Beyond Bell's theorem: correlation scenarios

Bell's theorem [Bel64, Shi04] shows that quantum phenomena cannot be modeled correctly by a theory satisfying the following natural assumptions:

• (I)  
  Realism: Any physical system can be described in terms of a probabilistic mixture of states (=hidden variable values). Composite systems are described by a joint probability distribution over the state spaces of its component systems.
• (II)  
  Locality: Physical systems have spatial components which can be described independently. They do not interact across space-like separated events.
• (III)  
  Free will: The parties in a Bell scenario have genuine randomness available which is independent of their environment. This is also known as λ-independence [BY08] and as measurement independence [Hal10].

Standard quantum theory fails (I) due to the way that joint systems are described. It is irrelevant whether (III) holds in quantum theory, since (III) is only used in combination with (I) and (II) in the derivation of the Bell inequalities, which are found to have quantum violations.

In this paper, we are concerned with assumption (III). More precisely, we are actually not concerned with (III), since we aim to replace it with a different property:

• (III')  
  Independence of sources [BGP10]: If an experiment contains several sources¹, then the theory describes these sources as independent. This means that the joint distribution of hidden variables is a product distribution.

Our observation is that (III) becomes obsolete when assuming (III'), so that one obtains:

Bell's theorem, new version. Quantum phenomena cannot be modeled correctly by a theory satisfying (I), (II) and (III').

Branciard et al already briefly considered scenarios in which each party has only one measurement setting [BRGP12, section V/VI]. These are a natural continuation of their earlier work [BGP10] that combined (III') with (III). Here, we build on their idea and set up a formal framework for multi-source 'correlation scenarios' in which each party has only one measurement setting available and derive more results within that framework. There are several advantages to this over the standard approach based on (III):

• One of the main goals of the hidden variable program was to resurrect a deterministic worldview [EPR35]. However, as has also been observed by ['H07] and probably others, determinism is at variance with (III) even without Bell's theorem since genuine randomness cannot be created in a deterministic world. This tension between determinism and free will has long been known to philosophers and led them to seek definitions of human free will compatible with determinism [McK04].
• Free will is an observer-centric notion which, depending on the theory, may require the observer to live outside that part of the universe described by the theory. In contrast, the property (III') concerns only observer-independent physical systems and has a clear physical meaning. Our formalism is best viewed as devoid of any conscious agents.
• Bell's theorem is often presented as a statement about theories satisfying realism (I) and locality (II) only. (III) is then tacitly assumed without explicit mention, either because one has failed to note it as an additional and crucial assumption or because it may be incorrectly regarded as self-evident. In contrast, (III') is more easily understood to be a non-trivial assumption.
• There has been speculation on the relation between quantum mechanics and free will. Our approach elucidates that this discussion is irrelevant to Bell's theorem (as is well known to experts, but possibly not to those just learning about Bell's theorem and assumption (III)).

Moreover, our formalism allows the consideration of (quantum) correlations which have no analogue in standard Bell scenarios and are genuinely new; see theorems 2.16 and 2.21. Our present results are not sufficient to tell what the meaning or relevance of such new kinds of correlations might be; ultimately, we hope for the development of quantum information protocols utilizing them in ways similar to those taking advantage of quantum correlations in standard Bell scenarios, e.g. quantum key distribution [Eke91] or certified randomness generation [PAM⁺10]. Another interesting direction might be to consider analogues of the amplification of free will [CR12] for the amplification of independence of sources.

Some of the mathematical problems we are going to discuss in this paper have been considered before in a totally different context. There is a work by Steudel and Ay [SA10] on the inference of common ancestors which concerns questions such as this: given three different languages, under which conditions can one derive the existence of a common antecedent language that influenced all three? Or, given the joint distribution of the prevalence of some diseases in a population, under which conditions can one conclude the existence of a certain pre-existing quantity or property (such as a genetic defect or a specific diet) having some influence on the occurrence of all the diseases considered? This is the question of existence of a common ancestor in a Bayesian network model [Pea09]. A variable in a Bayesian network typically has many ancestors, including itself. One then considers models of the given joint distribution of the observed variables in terms of Bayesian networks, in which each observed variable corresponds to a node, the other nodes represent unobserved variables and each edge represents a causal link. Then the question is whether one can find such a model without a node that is an ancestor of all the observed variables or whether such a Bayesian network model necessarily requires such a common ancestor.

For the special case of three observed variables a, b, c, the very general results of [SA10] show that if the single-variable Shannon entropies H(a), H(b), H(c) and the joint entropy H(abc) satisfy the inequality

$$\begin{equation} H(a) + H(b) + H(c) > 2 H(abc), \end{equation} \tag{ 1.1 }$$

then the existence of a common ancestor is necessary. In our example: if the vocabulary of three languages is correlated in such a way that the entropy of the joint distribution is so low that the inequality holds, then there needs to be a common precursor having influenced all three.

We will see that the inference of common ancestors is a special case of our formalism. A byproduct of our results will be an inequality similar to but strictly better than  (1.1), for the very particular case of three variables; see (2.14).

We hope that our ideas will spur new developments in several directions:

• Further study of classical, quantum and generalized correlations in correlation scenarios. The wealth of open problems we present shows that our results are nothing but a first step towards an understanding of correlation scenarios.
• What are the philosophical implications of our results? How do (III) and (III') compare from a philosophy of science perspective?
• Could our correlation scenarios have any relevance for applications such as quantum key distribution?

A further generalization of correlation scenarios to scenarios with arbitrary causal structure will be considered in [FS12]. Correlation scenarios are a natural intermediate step between Bell scenarios and the arbitrary causal structure of [FS12].

The interested reader should start with the next subsection on terminology and notation, otherwise the main text will not be comprehensible. The subsequent main part of the paper in sections 2 and 3 can be read in a linear way. Section 2 contains the most important material, namely the conceptual discussion and the examples we have considered so far. Those who do not care too much about abstract generalities may stop reading at any point at which they start losing interest. In particular, reading section 3, which contains an initial sketch of how an abstract approach to our formalism could look, is not required for understanding the main ideas. It is supposed to be an attempt at laying the formal basis for future work on the subject.

Due to the large amount of technical detail required for completely rigorous proofs, we restrict ourselves in several cases to the presentation of proof sketches. We hope that these make it clear how completely rigorous proofs can be constructed. In the cases where a general rigorous proof or definition involves measure theory, the main text provides the proof or definition for the case of discrete hidden variables; appendix A then treats the general case of hidden variables defined on arbitrary probability spaces.

Since the subject of this paper is relatively new, many questions remain open. In the main text, we mention a wealth of open problems of various difficulties. We warn the reader that trying to solve them can be quite frustrating; our own experience has been that the intuition we have developed for standard Bell scenarios is sometimes more of a hindrance than an asset. Many of our initially promising ideas have turned out to be misconceived. Those that have eventually worked are based on very different concepts ranging from entropic inequalities (lemma 2.14) via Hardy-type paradoxes (theorem 2.21) to Choquet's theorem (see appendix A.6). Nevertheless, we hope that our formalism will develop into an alternative approach to the study of non-locality and will continue to be studied not only from our mathematical point of view, but also from both the information processing and the philosophical perspective. For example, the recent 'PBR theorem' [PBR11] also considers hidden variable theories satisfying (III') and a comparison to our approach may be of interest.

Finally, appendix A contains measure-theoretical details of the consideration of non-discrete hidden variables. In the main text, all our definitions and proofs are rigorous only for the case of discrete hidden variables; without exception, the same ideas work in the general case, but the technicalities required are so much more laborious and obscure that we relegate them to the appendix.

A follow-up paper [FS12] will present an even more general formalism for device-independent physics in terms of hidden Bayesian networks. It will comprise not only standard Bell scenarios and the formalism we introduce here, but also other scenarios such as Popescu's 'hidden' non-locality [Pop95]. It will be conceptually similar to hidden Markov models [LA09].

From now on, we will restrain from using the misleading term non-locality and related terms such as local correlations. It is misleading terminology insofar as it suggests that non-local interactions would be the only way of escaping the conclusion of Bell's theorem; however, this is far from correct, since locality is only one of the assumptions (I)–(III). Moreover, despite the experimental verification of the existence of quantum 'non-locality' [AGR81], all known fundamental interactions in physics are of a local nature [CG07, Haa96, Jac96]; see also [Zeh06]. Consequently, we will rather speak of classical correlations in analogy with the commonly used term quantum correlations. We will use these notions both in the context of standard Bell scenarios and in our new correlation scenarios.

In the context of our correlation scenarios, we use typewriter-font upper-case letters A, B, C, ... to enumerate the measurements. Equivalently, one may think of these as observers or parties: since each observer or party gets assigned a fixed measurement which they carry out in each run of the experiment, this is the same. The corresponding measurement outcomes are denoted by lower-case letters a, b, c, .... We denote the joint probability distribution of outcomes of, for example, the joint measurement (A,Y) by p(a,y). This constitutes extensive abuse of notation, as it makes expressions such as p(97,−2) ambiguous: does this refer to the distribution p(a,y) or to another one such as p(w,z)? Notwithstanding, we use this notation here in order to keep clutter to a minimum, while making sure that it does not lead to ambiguous expressions. We also keep the order of the variables arbitrary: for example, p(x,a,b,y) stands for the same distribution as p(a,b,x,y), and the one we use depends on which one is more natural in that particular context. Moreover, notation such as p(a,b,x,y) makes sense, strictly speaking, only when all variables are discrete; while we do assume that all measurements have only a finite number of possible outcomes, we do not make any discreteness assumption on the hidden variables; see appendix A.

Any reader looking at this paper will probably already have the necessary understanding of Bell's theorem [Bel64, Shi04]. Moreover, we also need to assume good familiarity with the notions of (conditional) independence of random variables and conditioning of probabilities. A basic knowledge of the terminology of graphs and hypergraphs is required for section 3. Some background in Bayesian networks [KF09, Pea09] will be of advantage in order to understand the connection to [SA10]. Reading appendix A is not possible without some grasp of measure-theoretical probability theory and related subjects.

Let us consider an experimental setup as depicted abstractly in figure 1. There are four parties X, A, B, Y (circles) arranged in a linear way such that any pair of neighboring parties shares a source (square). Each of these three sources sends out, at time t_emit, one physical system to each adjacent party. As in the case of ordinary Bell scenarios, these two systems are typically correlated; in the classical case, this is shared randomness, while in the quantum case, such a correlation can also be entanglement. The parties receive these systems and each party carries out, at time t_meas > t_emit, a fixed measurement on the system(s) they have received; in the case of A and B, who receive two systems each, this will typically be a joint measurement operating on both systems simultaneously. In each run of the experiment, the parties obtain and register outcomes x, a, b and y. If the experiment is repeated many times, the parties will notice correlations between these outcomes and determine a joint probability distribution p(x,a,b,y). With the parties as vertices and the sources as edges, figure 1 has the structure of the path graph P₄, and therefore we will speak of the P₄ scenario. It was first studied in [BRGP12, section V].

