. The authors modify RBMs such that they can only represent separable states. This is achieved by separating the RBM into $K$ partitions that are only connected within themselves, but not with the other partitions. Each partition represents a (possibly) entangled subsystems, that however is not entangled with the other partitions. This choice enforces a specific $K$-separable Ansatz of the wavefunction. This RBM is trained to represent a target state. If the training converges, it must be representable by the Ansatz and thus be $K$-separable. However, if the training does not converge, the Ansatz is insufficient and the target state is either of a different $K^{\prime}$-separable form or fully entangled.

Can tools from machine learning help to distinguish entangled and separable states given the full quantum state (e.g. obtained by quantum tomography) as an input? Two recent studies address this question.

In Lu et al. lu2018separability , the authors detect entanglement by employing classic (i.e. non-deep learning) supervised learning. To simplify data generation of entangled and separable states, the authors approximate the set of separable states by a convex hull (CHA). States that lie outside the convex hull are assumed to be most likely entangled. For the decision process, the authors use ensemble training via bootstrap aggregating (bagging). Here, various supervised learning methods are trained on the data, and they form a committee together that decides whether a given state is entangled or not. The algorithm is trained with the quantum state and information encoding the position relative to the convex hull as inputs. The authors show that accuracy improves if bootstrapping of machine learning methods is combined with CHA.

In a different approach Goes et al. goes2020automated , the authors present an automated machine learning approach to classify random states of two qutrits as separable (SEP), entangled with positive partial transpose (PPTES) or entangled with negative partial transpose (NPT). For training, the authors elaborate a way to find enough samples to train on. The procedure is as follows: A random quantum state is sampled, then using the General Robustness of Entanglement (GR) and PPT criterion, it is classified to either SEP, PPTES or NPT. The GR measures the closeness to the set of separable states. The authors compare various supervised learning methods to distinguish the states. The input features fed into the machine are the components of the quantum state vector and higher-order combinations thereof, whereas the labels are the type of entanglement. Besides, they also train to estimate the GR with regression techniques and use it to validate the classifiers.