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Global sections over Alice and Bob: the sheaf-theoretic expression of non-locality and contextuality

03/04/2013 in Garabatos tricolores, Night science

The story is well known. It is a sunny Sunday and Alice and Bob decide that going out for a walk might be too mainstream so, instead, they decide to do some physics. They convince their friend Carlos, the third wheel, to prepare a pair of entangled photons, each to be sent to Alice or Bob. The couple can perform two different measurements on the photons sent by Carlos and the values that can be obtained for both measurements are either 0 or 1. In order to better enjoy the Sunday, they decide to perform the experiment separated from each other and Carlos. They take their longboards in opposite directions and decide to start taking measurements. Being separated, they will take randomly one of the two possible measurements and annotate it together with one of the two possible outcomes. After being the whole morning repeating the experience, they make a table with the obtained distributions of joint outcomes for each pair of measurements.

The two different measurements that Alice and Bob can make are distinguished by an apostrophe.

Empirical distribution of a Bell scenario experiment. The two different measurements that Alice and Bob can make are distinguished by an apostrophe.

After seeing the results, Alice finds herself surprised by the correlations obtained, as if some spooky action at a distance (spukhafte Fernwirkung) was relating both measurements. Bob, immediately accuses Carlos of being that spooky action. Alice points out that there is no way by which Carlos could have spied them while performing the measurements (getting to know the measurements they were going to make) and demands Bob for a detailed explanation of how Carlos could have produced the results. After thinking for a while, Bob points out that Carlos could, in principle, generate, in a deterministic way, all the possible assignments of their measurements to the outcomes. Hence, Carlos might be able to send the pair of photons with a previously uniquely specified assignment of outcomes for each of the two measurements Alice and Bob can make (for instance \lambda_{1} : a \mapsto 0, a' \mapsto 1, b \mapsto 0, b' \mapsto 0 ). To generate a stochastic behaviour, Carlos just has to specify a probability distribution on all the possible assignments to their measurements in such a way that, applied to the different contexts, it produces the observed probabilities on the joint measurements of the outcomes. Alice still holds that Carlos is innocent and what they have observed cannot be explained using Bob's reasoning. Bob refuses to believe that their result is a manifestation of a sort non-local and contextual behaviour. After being silent for most part of the discussion, Carlos decides to start a deeper conceptual investigation to prove his innocence.

In order to answer the question whether the explanation of data obtained requires a theory that violates the principle of locality , we would like to define a general abstract framework in which we can place our problem in a precise way. For the sake of generality, let us think of a system (like the entangled photons of Alice and Bob) as composed of several parts (spatially separated or not) upon which several individuals (Alice, Bob, Carlos, ..., Zed) can perform a set of measurements. In quantum mechanics, it usually happens that not all measurements are compatible (they cannot be made simultaneously). We will consider that the different parts of the systems are always compatible, and that incompatibilities, if they exist, arise within each part. These measurements have associated a set of possible outcomes that are the values we obtain (0 or 1, in our example). Given the different possible measurement contexts, a set of distributions on the joint outcomes obtained determines an empirical model, such as the one obtained by Alice and Bob.

Diagram showing the different elements composing a general measuring scenario (Abramsky et Bradenburguer, 2011).

Diagram showing the different elements composing a general measuring scenario (Abramsky et Bradenburguer, 2011).

For that empirical model, it might be possible to construct a hidden-variable model (such as the one proposed by Bob) that explains the results. In this model, a set \Lambda of independent unobserved variables $latex \lambda$ determine the outcomes of the different measurements. If an empirical model can be generated with a distribution of deterministic hidden-variables marginalized over the measurement contexts, it can serve as an explanation to the observed data.

The scenario described by Alice and Bob is that of the Bell inequalities, it will be henceforth referenced as a Bell scenario. To recapitulate, in the Bell scenario, we have two parts (Alice and Bob), two different measurements that are incompatible on the same part (a,a' for Alice or b,b' for Bob) and two different outcomes (0 or 1). The set of all maximal measurement contexts ( {a,b},{a',b},{a,b'},{a,b} ) can be seen as a topological space X and an assignment of outcomes can be thought as a section over its open sets U . Thus, a section over an open set U \subseteq X is map s such that  s : U \rightarrow O . The set of sections over a particular U can be noted as O^{U} . A presheaf F associates to each open set U' a set O^{U'} of sections on U' and to each open set U \subseteq U' a map F(U') \rightarrow F(U) that specifies the restrictions of the sections over U' to U :  F : U \rightarrow O^{U} .