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Ideally, the timing and geometry of the experiment should guarantee that the leftmost source cannot causally influence b or y in the time between t_emit and t_meas. A similar causal separation should hold between any other pair of source and measurement which do not share an arrow in figure 1. This ensures the validity of assumption (II).

Also, the sources should have been prepared in such a way that the correct quantum-mechanical description of the system will take the joint state of the sources to be a product state, and furthermore such that any correlation between them in a potential hidden variable description should be rendered very implausible. In other words, the experiment should try to guarantee that any hidden variable theory not satisfying (III') should be very unreasonable and contrived. This may be achieved, for example, by placing the sources at large spatial separation between each other and by using sources that employ different physical mechanisms. But of course, since the past light cones of the sources will always intersect, the requirement (III') can never be enforced. It will always be possible to explain all observations by, for example, a superdeterministic theory in which everything is predetermined since the beginning of the universe; compare with [Bel87, chapter 12].

As has already been noted in [BGP10], this discussion is completely analogous to the discussion of the validity of property (III): there exist hidden variable theories, such as superdeterminism, which do not allow free will and therefore evade the conclusion of Bell's theorem. However, these are generally so contrived that one cannot regard them as scientific theories of physics. Exactly the same applies to our assumption (III') in a suitably conducted experiment.

Now we imagine that many runs of such an experiment have been conducted and we are given the joint outcome statistics p(x,a,b,y). In the following, we work with the ideal case of infinite statistics, so that the outcome probabilities p(x,a,b,y) are known with perfect precision.

Then, due to the causal structure of the experiment, one should find that the outcome x is independent of y, since X and Y do not connect to a common source. Similarly, x should be independent of b; in fact, x should be independent of the pair (b,y). Similarly, y should be independent of the pair (x,a). Checking whether this is indeed the case amounts to a consistency check for the experiment.

More formally, these requirements mean that p(x,a,b,y) should be a correlation:

Definition 2.1. A correlation p in the P₄ scenario is a distribution p(x,a,b,y) whose marginals factorize as

$$\begin{equation} p(x,a,y) = p(x,a)p(y),\quad p(x,b,y) = p(x) p(b,y). \end{equation} \tag{ 2.1 }$$

Any of these two equations implies that p(x,y) = p(x)p(y). Upon using this, one finds that (2.1) is equivalent to p(a|x,y) = p(a|x) and p(b|x,y) = p(b|y) for all those values of x and y for which p(x) > 0 and p(y) > 0. Upon reinterpreting x and y as settings in a bipartite Bell scenario having outcomes a and b, these are the no-signaling equations. However, conceptually, (2.1) has nothing to do with the impossibility of communication between the parties: these cannot do anything else than apply their fixed measurement in each run of the experiment, which renders the very notion of communication meaningless.

We now ask under which conditions a given correlation p(x,a,b,y) is classical, i.e. consistent with the assumptions (I), (II) and (III'). What would it mean to have such a model? Due to (I), the state of the systems sent out by each of the three sources can be described in terms of a classical random variable; we will denote these 'hidden' variables by λ_XA, λ_AB, λ_BY, respectively, where the index specifies the source that the hidden variable models. For the precise definition of hidden variable, see appendix A.1.

Assumption (III') now means that the joint distribution of these hidden variables is a product distribution:

$$\begin{eqnarray*} &&p(\lambda_{{\tt X}{\tt A}},\lambda_{{\tt A}{\tt B}},\lambda_{{\tt B}{\tt Y}}) = p(\lambda_{{\tt X}{\tt A}}) p(\lambda_{{\tt A}{\tt B}}) p(\lambda_{{\tt B}{\tt Y}}) . \end{eqnarray*}$$

A sensible hidden variable model should also satisfy locality (II): each outcome should be a (deterministic or probabilistic) function of the hidden variables associated with the sources it interacts with and no others.

If such a hidden variable model exists for the correlation p, then we call p classical. A more precise statement is this:

Definition 2.2. [BRGP12] A correlation p(x,a,b,y) is classical in the P₄ scenario if and only if it can be written in the form

$$\begin{eqnarray} &&\fl p(x,a,b,y) = \int_{\lambda_{{\tt X}{\tt A}},\lambda_{{\tt A}{\tt B}},\lambda_{{\tt B}{\tt Y}}} p(x|\lambda_{{\tt X}{\tt A}}) p(\lambda_{{\tt X}{\tt A}}) p(a|\lambda_{{\tt X}{\tt A}},\lambda_{{\tt A}{\tt B}}) p(\lambda_{{\tt A}{\tt B}}) p(b|\lambda_{{\tt A}{\tt B}},\lambda_{{\tt B}{\tt Y}}) p(\lambda_{{\tt B}{\tt Y}}) p(y|\lambda_{{\tt B}{\tt Y}})\nonumber\\ \end{eqnarray} \tag{ 2.2 }$$

for some collection of (conditional) distributions

$$\begin{eqnarray*} &&\fl p(x|\lambda_{{\tt X}{\tt A}}),\quad p(\lambda_{{\tt X}{\tt A}}),\quad p(a|\lambda_{{\tt X}{\tt A}},\lambda_{{\tt A}{\tt B}}),\quad p(\lambda_{{\tt A}{\tt B}}),\quad p(b|\lambda_{{\tt A}{\tt B}},\lambda_{{\tt B}{\tt Y}}),\quad p(\lambda_{{\tt B}{\tt Y}}),\quad p(y|\lambda_{{\tt B}{\tt Y}}). \end{eqnarray*}$$

See appendix A.2 for an explanation of what these conditional distributions mean in case the hidden variables are not all discrete.

We take this to be a definition instead of a proposition or theorem because it is the first time that we have formalized the notion of the classical model in a mathematically rigorous way. The representation (2.2) can be informally derived from hypotheses (I), (II) and (III') as follows. Applying (I) and the definition of conditional probability gives

$$\begin{eqnarray*} &&\fl p(x,a,b,y) = \int_{\lambda_{{\tt X}{\tt A}},\lambda_{{\tt A}{\tt B}},\lambda_{{\tt B}{\tt Y}}} p(x,a,b,y|\lambda_{{\tt X}{\tt A}},\lambda_{{\tt A}{\tt B}},\lambda_{{\tt B}{\tt Y}}) p(\lambda_{{\tt X}{\tt A}},\lambda_{{\tt A}{\tt B}},\lambda_{{\tt B}{\tt Y}}). \end{eqnarray*}$$

By locality (II), the first factor in the integrand can be replaced by

$$\begin{eqnarray*} &&\fl p(a,b,x,y|\lambda_{{\tt X}{\tt A}},\lambda_{{\tt A}{\tt B}},\lambda_{{\tt B}{\tt Y}}) = p(x|\lambda_{{\tt X}{\tt A}}) p(a|\lambda_{{\tt X}{\tt A}},\lambda_{{\tt A}{\tt B}}) p(b|\lambda_{{\tt A}{\tt B}},\lambda_{{\tt B}{\tt Y}}) p(y|\lambda_{{\tt B}{\tt Y}}), \end{eqnarray*}$$

while independence of sources (III') guarantees that the second factor is equal to

$$\begin{eqnarray*} &&p(\lambda_{{\tt X}{\tt A}},\lambda_{{\tt A}{\tt B}},\lambda_{{\tt B}{\tt Y}}) = p(\lambda_{{\tt X}{\tt A}}) p(\lambda_{{\tt A}{\tt B}}) p(\lambda_{{\tt B}{\tt Y}}) , \end{eqnarray*}$$

and then (2.2) directly follows.

Remark 2.3. In the representation (2.2), it can be assumed without loss of generality that the four conditional distributions on the right-hand side are in fact deterministic, i.e. it can be assumed that the outcomes are functions

$$\begin{eqnarray*} &&x=x(\lambda_{{\tt X}{\tt A}}),\quad a=a(\lambda_{{\tt X}{\tt A}},\lambda_{{\tt A}{\tt B}}),\quad b=b(\lambda_{{\tt A}{\tt B}},\lambda_{{\tt B}{\tt Y}}),\quad y=y(\lambda_{{\tt B}{\tt Y}}). \end{eqnarray*}$$

In the case of discrete hidden variables, this can be seen as follows: if, for example, a is a probabilistic function of λ_XA and λ_AB, then the computation of this function can be regarded as the deterministic computation taking the values λ_XA,λ_AB and an additional random number r_A∈[0,1] as the input, calculating p(a|λ_XA,λ_AB) for each outcome a and then using r_A to determine which one of these finitely many outcomes occurs. But now we can redefine the hidden variable λ_XA to be the pair λ_XA' = (λ_XA,r_A) which contains the information about the original λ_XA as well as the additional random number r_A required in the computation; the party X will then also receive this new component of λ_XA', but can just ignore it. In this way, the function a(λ_XA',λ_AB) has become deterministic.

Upon applying this kind of hidden variable redefinition for each party, all the outcomes become deterministic functions of the hidden variables.

This reasoning not only applies to P₄, but in exactly the same way to any correlation scenario. We will make use of this in the proof of theorem 2.21. See appendix A.3 for a rigorous and general version of this argument.

It is also not difficult to define what quantum correlations are. Informally speaking, a quantum correlation is a correlation p(x,a,b,y) which can be modeled in terms of quantum resources: a bipartite quantum state for each source together with one measurement for each party operating jointly on all the systems received by that party. The Hilbert space dimension of the quantum systems can be arbitrary and will be infinite in general. We take the definition of quantum correlation to be sufficiently obvious that we need to go into detail here; see definition 3.16 for the technicalities.

The following theorem makes the connection to bipartite Bell scenarios. Its first part has also appeared in [BRGP12].

Theorem 2.4. 

• (1)  
  A correlation p(a,b,x,y) is classical in P₄ if and only if the associated conditional distribution p(a,b|x,y) is classical in the Bell scenario sense.
• (2)  
  A correlation p(a,b,x,y) is quantum in P₄ if and only if the associated conditional distribution p(a,b|x,y) is quantum in the Bell scenario sense.

Note that the use of conditional probabilities here, or in any other context, does not require any particular causal structure among the variables involved.

