56 P. D. Bruza and P. Wittek

functional identity it is impossible to assign a random variable to represent the outcomes of the same measurement protocol in di erent measurement contexts.

It is a requirement that the mapping adheres to the expected normalization condition: e E : v e p(v) = 1. By way of illustration, consider once again Fig. 3. This contextuality scenario has four edges. The normalization condition enforces the following constraints:

where pi, 1 ≤ i ≤ 4 and qj , 1 ≤ j ≤ 4 denote the probabilities of outcomes in the four hyperedges. A definition of contextuality can now be presented.

Definition 1 (Probabilistic contextuality). (General contextuality [2]). Let

X = (V, E) be a contextuality scenario. Let G(X) denote the set of probabilistic models on X. X is deemed “contextual” if G(X) = .

Probabilistic contextuality occurs when there is no probabilistic model p corresponding to composite contextuality scenario X. Determining whether X is contextual is computable by a linear program [2].

3Using Probabilistic Programs to Simulate Bell Scenario Experiments

One of the advantages of using a programming approach to develop probabilistic models is that experimental designs can be syntactically specified in a modular way. In this way, a wide variety of experimental designs across fields can potentially be catered for. For example, consider the situation where an experimenter wishes to determine whether a system S can validly be modelled compositionally in terms of two component subsystems A and B. Two di erent experiments can be carried out upon each of the two presumed components, which will answer a set of ‘questions’ with binary outcomes, leading to four measurement contexts. For example, one experimental context would be to ask A1 of component A and B1 of component B. In Bell scenario experiments, four measurement contexts are typically used: {{A1, B1}, {A1, B2}, {A2, B1}, {A2, B2}}. Bell scenario designs has been widely employed in cognitive psychology to test for contextuality in human cognition [3, 9, 14, 18].

One way to think about system S is that it is equivalent to a set of biased coins A and B, where the bias is local to a given measurement context. Figure 4 depicts a P-program that follows this line of thinking.

Fig. 4. Example “Bell scenario” P-program

The Bell scenario program first defines the components A and B together with the associated variables. Thereafter, the program features the four measurement associated contexts P1, P2, P3 and P4. Finally, the line model(design: ‘no-signal’,P1,P2,P3,P4) specifies that the measurement contexts are to be combined according to the no-signaling condition. The question now to be addressed is how the hypergraph semantics are to be formulated. Reference [2] provides the general semantics of the Bell scenarios by means of multipartite composition of contextuality scenarios.

As these semantics are compositional, it opens the door to map syntactically specified components in a P-program to contextuality scenarios and then to exploit the composition to provide the semantics of the program as a whole.

Consider the Bell scenario program depicted in Fig. 4. The syntactically defined components A and B are modelled as contextuality scenarios XA and XB respectively. The corresponding hypergraphs are depicted in Fig. 5.

58 P. D. Bruza and P. Wittek

signalling condition imposes constraints on the allowable probabilistic models on the combined hypergraph structure. Following Definition 3.1.2 in [2], a probabilistic model p G(XA × XB ) is a “no signalling” model if:

Reference [2] (p. 45) shows that not all probabilistic models of contextuality scenarios composed by a direct product are “no signalling” models. In order to guarantee that all probabilistic models of a combined contextuality scenario are “no signalling” models, the constituent contextuality scenarios XA and XB should be combined by the Foulis-Randall (FR) product denoted XAB = XA FR XB . As with the direct product XA × XB of contextuality scenarios, the vertices of the FR product are defined by V (XA FR XB ) = V (XA) × V (XB ). It is with respect to the hyperedges that there is a di erence between the FR product and the direct product:

XA FR XB = EA→B EB←A

where

EA→B := {v} × f (v) : ea E(XA), f : ea → E(XB )

v ea

EA←B := f (w) × {w} : eb E(XB ), f : eb → E(XA)

w eb

We are now in a position to illustrate the semantics of the P-program of Fig. 4 by the corresponding contextuality scenario depicted in Fig. 6. Observe how the FR product produces the extra edges that span the events across measurement contexts labeled P1, P2, P3 and P4. At first these spanning edges may seem arbitrary, but they happen to guarantee that the allowable probabilistic models over the composite contextuality scenario XA FR XB satisfy the “no signalling” condition [22]. By way of illustration, the normalization condition on edges imposes the following constraints (see Fig. 6):

where pi, 1 ≤ i ≤ 4 and qj , 1 ≤ j ≤ 4 denote the probabilities of events in the respective hyperedges. A consequence of constraints (5) and (7) is that p3 + p4 = q3 + q4. When considering the associated outcomes this means

In other words, the marginal probability p(A1 = 1) does not di er across the measurement contexts P1 and P2 specified in the P-program of Fig. 4. In a similar vein, Eqs. (5) and (8) imply that the marginal probability p(A1 = 0) does not di er across measurement contexts P1 and P2. The stability of marginal probability ensures that no signalling is occurring from component B to component A. In quantum physics, the FR product is used to compose contextuality scenarios because this product ensures that there is no signalling between the systems. As a consequence, the operational semantics of the P-program must compute the FR product as some component hyperedges derive from measurement contexts, which have been syntactically specified in the P-program, and other edges express the no-signalling constraint. When the FR product is part of the operational semantics, it provides an underlying data structure which allows both classical and non-classical statistical correlations to be simulated [20]. For example, non-classical correlations between variables such as A1B1 can be produced by the P-program using standard Bernoulli samplers to produce (biased) coin flips and the underlying hypergraph data structure constrains the sampling to allow quantum-like correlations to emerge.

To illustrate a Bell scenario experiment in human information processing, consider the information fusion model depicted in Fig. 7. The variable S is a random variable which ranges over a set of image stimuli. Human subjects must decide whether an image is trustworthy [8]. Bivalent random variables C1, C2 relate to features associated with the content of the image. For example, C1 may model the decision whether a subject deems a person portrayed in an image to be honest. Conversely, R1 and R2 are bivalent random variables that relate to representational aspects of the image. For example, R1 may model the decision whether the image has been manipulated, or not. Variable R2 might model the decision whether there was something unexpected perceived in the image. The latent variable γ models the decision whether the content of the image is trustworthy, and depends on variables related to the content C1, C2. Conversely the latent ρ models the decision whether the image is deemed to be authentic, i.e., a true and accurate depiction of reality. Finally, the variable T corresponds to the decision whether the human subject trusts what they have been shown by fusing the assessments regarding the content and representational aspects of the image.

Fig. 6. Contextuality scenario of the P-program of Fig. 4. In total the hypergraph comprises 12 edges of four events. The nodes in rectangles represent events in a probability distribution returned by a given scope: P1, P2, P3, and P4. Note this figure depicts a single hypergraph. Two copies have been made to depict the spanning edges more clearly. This figure corresponds to Figure 7f in [2].

Probabilistic programming languages (PPLs) hav already proved useful in cognitive science [16], but, to our knowledge, they have yet to be seriously taken

up by quantum physics. PPLs o er quantum physicists a convenient way to describe specify experiments, and enable a new tool for analyzing statistical correlations based both on simulated as well as actual experimental results. Their potential use is not restricted to Bell scenarios.

Since any PPL is based on random variables, we can ask the question what exactly is a random variable in quantum physics. If we restrict our attention to a single measurement context, due to the normalization constraint, we can think of the measurement context as a (conditional) probability distribution over random variables, which describe the measurement outcomes. More formally, this