In the Bell scenario, the set of possible assignments from the different possible measurement pairs is given by the following table. Each row constitutes a set of sections O^{U} over the particular measurement context.

Sections over de different maximal measurement contexts (Abramsky et Brandenburguer, 2011).

Sections over de different maximal measurement contexts (Abramsky et Brandenburguer, 2011).

Given a family of sets U_{i} that form a cover of X , their sections are locally determined whenever two sections s_1 and s_2 over X coincide when they are restricted to each U_{i} . A locally determined presheaf is called separated. A sheaf is a separated presheaf that can be glued together when there is a unique section s of X such that s|U_{i} = s_{i} . For the previously described presheaf of the outcomes sections, both the compatibility and gluing conditions hold so that we obtain a sheaf.

The category theory generalization of the previously described mathematical structure is the presheaf functor. For a category \mathcal{C} the S-valued presheaf F is a functor such that: F : \mathcal{C}^{opp} \rightarrow \textit{S} . Here, S can be any category like the category of sets ( Set ), commutative rings ( CRing ) or the category of Abelian groups ( Ab ). In order to recover the notion of presheaf in topological spaces, we have to take the Set valued presheaf of the category \mathcal{C} constituted by the posets of open sets of a topological space.

Functor operating on the category constituted by the posets of open sets of a topological space.

Functor operating on the category constituted by the posets of open sets of a topological space.

Then, the presheaf functor takes the open subsets of \mathcal{C} and maps them to set of sections on the open subset O^{U} . The morphisms of the \mathcal{C}^{opp} category are indeed retractions and are mapped by the presheaf functor into restrictions of sections: res_{U,U'}: F(U') \rightarrow F(U) . Such that the composition rules and identities are satisfied in order to constitute a category.

In order to recover the statistical behaviour of the system, we would like to assign a probability distribution to each set of sections O^{U} . Given a set X and a commutative semiring R (like the non-negative reals), we can define an R-distribution as a function d : X \rightarrow R if the function has a finite support such that  \sum_{x \in X}d(x)=1 . The set of R-distributions on X can be written as D_{R} (X) and for the non-negative reals corresponds to the set of probability distributions. It can be seen that D_{R} is a functor D_{R}: Set \rightarrow Set and can be composed with the previous event sheaf F : \mathcal{C}^{opp} \rightarrow \textit{Set} to form a presheaf D_{R}F: \mathcal{C}^{opp} \rightarrow \textit{Set} .

Our aim is to find if the sheaf condition holds for this new presheaf D_{R}F . So , we want to find the global section d \in D_{R}F(X) exists for the entire set of measurements X = {a,a',b,b'} defining a distribution on the set F(X)=O^{X} of possible assignments \lambda_{i} of outcomes to X . Each of this global sections of $F$ can be considered a deterministic hidden variable (for instance \lambda_{1} : a \mapsto 0, a' \mapsto 1, b \mapsto 0, b' \mapsto 0 ) assigning a particular outcome to a particular measurement independent of the context. The global section d of D_{R}F would define a distribution on the set of deterministic hidden variables that marginalizes on the different measurement contexts to yield the observed probabilities (modelling the role of Carlos suggested by Bob in our initial story).

If we can find the sheaf, Bob will be right , however if there is an obstruction to the formation of it, we will have to accept the non-local and contextual behaviour. Given the empirical results, this can be done by solving the following linear system \textbf{Md}=\textbf{e} where \textbf{d} stands for the global section, \textbf{e} for the empirical model and \textbf{M} for the incidence matrix. The latter is defined by the relation between the global assignments \lambda_{j} and the sections s_{i} relating the contexts with the outcomes. The elements m_{i,j} are 1 if the \lambda_{j} , when restricted to that measurement context, is equal to the section and 0 otherwise. For the Bell scenario the incidence matrix is:

Incidence matrix for the Bell scenario  (Abramsky et Brandenburguer, 2011).

Incidence matrix for the Bell scenario (Abramsky et Brandenburguer, 2011).

It can be shown that there is no \textbf{d} that solves the equation such that all terms are non-negative and add up to one. So, for this empirical distribution obtained in the sunny Sunday there is not a mischievous Carlos or a hidden variable model that explains it

Abramsky S. and Brandenburger A. 2011. The Sheaf-Theoretic Structure of Non-Locality and Contextuality. 13: 113036.