In forming p(a,b|x,y), it is implicitly assumed that all outcomes for x and y have strictly positive probability; this can always be achieved by redefining the set of outcomes to consist of only those values which occur with positive probability.

Thus, we can roughly summarize our present results as follows: by definition 2.2, a correlation p(a,b,x,y) can be interpreted in a conventional bipartite Bell scenario as a no-signaling box together with a specification of input distributions p(x) and p(y); and the correlation is classical (resp. quantum) if and only if the associated no-signaling box is classical (resp. quantum).

Proof of theorem 2.4. 

• (1)  
  Suppose that p is classical. Then

  $$\begin{eqnarray*} &&p(a,b|x,y) = \int_{\lambda_{{\tt A}{\tt B}}} p(a,b|x,y,\lambda_{{\tt A}{\tt B}}) p(\lambda_{{\tt A}{\tt B}}) . \end{eqnarray*}$$

  By the assumption (2.2), upon conditioning on λ_AB, the variables (a,x) are independent of the variables (b,y); therefore, p(a,b|x,y,λ_AB) = p(a|x,λ_AB)p(b|y,λ_AB), and

  $$\begin{equation} p(a,b|x,y) = \int_{\lambda_{{\tt A}{\tt B}}} p(a|x,\lambda_{{\tt A}{\tt B}}) p(b|y,\lambda_{{\tt A}{\tt B}}) p(\lambda_{{\tt A}{\tt B}}) . \end{equation} \tag{ 2.3 }$$

  This is the standard representation of the conditional probabilities obtained from local hidden variables in a bipartite Bell scenario. In particular, p(a,b|x,y) will have to satisfy all Bell inequalities.Conversely, we start from a correlation p(a,b,x,y) for which p(a,b|x,y) satisfies all Bell inequalities. This means in particular that there is a hidden variable λ such that

  $$\begin{eqnarray*} &&p(a,b|x,y) = \int_{\lambda} p(a|x,\lambda)p(b|y,\lambda)p(\lambda). \end{eqnarray*}$$

  Defining λ_AX = x, λ_BY = y and λ_AB = λ now yields a hidden variable model in the P₄ correlation scenario, i.e. the right-hand side of (2.3).
• (2)  
  Suppose that p(a,b,x,y) is quantum. Then one has one bipartite quantum state at each source and one quantum measurement at each party. We think of the measurement X as remotely preparing, via steering depending on the outcome x, a quantum system for A. In order to ease notation, we may assume, without loss of generality, the shared state to be pure and X's measurement to be projective. Furthermore, we may take X's projective measurement to be non-degenerate; going to a degenerate measurement amounts to a coarse-graining of X, which preserves the quantum-mechanical realizability of p(a,b,x,y). By these assumptions, the steered states for A are a family {|χ_x〉} of pure states. Using the same assumptions for Y, we end up with a family {|μ_y〉} of pure steered states for B.We now replace the source between X and A by a hidden variable defined to be λ_AX = x; then the new measurement protocol of X simply consists in announcing λ_AX's value as his outcome. The new protocol of A consists in receiving λ_AX, preparing the quantum state which X would have steered to given the outcome λ_AX and then proceeding with the measurement specified in the original protocol. This replacement preserves the overall correlation p(a,b,x,y). The same procedure can be applied in order to replace the source between Y and B by a hidden variable λ_BY and the measurement of Y by the protocol of simply announcing λ_BY's value as the outcome y.Let {A_a} (resp. {B_b}) denote the POVM employed by A (resp. B). Then

  $$\begin{equation} p(a,b|x,y) = \left( \langle\chi_x|\otimes \langle\psi| \otimes \langle\mu_y| \right) \left(A_a\otimes B_b\right) \left( |\chi_x\rangle\otimes |\psi\rangle \otimes |\mu_y\rangle \right) , \end{equation} \tag{ 2.4 }$$

  or, in graphical notation [Coe10],

  

  Here, the dashed line indicates how to consider $A^x_a\stackrel{\mathrm{def}}{=}\langle\chi_x|A_a|\chi_x\rangle$ as well as $B^y_b\stackrel{\mathrm{def}}{=}\langle\mu_y|B_b|\mu_y\rangle$ as operators acting on one part of the bipartite state |ψ〉. By $\sum _a A_a=\mathbbm {1}$ and normalization of |χ_x〉, it follows that $\sum _a A^x_a=\mathbbm {1}$ for all x; similarly, $\sum _y B^y_b=\mathbbm {1}$ for all y. By definition, (2.4) can then be written as

  $$\begin{equation} p(a,b|x,y) = \langle\psi|A^x_a\otimes B^y_b|\psi\rangle. \end{equation} \tag{ 2.5 }$$

  This is the desired quantum representation of p in a bipartite Bell scenario.Conversely, we start from a correlation p(a,b,x,y) of the form (2.5). As sources between A and X and between B and Y, we again take hidden variables defined by λ_XA = x and λ_BY = y; again, the protocol of X and Y is simply to announce the values of these variables as their outcome. Only the source between A and B is taken to be quantum and produces the bipartite state |ψ〉 of (2.5). The measurement protocol conducted by A is similar to the above: measure λ_XA, use the result as the choice of setting for the subsequent measurement on |ψ〉 and then announce both outcomes as the total outcome. This protocol can be interpreted as measuring a single POVM given by

  $$\begin{eqnarray*} &&|x\rangle\langle x|\otimes A^x_a \mapsto (x,a), \end{eqnarray*}$$

  where the left-hand side is a POVM element indexed by x and a, and the right-hand side denotes the resulting outcome announced by A. The analogous POVM is measured by B. By construction, this reproduces both the desired conditional distribution (2.5) and the marginal distribution p(x,y) = p(x)p(y), and therefore also the whole distribution p(x,a,b,y).   □

Corollary 2.5. 

• (1)  
  There exist non-classical quantum correlations in P₄.
• (2)  
  There exist non-quantum correlations in P₄.

Proof. This follows from the existence of Bell inequality violations and no-signaling violations of Tsirelson's bound [Cir80], respectively.   □

Remark 2.6. Due to theorem 2.4, we can regard P₄ as the analogue of a bipartite Bell scenario within our formalism. Nevertheless, there are several important differences. For one, the correlations live in completely different spaces: in a Bell scenario, one works in the space of conditional distributions p(a,b|x,y), which results in the convexity of the sets of classical and quantum correlations. In contrast, in the case of P₄, we work on the level of unconditional distributions p(x,a,b,y), which contain, from the point of view of Bell scenarios, also information about the distributions of settings p(x) and p(y). The sets of classical and quantum correlations in this formulation are not convex, which can be seen as follows: firstly, the set of classical correlations contains all the deterministic distributions p(x,a,b,y) in which all measurements always produce the same outcome. Secondly, any probability distribution p(x,a,b,y), and in particular every correlation, is a convex combination of deterministic ones. Thirdly, not every correlation is classical. Thus, not every convex combination of classical correlations is a classical correlation; for that matter, most convex combinations of classical correlations are not even correlations! The same reasoning shows that the set of quantum correlations is not convex. Analogous arguments apply to any other correlation scenario in which non-classical (resp. non-quantum) correlations exist.

We proceed to the second example of a correlation scenario. It is depicted in figure 2. With parties as vertices and sources as edges, this is the path graph P₅, and therefore we will speak of the P₅ scenario; the conceptual discussion we gave of the P₄ scenario applies here and to all the following examples just as well. We will see that the P₅ scenario relates to the 'bilocality' scenarios of Branciard, Gisin and Pironio [BGP10] (BGP scenarios) just as we have seen the P₄ scenario to relate standard bipartite Bell scenarios.

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Given the five-variable distribution p(x,a,b,c,z), under which conditions would we expect it to arise from a configuration like figure 2? In other words, what is the analogue of definition 2.1? Following a reasoning analogous to the P₄ case, the answer is straightforward:

Definition 2.7. A correlation p in the P₅ scenario is a distribution p(x,a,b,c,z) whose marginals factorize as

$$\begin{eqnarray*} &&\fl p(x,a,b,z) = p(x,a,b) p(z),\!\quad\!\! p(x,a,c,z) = p(x,a) p(c,z),\!\quad\!\! p(x,b,c,z) = p(x) p(b,c,z). \end{eqnarray*}$$

Any of these three equations implies p(x,z) = p(x)p(z). Upon using this, the first and third conditions can also be written as $\sum _c p(a,b,c|x,z)=\sum _c p(a,b,c|x)$ and $\sum _a p(a,b,c|x,z) = \sum _a p(a,b,c|z)$, respectively, which are formally identical to the no-signaling equations of the BGP scenario. Similarly, the second condition is then equivalent to p(a,c|x,z) = p(a|x)p(c|z), which is also formally identical to a consistency constraint in the BGP scenario [CF12].

The classicality assumptions (I), (II) and (III') now yield the following characterization:

Definition 2.8. A correlation p(x,a,b,c,z) is classical in the P₅ scenario if and only if it can be written in the form

$$\begin{eqnarray} &&\fl p(a,b ,c,x,z) = \int_{\lambda_{{\tt X}{\tt A}},\lambda_{{\tt A}{\tt B}},\lambda_{{\tt B}{\tt C}},\lambda_{{\tt C}{\tt Z}}}\nonumber\\ &&\fl \times p(x|\lambda_{{\tt X}{\tt A}}) p(\lambda_{{\tt X}{\tt A}}) p(a|\lambda_{{\tt X}{\tt A}},\lambda_{{\tt A}{\tt B}}) p(\lambda_{{\tt A}{\tt B}}) p(b|\lambda_{{\tt A}{\tt B}},\lambda_{{\tt B}{\tt C}}) p(\lambda_{{\tt B}{\tt C}}) p(c|\lambda_{{\tt B}{\tt C}},\lambda_{{\tt C}{\tt Z}}) p(\lambda_{{\tt C}{\tt Z}}) p(z|\lambda_{{\tt C}{\tt Z}})\nonumber\\ \end{eqnarray} \noindent \tag{ 2.6 }$$

for some collection of (conditional) distributions

$$\begin{eqnarray*} p(x|\lambda_{{\tt X}{\tt A}}&),\quad p(a|\lambda_{{\tt X}{\tt A}},\lambda_{{\tt A}{\tt B}}),\quad p(b|\lambda_{{\tt A}{\tt B}},\lambda_{{\tt B}{\tt C}}),\quad p(c|\lambda_{{\tt B}{\tt C}},\lambda_{{\tt C}{\tt Z}}),\quad p(z|\lambda_{{\tt C}{\tt Z}}).\\ & p(\lambda_{{\tt X}{\tt A}}),\quad\quad\:\: p(\lambda_{{\tt A}{\tt B}}),\quad\quad\:\: p(\lambda_{{\tt B}{\tt C}}),\quad\quad\:\: p(\lambda_{{\tt C}{\tt Z}}) . \end{eqnarray*}$$

As before, we regard the analogous definition of quantum correlations as straightforward and refer to 3.16 for the details.

Theorem 2.9. 

• (1)  
  A correlation p(a,b,c,x,z) is classical in P₅ if and only if the associated conditional distribution p(a,b,c|x,z) is classical in the BGP scenario sense.
• (2)  
  A correlation p(a,b,c,x,z) is quantum in P₅ if and only if the associated conditional distribution p(a,b,c|x,z) is quantum in the BGP scenario sense.

We abbreviate the proof a bit because it is completely analogous to the proof of theorem 2.4.

Proof. 

• (1)  
  Suppose that p is classical, i.e. can be written in the form (2.6). Then,

  $$\begin{eqnarray*} &&p(a,b,c|x,z) = \int_{\lambda_{{\tt A}{\tt B}},\lambda_{{\tt B}{\tt C}}} p(a,b,c|x,z,\lambda_{{\tt A}{\tt B}},\lambda_{{\tt B}{\tt C}}) p(\lambda_{{\tt A}{\tt B}}) p(\lambda_{{\tt B}{\tt C}}) . \end{eqnarray*}$$

  Upon conditioning on λ_AB and λ_BC, we have

  $$\begin{eqnarray*} &&p(a,b,c|x,z,\lambda_{{\tt A}{\tt B}},\lambda_{{\tt B}{\tt C}}) = p(a|x,\lambda_{{\tt A}{\tt B}}) p(b|\lambda_{{\tt A}{\tt B}},\lambda_{{\tt B}{\tt C}}) p(c|\lambda_{{\tt B}{\tt C}},z), \end{eqnarray*}$$

  and therefore,

  $$\begin{eqnarray*} &&p(a,b,c|x,z) = \int_{\lambda_{{\tt A}{\tt B}},\lambda_{{\tt B}{\tt C}}} p(a|x,\lambda_{{\tt A}{\tt B}}) p(b|\lambda_{{\tt A}{\tt B}},\lambda_{{\tt B}{\tt C}}) p(c|\lambda_{{\tt B}{\tt C}},z) p(\lambda_{{\tt A}{\tt B}}) p(\lambda_{{\tt B}{\tt C}}), \end{eqnarray*}$$

  which is the standard representation of a classical correlation in the BGP scenario [BGP10]. Conversely, upon starting from such a representation, one can again take λ_XA = x and λ_BY = y, and (2.6) also holds.
• (2)  
  We start with a quantum correlation p(a,b,c|x,z). Upon applying the same steering argument as in the proof of theorem 2.4, we may assume, in the obvious notation

  $$\begin{eqnarray} &&\fl p(a,b, c|x,z)\nonumber\\ &&\fl \quad = \left( \langle\chi_x|\otimes \langle\psi_{{\tt AB}}| \otimes \langle\psi_{{\tt BC}}| \otimes \langle\mu_y| \right) \left(A_a\otimes B_b\otimes C_c\right) \left( |\chi_x\rangle\otimes |\psi_{{\tt AB}}\rangle \otimes |\psi_{{\tt BC}}\rangle \otimes |\mu_y\rangle \right), \end{eqnarray} \tag{ 2.7 }$$

  

  Here, the dashed line indicates how to consider $A^x_a\stackrel{\mathrm{def}}{=}\langle\chi_x|A_a|\chi_x\rangle$, respectively $C^z_c\stackrel{\mathrm{def}}{=}\langle\zeta_z|C_c|\zeta_z\rangle$, as operators acting on one part of the bipartite state |ψ_AB〉, respectively |ψ_BC〉. By $\sum _a A_a=\mathbbm {1}$ and normalization of |χ_x〉, it follows that $\sum _a A^x_a=\mathbbm {1}$ for all x; similarly, $\sum _z C^z_c=\mathbbm {1}$ for all z. By definition, (2.4) can then be written as

  $$\begin{equation} p(a,b,c|x,z) = \left(\langle\psi_{{\tt AB}}| \otimes \langle\psi_{{\tt BC}}|\right)\left(A^x_a\otimes B_b\otimes C^z_c\right)\left(|\psi_{{\tt AB}}\rangle\otimes|\psi_{{\tt BC}}\rangle\right) . \end{equation} \tag{ 2.8 }$$

  This is the desired quantum representation of p in a BGP scenario.Conversely, we start from a correlation p(a,b,c,x,z) of the form (2.8). As the sources between A and X and between C and Z, we again take hidden variables defined by λ_XA = x and λ_CZ = z; again, the protocol of X and Z is simply to announce the values of these variables as their outcome. Only the sources between A and B and between B and C are taken to be quantum and produce, respectively, the bipartite states |ψ_AB〉 and |ψ_BC〉 of (2.8). The measurement protocol conducted by A is similar to the above: measure λ_XA, use the result as the choice of setting for the subsequent measurement on |ψ_AB〉 and then announce both outcomes as the total outcome. This protocol can be interpreted as measuring a single POVM given by

  $$\begin{eqnarray*} &&|x\rangle\langle x|\otimes A^x_a \mapsto (x,a), \end{eqnarray*}$$

  where the left-hand side is a POVM element indexed by x and a and the right-hand side denotes the resulting outcome announced by A. The analogous POVM is measured by C. By construction, this reproduces both the desired conditional distribution (2.8) and the marginal distribution p(x,z) = p(x)p(z), and therefore also the whole distribution p(x,a,b,c,z).   □

Due to this theorem, we can regard P₅ as the analogue of the BGP scenario within our formalism.

However, this is not yet the end of the story; our new point of view provides more than just a reformulation of familiar things. Let us imagine that party Z, in the P₅ scenario, has failed to collect data. Or that we disregard Z's measurement for some other reason. Then, we can regard the remaining parties X, A, B and C as forming a P₄ scenario and apply theorem 2.4 to the distribution p(x,a,b,c), with c now playing the role of y. In this way, the P₄ scenario is a natural subscenario of P₅. This is an observation that does not make sense in the standard formalism.

Our next example, first proposed in [BRGP12, section VI], is the correlation scenario illustrated in figure 3. It consists of three parties of which each two share a common source. We will see in corollary 3.10 that it is the smallest scenario in which non-classical correlations exist. In this subsection, we prove the existence of non-classical quantum correlations in C₃.

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We find this scenario especially appealing both due to its symmetry and due to its appearance in the study of inference of common ancestors [SA10]; see below. Since the main ideas concerning correlation scenarios should have already become clear in the last two examples, we now increase the pace a bit.

Definition 2.10. A correlation in C₃ is a distribution p(a,b,c). (It is not required to satisfy any particular constraint.)

This definition seems reasonable to us since, in general, one cannot expect any two of the variables (a,b,c) to be independent.

Example 2.11. If all three variables take values in {0,1}, then

$$\begin{eqnarray*} &&p(a=b=c=0)=\textstyle\frac{1}{2},\quad p(a=b=c=1)=\textstyle\frac{1}{2} \end{eqnarray*} \noindent$$

defines a correlation. We call this the perfect correlation since all three variables are random, but perfectly correlated.

Definition 2.12. A correlation p(a,b,c) is classical in the C₃ scenario if and only if it can be written in the form

$$\begin{equation} \fl p(a,b,c) = \int_{\lambda_{{\tt A}{\tt B}},\lambda_{{\tt B}{\tt C}},\lambda_{{\tt C}{\tt A}}} p(a|\lambda_{{\tt C}{\tt A}},\lambda_{{\tt A}{\tt B}}) p(b|\lambda_{{\tt A}{\tt B}},\lambda_{{\tt B}{\tt C}}) p(c|\lambda_{{\tt B}{\tt C}},\lambda_{{\tt C}{\tt A}}) p(\lambda_{{\tt A}{\tt B}}) p(\lambda_{{\tt B}{\tt C}}) p(\lambda_{{\tt C}{\tt A}}), \end{equation} \noindent \tag{ 2.9 }$$

for appropriate (conditional) distributions p(a|λ_CA, λ_AB), p(b|λ_AB, λ_BC), p(c|λ_BC,λ_CA), p(λ_AB), p(λ_BC), p(λ_CA).

Classical correlations in C₃ are monogamous in the following sense:

Proposition 2.13. Let p(a,b,c) be classical. If p(a = c) = 1, then a is independent of λ_AB.

Intuitively, this is because in order to create these perfect correlations between a and c, the outcome a cannot depend on λ_AB. In particular, this implies that there cannot be any correlations between a and b. Rigorously, the proof technique is the same as the one used in the proof of this inequality relating Shannon entropy and mutual information, which can be regarded as a monogamy inequality:

Lemma 2.14. Let p(a,b,c) be classical. Then

$$\begin{equation} I(a:b) + I(a:c) \leqslant H(a) . \end{equation} \noindent \tag{ 2.10 }$$

The interpretation of this is a kind of monogamy: a can share strong correlations with only b or c, but not with both. In particular, this inequality shows that the perfect correlation of example 2.11 is not classical.

Proof of proposition 2.13 and lemma 2.14. The present proof concerns the case that the hidden variables are discrete; see appendix A.4 for the general case.

Since a and b are conditionally independent given λ_AB, and similarly for a and c, the data processing inequality can be used to bound the left-hand side of (2.10) by

$$\begin{eqnarray*} &&\fl I(a:b) + I(a:c) \leqslant I(a:\lambda_{{\tt A}{\tt B}}) + I(a:\lambda_{{\tt C}{\tt A}}) = 2H(a) + H(\lambda_{{\tt A}{\tt B}}) + H(\lambda_{{\tt C}{\tt A}}) - H(a\lambda_{{\tt A}{\tt B}}) - H(a\lambda_{{\tt C}{\tt A}}). \end{eqnarray*}$$

Submodularity of Shannon entropy guarantees that H(aλ_AB) + H(aλ_CA) ⩾ H(a) + H(aλ_ABλ_CA), which can be applied here to obtain

$$\begin{eqnarray*} &&\fl I(a:b) + I(a:c) \leqslant H(a) + H(\lambda_{{\tt A}{\tt B}}) + H(\lambda_{{\tt C}{\tt A}}) - H(a\lambda_{{\tt A}{\tt B}}\lambda_{{\tt C}{\tt A}}) \leqslant H(a) + I(\lambda_{{\tt A}{\tt B}}:\lambda_{{\tt C}{\tt A}}) . \end{eqnarray*}$$

Since I(λ_AB:λ_CA) = 0, the claim of lemma 2.14 follows.

Concerning proposition 2.13, its assumption implies I(a:c) = H(a); the sequence of inequalities derived in this proof then guarantees that I(a:λ_AB) = 0, as was to be shown.   □

Corollary 2.15. Let p(a,b,c) be classical and f,g functions such that f(a) and g(c) are defined. If $p\left (f(a)=g(c)\right )=1$, then f(a) and g(c) are independent of λ_AB.

Proof. The assumptions imply that p(f(a),b,g(c)) is also a classical correlation in C₃. Now the claim follows from proposition 2.13.   □

Theorem 2.16. There exist non-classical quantum correlations in C₃.

Proof. We take |ψ〉 to be a bipartite two-qubit state which violates the CHSH inequality [CHSH69] with respect to measurements in the two bases {|ϕ₀,〉,|ϕ₁〉}, {|ω₀,〉,|ω₁〉}, which are the same for both parties.

The quantum correlations we consider in C₃ are obtained as follows. We take A and B to share |ψ〉, while A and C as well as B and C share either a maximally entangled state

$$\begin{equation} \frac{|00\rangle + |11\rangle}{\sqrt{2}} \end{equation} \tag{ 2.11 }$$

or, equivalently, a classically correlated mixed state

$$\begin{equation} \textstyle\frac{1}{2}\left(|00\rangle\langle 00| + |11\rangle\langle 11|\right) \end{equation} \tag{ 2.12 }$$

of two qubits. The purpose of these states is simple: it obsoletes free will in that A and B first measure the system they receive from the source shared with C in the {|0〉,|1〉}-basis and use the resulting outcome as a measurement setting on |ψ〉; this is similar to how the proofs of theorems 2.4 and 2.9 work. A and B announce the outcomes of both measurements as their total outcome. Similarly, we take C to apply the {|0〉,|1〉}-measurement on each of his qubits, so that C knows the measurement 'setting' used by A and B. He announces both of them as his outcome c. We regard the two bits announced by each party as the outcome of a single four-outcome measurement. The resulting correlation p(a,b,c) is a probability distribution on 4³ outcomes which does not depend on whether (2.11) or (2.12) is used.

More formally, we can define the measurements as follows: both A and B measure in the following basis and announce respective outcomes:

$$\begin{eqnarray*} &&|0\rangle\langle 0| \otimes |\phi_0\rangle\langle\phi_0| \mapsto (0,0),\quad |0\rangle\langle 0| \otimes |\phi_1\rangle\langle\phi_1| \mapsto (0,1), \\ &&|1\rangle\langle 1| \otimes |\omega_0\rangle\langle\omega_0| \mapsto (1,0),\quad |1\rangle\langle 1| \otimes |\omega_1\rangle\langle\omega_1| \mapsto (1,1), \end{eqnarray*}$$

while C simply measures both his qubits in the standard basis and announces both results.

It needs to be proven that these correlations are non-classical in C₃. This is guaranteed by the monogamy property of corollary 2.15: since C has perfect information about the 'settings' employed by A and B, these 'settings' are necessarily independent of λ_AB. This simulates the free will ('λ-independence') required for a standard Bell test to apply. The hidden variable λ_AB in any potential classical model would therefore have to function exactly like a hidden variable in a standard Bell scenario, which is guaranteed to be impossible due to the Bell inequality violation.   □

These arguments apply in the same way to a construction of a non-classical quantum correlations from a Bell inequality violation in any bipartite Bell scenario. Although this class of examples proves the theorem, we do not find such examples satisfying since they are again based on a Bell test in the standard sense. It is difficult to regard them as entirely new kinds of non-classicality. Nevertheless, we find it surprising that non-classical quantum correlations exist in C₃ even in the case when only one of the sources produces entanglement. We had not expected this at all when we started thinking about the C₃ scenario.

Problem 2.17. Find an example of non-classical quantum correlations in C₃ together with a proof of its non-classicality which does not hinge on Bell's theorem.

In order to find more examples of non-classical quantum correlations in C₃, it would be helpful to have inequalities bounding the set of classical correlations and violated by some quantum correlations. Unfortunately, our proof of theorem 2.16 does provide inequalities only conditional on the perfect correlations required between A and C and between B and C. However, we expect that our idea can be used to derive unconditional inequalities if one knows bounds on the maximal classical value of a Bell inequality as a function of the correlation between the measurement settings and the hidden variable. We expect that such bounds can be derived by considerations similar to those of [BG11] and/or [CR12] or may even be implicitly contained in these works.

Before moving on to the next example of a correlation scenario, we return briefly to the work of Steudel and Ay [SA10] on the inference of common ancestors. So, what is a 'common ancestor'?

If one makes certain (say, real-world macroscopic) observations a and b, repeats them many times in order to gather statistics and detects a correlation between these, then one can conclude that a and b need to have a common ancestor: there needs to be some quantity or property λ such that both a and b depend on λ, and λ is not deterministic; this includes the possibilities λ = a and λ = b as degenerate cases. This λ is a common ancestor of a and b in the sense of a pre-existing condition on which both a and b depend.

This is Reichenbach's principle of common cause [RR56, Ebe08]; it is based on the premise that good models of the world adhere to assumption (III') in the sense that a good model should predict a and b to be independent, unless there is some previously occurring event causally connected to both variables, i.e. a common ancestor.

Now what if one does the same for three observations a, b and c? How can one conclude that there is a common ancestor λ on which all three of them depend? Or for any number $n\in \mathbb {N}$ of observations? Among other things, it has been shown in [SA10] that the entropy of the common ancestors is lower bounded by a certain linear combination of the joint entropy and the single-variable entropies; therefore, strict positivity of that linear combination witnesses the necessity of a common ancestor. See also [Ay09] for related work providing a generalization and quantification of Reichenbach's principle.

Let us consider the particular case of n = 3 variables. Then the main observation is that the causal structure of the C₃ scenario is precisely the null hypothesis: if no common ancestor exists, then there can at most be common ancestors for every pair of variables, but not for all three variables together. Therefore, if no common ancestor exists, then p(a,b,c) is a classical correlation in the C₃ scenario. Figure 3 coincides with [SA10, figure 1]. The results of Steudel and Ay for this particular case state that if p(a,b,c) is classical, then

$$\begin{equation} H(a) + H(b) + H(c) \leqslant 2 H(abc), \end{equation} \tag{ 2.13 }$$

where H(abc) is the entropy of the joint distribution. Intuitively, if this inequality is violated, then the joint entropy is relatively small in comparison to the single-variable entropies, implying the existence of strong correlations between the variables and therefore of a common ancestor.

Writing out our inequality (2.10) in terms of joint entropies, one obtains

$$\begin{eqnarray*} &&H(a) + H(b) + H(c) \leqslant H(ab) + H(ac), \end{eqnarray*}$$

which is an improvement over (2.13) since the right-hand side is bounded by 2H(abc). In particular, a violation

$$\begin{equation} H(a) + H(b) + H(c) > H(ab) + H(ac) \end{equation} \noindent \tag{ 2.14 }$$

successfully witnesses the necessity of a common ancestor in strictly more cases than (1.1).

In the case of n > 3 variables, it is still true that the null hypothesis of non-existence of a common cause corresponds to classicality in the appropriate correlation scenario: for the necessity of a common ancestor of some (k + 1)-element subset of n variables, the null hypothesis is that at most each k-tuple has common ancestor(s). Roughly speaking, it is enough to consider only those ancestors which themselves do not have any parents: all the randomness creation can be delegated to those without changing the distribution of the observed variables, while all other nodes then carry out deterministic information processing; compare remark 2.3 and appendix A.3. Then each such initial node can be replaced by a source connecting to at most k observed variables, and the deterministic information processing can as well be delegated to the measurement nodes, again without changing the distribution of outcomes. Therefore, this corresponds to a classical correlation in the correlation scenario defined by n measurements in which each k-tuple of measurements is allowed to share a source. Conversely, it is clear that every such classical correlation represents a joint distribution of n variables which can be modeled without a common ancestor for any (k + 1)-tuple. To summarize, the given joint distribution is a classical correlation in this scenario if and only if the joint distribution can be obtained from a Bayesian network in which no (k + 1)-element subset of the given variables has a common ancestor.

However, at the moment we do not know how to generalize our inequality (2.14) to these cases, and refer once again to [SA10] for the current state of the art.

Another interesting correlation scenario is the square scenario illustrated in figure 4. In this case, the underlying graph is C₄, the cycle graph on four vertices. It can be regarded as P₄ (figure 1) equipped with an additional source between X and Y. Along the lines of theorem 2.4, this would suggest that correlations p(a,b,x,y) in C₄ should be interpretable as arising from a Bell scenario together with correlations between the measurement settings. However, the forthcoming proposition 2.20 will show that this intuition is false.

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Definition 2.18. A correlation p in the C₄ scenario is a distribution p(a,b,x,y) whose marginals factorize as

$$\begin{eqnarray*} &&p(a,y) = p(a)p(y),\quad p(b,x) = p(b) p(x). \end{eqnarray*}$$

Definition 2.19. A correlation p(a,b,x,y) is classical in the C₄ scenario if and only if it can be written in the form

$$\begin{eqnarray} &&\fl p(a,b,x,y) = \int_{\lambda_{{\tt A}{\tt B}},\lambda_{{\tt B}{\tt Y}},\lambda_{{\tt Y}{\tt X}},\lambda_{{\tt X}{\tt A}}} p(a|\lambda_{{\tt X}{\tt A}},\lambda_{{\tt A}{\tt B}}) p(b|\lambda_{{\tt A}{\tt B}},\lambda_{{\tt B}{\tt Y}}) p(y|\lambda_{{\tt B}{\tt Y}},\lambda_{{\tt Y}{\tt X}}) p(x|\lambda_{{\tt Y}{\tt X}},\lambda_{{\tt X}{\tt A}}) p(\lambda_{{\tt A}{\tt B}})\nonumber\\ &&\times p(\lambda_{{\tt B}{\tt Y}}) p(\lambda_{{\tt Y}{\tt X}}) p(\lambda_{{\tt X}{\tt A}}) \end{eqnarray} \tag{ 2.15 }$$

for appropriate (conditional) distributions p(a|λ_XA,λ_AB), p(b|λ_AB,λ_BY), p(y|λ_BY,λ_YX), p(x|λ_YX,λ_XA), p(λ_AB), p(λ_BY), p(λ_YX) and p(λ_XA).

Proposition 2.20. There are classical correlations p(a,b,x,y) in the C₄ scenario such that the associated conditional distribution p(a,b|x,y) is signaling.

Proof. We start from any classical correlation p₀(x,a,b,y) in the P₄ scenario. In particular, by theorem 2.4, p₀(a,b|x,y) does not violate any Bell inequality. We now apply the relabeling

$$\begin{eqnarray*} &&a\longleftrightarrow x,\quad b\longleftrightarrow y \end{eqnarray*}$$

and take the resulting correlation to be p(a,b,x,y). By construction, the resulting correlation p(a,b,x,y) is classical in C₄. By construction, p(x,y|a,b) does not violate a Bell inequality. The conditional distribution

$$\begin{eqnarray*} &&p(a,b|x,y) = p(x,y|a,b) \cdot\frac{p(x,y)}{p(a)p(b)} \end{eqnarray*}$$

then is precisely the time reversal, in the sense of Coecke and Lal [CL12], of the classical no-signaling box p(x,y|a,b) with respect to p(a,b) = p(a)p(b) as its distribution of settings. It was shown in [CL12] that there exist p₀(a,b|x,y) for which this time reversal is necessarily signaling.   □

In particular, proposition 2.20 shows that the conditional distribution p(a,b|x,y) associated with a classical correlation p(a,b,x,y) in C₄ may violate Bell inequalities.

Any classical (resp. quantum) correlation in a bipartite Bell scenario can be turned into a classical (resp. quantum) correlation in the C₄ scenario in four different ways: one of the four edges of C₄ needs to be designated as the Bell scenario's source, while the source corresponding to the opposite edge does nothing at all.

Theorem 2.21. There exist non-classical correlations in C₄.

Proof. We define the correlation p(a,b,x,y) by taking p(a,b|x,y) to be a Popescu–Rohrlich box [PR94] and p(x,y) to be the uniform distribution. More concretely, all four outcomes are bits a,b,x,y∈{0,1} with the table of joint probabilities given by

We now use a Hardy-type [Har93] argument in order to show that this correlation is not classical. For the sake of contradiction, let us assume p(a,b,x,y) to be classical with hidden variable distributions p(λ_AB), p(λ_BY), p(λ_YX), p(λ_XA); thanks to remark 2.3, we can take the four outcomes to be deterministic functions of the hidden variables. We start by considering the case of discrete hidden variables. Then, there has to be a hidden variable combination

$$\begin{eqnarray*} &&(\lambda_{{\tt A}{\tt B}},\lambda_{{\tt B}{\tt Y}},\lambda_{{\tt Y}{\tt X}},\lambda_{{\tt X}{\tt A}}) = (\ell_{{\tt A}{\tt B}},\ell_{{\tt B}{\tt Y}},\ell_{{\tt Y}{\tt X}},\ell_{{\tt X}{\tt A}}) \end{eqnarray*}$$

occurring with positive probability, which produces the outcome (a,b,x,y) = (0,0,0,0); similarly, there has to be a hidden variable combination

$$\begin{eqnarray*} &&(\lambda_{{\tt A}{\tt B}},\lambda_{{\tt B}{\tt Y}},\lambda_{{\tt Y}{\tt X}},\lambda_{{\tt X}{\tt A}}) = (\kappa_{{\tt A}{\tt B}},\kappa_{{\tt B}{\tt Y}},\kappa_{{\tt Y}{\tt X}},\kappa_{{\tt X}{\tt A}}), \end{eqnarray*}$$

occurring with positive probability, which produces the outcome (a,b,x,y) = (1,0,1,1). Then, the independence of sources guarantees that the hidden variable combination

$$\begin{eqnarray*} &&(\lambda_{{\tt A}{\tt B}},\lambda_{{\tt B}{\tt Y}},\lambda_{{\tt Y}{\tt X}},\lambda_{{\tt X}{\tt A}}) = (\kappa_{{\tt A}{\tt B}},\kappa_{{\tt B}{\tt Y}},\ell_{{\tt Y}{\tt X}},\kappa_{{\tt X}{\tt A}}) \end{eqnarray*}$$

also has positive probability. Because of locality and determinism, it necessarily produces an outcome $(1,0,\hat {x},\hat {y})$; by (2.16), $\hat {x}=\hat {y}=1$. Likewise, the hidden variable combination (ℓ_AB,ℓ_BY,ℓ_YX,κ_XA) has positive probability, and produces some outcome of the form (a',0,1,0). Thanks to the form of (2.16), necessarily a' = 0. Similarly, the hidden variable combination (ℓ_AB,κ_BY,ℓ_YX,ℓ_XA) must give the outcome (0,0,0,1). However, the hidden variable combination (ℓ_AB,κ_BY,ℓ_YX,κ_XA) then gives the outcome (0,0,1,1) with positive probability, a contradiction with (2.16).

In the case of general (non-discrete) hidden variables, the same proof idea can be used, although the technical details are quite involved; see appendix A.5.   □

Problem 2.22. 

• (1)  
  Are there non-classical quantum correlations in C₄?
• (2)  
  Is there a simple way to characterize the classical correlations in C₄?

So far, we have only considered example scenarios in which each source produces a pair of systems which it distributes among two parties. However, it is quite common to consider Bell scenarios involving a source that distributes systems among several parties [GHZ90]. The same can be easily done in our framework; an example scenario of this type is illustrated in figure 5.

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More generally, we want to consider the family of multiarm scenarios A_k indexed by the number of arms $k\in \mathbb {N}$; each arm consists of two parties sharing a bipartite source, and there is one k-partite source shared by all the parties obtained by choosing one party in each arm. Figure 5 represents the case k = 5, while k = 2 is the P₄ scenario of figure 1.

The following considerations are immediate generalizations of those of the P₄ scenario. Just as P₄ corresponds to a bipartite Bell scenario, A_k corresponds to a k-partite Bell scenario. We use 'hat' notations like $a_1,\ldots ,\hat {a}_i,\ldots ,a_k$ as short for a₁,...,a_i−1,a_i+1,...,a_k.

Definition 2.23. A correlation in the A_k scenario is a probability distribution p(a₁,...,a_k,x₁...,x_k) whose marginals factorize as

$$\begin{eqnarray} &&\fl p(a_1,\ldots,\hat{a}_i,\ldots,a_k,x_1,\ldots,x_k) = p(x_i) p(a_1,\ldots,\hat{a}_i,\ldots,a_k,x_1,\ldots,\hat{x}_i,\ldots,x_k), \quad \forall i.\nonumber \end{eqnarray} \tag{ 2.17 }$$

Repeated application of (2.17) implies p(x₁,...,x_n) = p(x₁)···p(x_n). Upon using this, and considering only those values x_i for which p(x_i) > 0, the condition (2.17) becomes equivalent to the equations

$$\begin{eqnarray*} &&p(a_1,\ldots,\hat{a}_i,\ldots,a_n|x_1,\ldots,x_n) = p(a_1,\ldots,\hat{a}_i,\ldots,a_n|x_1,\ldots,\hat{x}_i,\ldots,x_n) , \end{eqnarray*} \noindent$$

which are formally identical to the no-signaling equations in a k-partite Bell scenario.

Theorem 2.24. 

• (1)  
  A correlation p is classical in A_k if and only if the associated conditional distribution p(a₁,...,a_k|x₁,...,x_k) is classical in the Bell scenario sense.
• (2)  
  A correlation p(a,b,x,y) is quantum in A_k if and only if the associated conditional distribution p(a₁,...,a_k|x₁,...,x_k) is quantum in the Bell scenario sense.

Proof. Analogous to the proof of theorem 2.4.   □

The literature shows examples of discrete hidden variables and continuous hidden variables. In standard Bell scenarios, Carathéodory's theorem guarantees that considering discrete hidden variables is enough; unfortunately, we do not know whether this also holds for our case (problem 3.11). Therefore, we should allow hidden variables that are as general as possible and require a definition that not only comprises discrete and continuous hidden variables, but also allows intermediate possibilities and even hidden variables with more than continuously many values.

Since the only successful general theory of (classical) randomness is the one based on the Kolmogorov axioms for probability measures and probability spaces, this is what seems to us to be the only reasonable general definition of a hidden variable:

Definition. A hidden variable is a probability space $(\Omega ,\mathcal {E},P)$.

We think of the actual value of the hidden variable to be a random element λ∈Ω with distribution P. This is the most general kind of classical hidden variable we can imagine. It comprises both discrete and continuous variables as special cases as well as everything else, for example hidden variables with so many values that Ω has cardinality greater than the continuum.

Definitions 2.2, 2.8, 2.12, 2.19 and 3.5 talk about outcome distributions conditional on one or several hidden variables. What does a conditional distribution, such as p(a|λ), mean when λ is not discrete?

There are several equivalent ways to answer this question. We have chosen the following one which is convenient in that it is partly formulated in terms familiar from quantum theory.

Definition. Let $L^\infty (\Omega ,\mathcal {E},P)$ be the von Neumann algebra associated with $(\Omega ,\mathcal {E},P)$. A distribution of a conditional on λ∈Ω is an assignment of some positive operator $\mathcal {O}_a^*=\mathcal {O}_a\in L^\infty (\Omega ,\mathcal {E},P)$, $\mathcal {O}_a\geqslant 0$, to every a such that $\sum _{a} \mathcal {O}_a = \mathbbm {1}$.

The attentive reader will have noticed that this is nothing but a POVM in $L^\infty (\Omega ,\mathcal {E},P)$ indexed by a. Roughly speaking, each $\mathcal {O}_a$ is a real-valued function on Ω whose values $\mathcal {O}_a(\lambda )$ represent the conditional probabilities p(a|λ). For finite Ω with $\mathcal {E}=2^\Omega$ and P(λ) > 0 for every λ∈Ω, this intuition is exact; in general however, it has to be kept in mind that $\mathcal {O}_a$ is not a single function, but rather a whole equivalence class of functions, such that expressions such as

$$\begin{eqnarray*} &&p(a) = \int_{\lambda} \mathcal{O}_a(\lambda) P(\lambda), \end{eqnarray*}$$

are well defined in the sense that the value of the integral is independent of the choice of representative.

In general, a measurement a will depend on several hidden variables given by probability spaces $(\Omega _1,\mathcal {E}_1,P_1),\:\ldots ,\:(\Omega _n,\mathcal {E}_n,P_n)$. In this case, $\mathcal {O}$ should be a POVM in the von Neumann algebra of the product probability space $\left (\prod _i\Omega _i,\prod _i\mathcal {E}_i,\prod _i P_i\right )$.

We now state definition 3.5 again in the present language.

Definition. Let G = (V,E) be a correlation scenario. A correlation p(v₁,...,v_n) in G is classical if the following data exist:

• (1)  
  for every e∈E, a hidden variable λ_e given in terms of a probability space $(\Omega _e,\mathcal {E}_e,P_e)$;
• (2)  
  conditional probabilities $\mathcal {O}_a\in L^\infty (\Omega _{\Lambda _i},\mathcal {E}_{\Lambda _i},P_{\Lambda _i})$ where Λ_i = {λ_e;v_i∈e} is the collection of hidden variables associated with all the sources connected to v_i, and $(\Omega _{\Lambda _i},\mathcal {E}_{\Lambda _i},P_{\Lambda _i})$ is the corresponding product probability space;

such that

$$\begin{equation} p(v_1,\ldots,v_n) = \int_{\{\lambda_e;\: e\in E\}} \prod_{v_i\in V} \mathcal{O}_{v_i}(\Lambda_i) \prod_{e\in E} \mathrm{d}P(\lambda_e). \end{equation} \noindent \tag{ A.1 }$$

In particular, this clarifies also the definitions of classical correlation in our example scenarios, definitions 2.2, 2.8, 2.12 and 2.19

We have outlined in remark 2.3 why the conditional distributions $\mathcal {O}_a$ as used above can in fact be taken to be deterministic. In our present picture, determinism means $\mathcal {O}_a^2=\mathcal {O}_2$, i.e. that $\mathcal {O}_a$ is a projection. This is equivalent to $\mathcal {O}_a(\lambda )\in \{0,1\}$ for almost all λ∈Ω which corresponds to determinism in the form p(a|λ)∈{0,1}.

We now turn the intuitive argument of remark 2.3 into a rigorous proof sketch.

Proposition. Let G = (V,E) be a correlation scenario. If p is classical, then there exists a classical model for p in which all $\mathcal {O}_{v_i}$ are projections.

Proof. We show how to replace the $\mathcal {O}_{w}$'s by a projection for some fixed w∈V ; the claim then follows from applying this procedure to every vertex w∈V . We start by choosing a source e∈E which connects to v₁ and replace the given probability space $(\Omega _e,\mathcal {E}_e,P_e)$ by $\Omega'_e\stackrel{\mathrm{def}}{=} \Omega_e\times[0,1]$, which we take to be equipped with the product σ-algebra $\mathcal {E}'_e$ and the product measure P'_e, where [0,1] carries the Lebesgue σ-algebra and measure; the second factor in this product represents the additional random number mentioned in remark 2.3. We enumerate the possible outcomes as w∈{1,...,d} for some $d\in \mathbb {N}$, and define

$$\begin{eqnarray*} \fl \mathcal{O}'_{w} \;:\; \Omega_e\times [0,1] \to \{0,1\},\quad (\lambda,x) \mapsto \left\{ \begin{array}{@{}ll@{}} 1, & \textrm{if } \sum_{w'=1}^{w-1} \mathcal{O}_{w'}(\lambda) \leqslant x < \sum_{w'=1}^{w} \mathcal{O}_{w'}(\lambda), \\ 0, & \textrm{otherwise}, \end{array}\right. \end{eqnarray*}$$

which is easily seen to represent a projection in $L^\infty (\Omega '_e,\mathcal {E}'_e,P'_e)$. The requirement $\sum _{w=1}^d \mathcal {O}'_w = \mathbbm {1}$ holds by construction in $L^\infty (\Omega '_e,\mathcal {E}'_e,P'_e)$, i.e. up to a set of measure zero.

All $\mathcal {O}_{v_i}$ with v_i ≠ w connecting to e we take to operate as before in the sense that we replace them by $\mathcal {O}'_{v_i}(\lambda ,x) = \mathcal {O}_{v_i}(\lambda )$; all other sources ≠e and all measurements not connected to e remain completely unchanged.

We leave it to the reader to verify that these replacements preserve the correlation.   □

We follow essentially the same lines as in the discrete-variable proof of the main text. Since we do not know of a formulation of the data processing inequality for (relative) Shannon entropy on arbitrary probability spaces, and similarly for submodularity of entropy, we make our own definitions and derive our inequalities in analogy with the discrete case. We start with the first argument involving the data processing inequality. In order to obtain finite quantities, we need to work with conditional entropies, in which the hidden variables appear only as conditioning variables. For the sake of illustration, we start with the discrete-variable case in which

$$\begin{eqnarray*} &&H(a|\lambda_{{\tt A}{\tt B}}) = \sum_{a,\lambda_{{\tt A}{\tt B}}} f\left(p(a|\lambda_{{\tt A}{\tt B}})\right) p(\lambda_{{\tt A}{\tt B}}), \end{eqnarray*}$$

where we abbreviated f(x) = −x·logx, with $f(0)\stackrel{\mathrm{def}}{=} 0$ as usual. Thanks to the conditional independence p(a|λ_AB) = p(a|λ_AB,b) and concavity of f,

$$\begin{eqnarray*} &&\fl H(a|\lambda_{{\tt A}{\tt B}}) = \sum_{a,b,\lambda_{{\tt A}{\tt B}}} f\left(p(a|\lambda_{{\tt A}{\tt B}})\right) p(\lambda_{{\tt A}{\tt B}})p(b|\lambda_{{\tt A}{\tt B}}) = \sum_{a,b} \sum_{\lambda_{{\tt A}{\tt B}}} f\left(p(a|\lambda_{{\tt A}{\tt B}},b)\right) p(\lambda_{{\tt A}{\tt B}}|b) p(b) \\ &&\leqslant \sum_{a,b} f\left(\sum_{\lambda_{{\tt A}{\tt B}}} p(a|\lambda_{{\tt A}{\tt B}},b) p(\lambda_{{\tt A}{\tt B}}|b)\right) p(b) = \sum_{a,b} f\left( p(a|b)\right) p(b) = H(a|b). \end{eqnarray*}$$

We now emulate this estimate in the general case by defining

$$\begin{eqnarray*} &&H(a|\lambda_{{\tt A}{\tt B}}) \stackrel{\mathrm{def}}{=} \sum_a \int_{\lambda_{{\tt A}{\tt B}}} f\left( \mathcal{O}_a(\lambda_{{\tt A}{\tt B}}) \right) \mathrm{d}P_{{\tt A}{\tt B}}(\lambda_{{\tt A}{\tt B}}) \end{eqnarray*}$$

and noting that this is well defined, thanks to $\mathcal {O}_a(\lambda _{{\tt A}{\tt B}})\in [0,1]$ a.s., and coincides with the standard definition in the discrete case. We rewrite this as

$$\begin{eqnarray*} &&H(a|\lambda_{{\tt A}{\tt B}}) = \sum_{a,b} \int_{\lambda_{{\tt A}{\tt B}}} f\left( \mathcal{O}_a(\lambda_{{\tt A}{\tt B}}) \right) \frac{\mathcal{O}_b(\lambda_{{\tt A}{\tt B}})\mathrm{ d}P_{{\tt A}{\tt B}}(\lambda_{{\tt A}{\tt B}})}{p(b)} p(b) . \end{eqnarray*} \noindent$$

Now for p(b) > 0, the fraction in the integrand is again a measure on $(\Omega _{{\tt A}{\tt B}},\mathcal {E}_{{\tt A}{\tt B}})$, and Jensen's inequality gives

$$\begin{eqnarray*} &&H(a|\lambda_{{\tt A}{\tt B}}) \leqslant \sum_{a,b} f\left( \int_{\lambda_{{\tt A}{\tt B}}} \mathcal{O}_a(\lambda_{{\tt A}{\tt B}}) \cdot \frac{\mathcal{O}_b(\lambda_{{\tt A}{\tt B}})\mathrm{ d}P_{{\tt A}{\tt B}}(\lambda_{{\tt A}{\tt B}})}{p(b)} \right) p(b). \end{eqnarray*}\noindent$$

Since $p(a,b) = \int _{\lambda _{{\tt A}{\tt B}}} \mathcal {O}_a(\lambda _{{\tt A}{\tt B}})\mathcal {O}_b(\lambda _{{\tt A}{\tt B}})\mathrm { d}P_{{\tt A}{\tt B}}$, the integral inside f evaluates to p(a|b), so that

$$\begin{equation} H(a|\lambda_{{\tt A}{\tt B}}) \leqslant \sum_{a,b} f\left( p(a|b) \right) p(b) = H(a|b), \end{equation} \noindent \tag{ A.2 }$$

which is the data processing inequality we wanted to prove.

We now make the usual estimates known from proofs of non-negativity of conditional mutual information or non-negativity of the Kullback–Leibler divergence [CT06, theorem 8.6.1],

$$\begin{eqnarray*} &&\fl H(a|\lambda_{{\tt A}{\tt B}}) + H(a|\lambda_{{\tt A}{\tt C}}) - H(a) - H(a|\lambda_{{\tt A}{\tt B}}\lambda_{{\tt A}{\tt C}}) \\ &&= \sum_a \left[ \int_{\lambda_{{\tt A}{\tt B}},\lambda_{{\tt A}{\tt C}}} \mathcal{O}_a(\lambda_{{\tt A}{\tt B}},\lambda_{{\tt A}{\tt C}}) \bigg( - \log(\mathcal{O}_a(\lambda_{{\tt A}{\tt B}})) - \log(\mathcal{O}_a(\lambda_{{\tt A}{\tt C}}))\right.\\ &&\left. + \log(p(a)) + \log(\mathcal{O}_a(\lambda_{{\tt A}{\tt B}},\lambda_{{\tt A}{\tt C}})) \bigg)\, \mathrm{d}P_{{\tt A}{\tt B}}\, dP_{{\tt A}{\tt C}} \right] \\ &&= - \sum_a \left[ \int_{\lambda_{{\tt A}{\tt B}},\lambda_{{\tt A}{\tt C}}} \mathcal{O}_a(\lambda_{{\tt A}{\tt B}},\lambda_{{\tt A}{\tt C}}) \log \left( \frac{\mathcal{O}_a(\lambda_{{\tt A}{\tt B}})\mathcal{O}_a(\lambda_{{\tt A}{\tt C}})}{p(a)\mathcal{O}_a(\lambda_{{\tt A}{\tt B}},\lambda_{{\tt A}{\tt C}})} \right) \, \mathrm{d}P_{{\tt A}{\tt B}}\, \mathrm{d}P_{{\tt A}{\tt C}} \right] \\ &&\geqslant - \log \left[ \sum_a \int_{\lambda_{{\tt A}{\tt B}},\lambda_{{\tt A}{\tt C}}} \mathcal{O}_a(\lambda_{{\tt A}{\tt B}},\lambda_{{\tt A}{\tt C}}) \cdot \frac{\mathcal{O}_a(\lambda_{{\tt A}{\tt B}})\mathcal{O}_a(\lambda_{{\tt A}{\tt C}})}{p(a)\mathcal{O}_a(\lambda_{{\tt A}{\tt B}},\lambda_{{\tt A}{\tt C}})} \mathrm{d}P_{{\tt A}{\tt B}}\, \mathrm{d}P_{{\tt A}{\tt C}} \right]\\ &&= - \log\left[ \sum_a \frac{p(a)p(a)}{p(a)} \right] = 0. \end{eqnarray*}$$

Since H(a|λ_ABλ_AC) is defined as the integral of an a.s. non-negative function, it is itself nonnegative, and therefore

$$\begin{equation} H(a|\lambda_{{\tt A}{\tt B}}) + H(a|\lambda_{{\tt A}{\tt C}}) \geqslant H(a) . \end{equation} \noindent \tag{ A.3 }$$

Piecing finally the two ingredients (A.2) and (A.3) together, we find that

$$\begin{eqnarray*} &&\fl I(a:b) + I(a:c) = 2H(a) - H(a|b) - H(a|c) \leqslant 2H(a) - H(a|\lambda_{{\tt A}{\tt B}}) - H(a|\lambda_{{\tt A}{\tt C}}) \leqslant H(a), \end{eqnarray*}$$

as was to be shown.

In the discrete-variable case, we started with the assumption that the measurements were deterministic and noted that if a certain combination of outcomes has positive probability, then there has to be a combination of hidden variable values, each occurring with positive probability, which produces that outcome combination.

This reasoning needs to be modified in order to apply in the general case; when dealing with non-atomic probability spaces, no single hidden variable combination has positive probability. It is therefore necessary to consider combinations of sets of hidden variable values, which is unfortunately somewhat technical.

Lemma. Let $(\Omega _1,\mathcal {E}_1,P_1),\:\ldots ,\:(\Omega _n,\mathcal {E}_n,P_n)$ be probability spaces and let $\Omega =\prod _{i=1}^n\Omega _i$ be equipped with the product σ-algebra $\mathcal {E}=\sigma \left (\prod _{i=1}^n \mathcal {E}_i\right )$ and the product measure $P=\prod _{i=1}^n P_i$, so that $(\Omega ,\mathcal {E},P)$ is a probability space.

Then, for a measurable function f:Ω → {0,1} with P(f = 1) > 0 and any ε > 0, there exist measurable subsets Ξ_i⊆Ω_i, with P_i(Ξ_i) > 0, such that

$$\begin{eqnarray*} &&P\left(\,f=1\,|\,\Xi_1\times\cdots\times\Xi_n\,\right) > 1 - \varepsilon. \end{eqnarray*}$$

Proof.  This lemma can be reformulated as saying that if Θ⊆Ω has positive measure, then there exist Ξ_i⊆Θ of positive measure such that $P(\Theta |\prod _{i=1}^n \Xi _i) > 1-\varepsilon$.

We start to prove this reformulation by noting that the collection of sets which are finite disjoint unions of product sets is an algebra of sets [Hal50, 33.E]. It then follows from the approximation lemma of measure theory [Hal50, 13.D] that Θ, a set of positive measure, can be δ-approximated by a set S(δ) which is a finite union of product sets, i.e. for every δ > 0 we can find such S(δ) with P(Θ\S(δ)) < δ and P(S(δ)\Θ) < δ. We assume that δ < P(Θ), so that P(S(δ)) > 0 is guaranteed.

Decomposing this S(δ) into a finite union of disjoint product sets gives

$$\begin{eqnarray*} &&S(\delta) = \bigcup_{j=1}^{k(\delta)} \Xi^j(\delta), \end{eqnarray*}$$

for product sets Ξ^j(δ) = Ξ^j₁(δ) × ··· × Ξ^j_n(δ), which we may assume to be of positive measure (if some Ξ^j(δ) has zero measure, then it may as well be omitted). By construction, we know that

$$\begin{eqnarray*} &&\sum_{j=1}^{k(\delta)} P(\Xi^j(\delta)\cap \Theta) > P(\Theta) - \delta,\quad \sum_{j=1}^{k(\delta)} P(\Xi^j(\delta)\setminus\Theta) < \delta. \end{eqnarray*}$$

Since the second inequality states that

$$\begin{eqnarray*} &&\sum_{j=1}^k \frac{P(\Xi^j(\delta)\cap\Theta)}{P(S(\delta))} \cdot \frac{P(\Xi^j(\delta)\setminus\Theta)}{P(\Xi^j(\delta)\cap\Theta)} < \frac{\delta}{P(S(\delta))} , \end{eqnarray*}$$

and this sum is a convex combination, we conclude that there is at least one index j for which

$$\begin{eqnarray*} &&\frac{P(\Xi^j(\delta)\setminus\Theta)}{P(\Xi^j(\delta)\cap\Theta)} < \frac{\delta}{P(S(\delta))} < \frac{\delta}{P(\Theta)-\delta} . \end{eqnarray*}$$

We define $\Xi = \prod _{i=1}^n \Xi _i$ to be equal to this Ξ^j(δ). Then

$$\begin{eqnarray*} &&\fl P(\Theta|\Xi) = \frac{P(\Theta\cap\Xi)}{P(\Xi\setminus\Theta)+P(\Xi\cap\Theta)} = \left(1 + \frac{P(\Xi\setminus\Theta)}{P(\Xi\cap\Theta)}\right)^{-1} > \left(1 + \frac{\delta}{P(\Theta)-\delta}\right)^{-1}. \end{eqnarray*}$$

For δ sufficiently small, this is >1 − ε, as has been claimed.   □

We return to the main line of the proof of theorem 2.21 and fix ε > 0. In a hidden variable combination such as (ℓ_AB,ℓ_BY,ℓ_YX,ℓ_XA), each component now becomes a set of hidden variable values having positive probability. By the lemma, we can choose these sets in such a way that if such a combination of hidden variables occurs, then the joint outcome is (0,0,0,0) with probability >1 − ε. In particular, if a hidden variable combination in (ℓ_AB,ℓ_BY,*,*) occurs, where the last two components are unspecified, then b = 0 with probability >1 − ε. Similarly, we find a combination of sets (κ_AB,κ_BY,κ_YX,κ_XA) producing (1,0,1,1) with probability >1 − ε. Therefore, the combination (κ_AB,κ_BY,ℓ_YX,κ_XA) yields (1,0,1,1) with probability >1 − 2ε; it should now be clear how to complete the proof, following the steps of the discrete-variable case and bounding the probabilities in each step. Choosing ε small enough then shows that the probability to get the outcome (0,0,1,1) is strictly positive in contradiction with (2.16).

Here, we lift the restriction of finite dimensionality from the proof of proposition 3.17. First of all, what does separability even mean in the infinite-dimensional case? In the following, we work with arbitrary Hilbert spaces $\mathcal{H}$ which are not necessarily separable, and put the usual trace-norm topology on $S(\mathcal{H})$; upon interpreting a quantum state on $\mathcal{H}$ as a normal positive linear functional on $\mathcal {B}(\mathcal{H})$, this is the weak *-topology. Moreover, $S(\mathcal{H})$ carries the Borel σ-algebra induced by this (metrizable) topology.

Definition. Let $\mathcal{H}_1,\ldots ,\mathcal{H}_k$ be Hilbert spaces. A state $\rho \in S(\mathcal{H}_1\otimes \cdots \otimes \mathcal{H}_k)$ is separable if it lies in the closed convex hull of the set of product states.

In general, one cannot expect a separable state to have a decomposition into a finite or infinite convex combination of product states; rather, integrals are needed [HSW05].

Lemma. Let $\rho \in S(\mathcal{H}_1\otimes \cdots \otimes \mathcal{H}_k)$ be separable. Then there exists a probability measure P on the set of product states such that

$$\begin{eqnarray*} &&\rho = \int_{S(\mathcal{H}_1)\otimes\cdots\otimes S(H_k)} \left(\rho_1\otimes\cdots \otimes\rho_k \right) \mathrm{d}P(\rho_1\otimes\cdots\otimes\rho_k). \end{eqnarray*}$$

In the finite-dimensional case, one can take the measure P to have finite support, so that the integral becomes a finite convex combination.

Proof. Since the set of product states is compact, Milman's converse to the Krein–Milman theorem guarantees that every extreme point of the set of separable states is a product state. Then the assertion follows from Choquet's theorem [Phe01].   □

This should make it clear how to prove proposition 3.17 in the general case: to a source s sending out a separable state

$$\begin{eqnarray*} &&\rho_s = \int_{\prod_{\{m\::\: sCm\}} S(\mathcal{H}_{(s,m)})} \left( \bigotimes_{\{m\::\: sCm\}} \rho_{(s,m)} \right) \mathrm{d}P_s\left(\bigotimes_{\{m\::\: sCm\}} \rho_{(s,m)}\right) \end{eqnarray*}$$

we associate the hidden variable probability space $\Omega _s=\prod _{\{m\::\: sCm\}} S(\mathcal{H}_{(s,m)})$ equipped with its Borel σ-algebra and its probability measure P_s, so that the hidden variable λ_s ranges over all product states $\lambda _s=\bigotimes _{\{m\::\: sCm\}} \rho _{(s,m)}$. Concerning the conditional probabilities, (3.6) now reads

$$\begin{eqnarray*} &&\mathcal{O}_{m} \left(\left\{\lambda_s = \bigotimes_{\{m'\::\: sCm'\}} \rho_{(s,m')} : sCm\right\}\right) \stackrel{\mathrm{def}}{=} \mathrm{tr}\left[ \left(\bigotimes_{s\::\: sCm} \rho_{(s,m)}\right) \mathcal{F}_{m} \right] . \end{eqnarray*}$$

This is a continuous, and therefore measurable, function on $\Omega _s=\prod _{\{m\::\: sCm\}} S(\mathcal{H}_{(s,m)})$.

The intuition about how this classical model works is similar to the finite-dimensional case. One may think of the hidden variable λ_s as an abstract classical description of the product state sent out by the source; each party m then receives all descriptions from all the sources it connects to, for each of these product states retains only the information concerning its system while throwing away the rest, and uses that information to calculate its required outcome distribution, which can then get sampled in order to obtain its outcome. By construction, this produces the desired joint distribution of outcomes.