Sheaf-theoretic framework The
sheaf-theoretic, or Abramsky–Brandenburger, approach to contextuality initiated by
Samson Abramsky and
Adam Brandenburger is theory-independent and can be applied beyond quantum theory to any situation in which empirical data arises in contexts. As well as being used to study forms of contextuality arising in quantum theory and other physical theories, it has also been used to study formally equivalent phenomena in
logic,
relational databases,
natural language processing, and
constraint satisfaction. In essence, contextuality arises when empirical data is
locally consistent but globally inconsistent. This framework gives rise in a natural way to a qualitative hierarchy of contextuality: •
(Probabilistic) contextuality may be witnessed in measurement statistics, e.g. by the violation of an inequality. A representative example is the
KCBS proof of contextuality. •
Logical contextuality may be witnessed in the "possibilistic" information about which outcome events are possible and which are not possible. A representative example is
Hardy's nonlocality proof of nonlocality. •
Strong contextuality is a maximal form of contextuality. Whereas (probabilistic) contextuality arises when measurement statistics cannot be reproduced by a mixture of global value assignments, strong contextuality arises when no global value assignment is even compatible with the possible outcome events. A representative example is the original Kochen–Specker proof of contextuality. Each level in this hierarchy strictly includes the next. An important intermediate level that lies strictly between the logical and strong contextuality classes is
all-versus-nothing contextuality, Within this framework experimental scenarios are described by graphs, and certain
invariants of these graphs were shown have particular physical significance. One way in which contextuality may be witnessed in measurement statistics is through the violation of noncontextuality inequalities (also known as generalized Bell inequalities). With respect to certain appropriately normalised inequalities, the
independence number,
Lovász number, and fractional packing number of the graph of an experimental scenario provide tight upper bounds on the degree to which classical theories, quantum theory, and generalised probabilistic theories, respectively, may exhibit contextuality in an experiment of that kind. A more refined framework based on
hypergraphs rather than graphs is also used. developed by Ehtibar Dzhafarov, Janne Kujala, and colleagues, (non)contextuality is treated as a property of any
system of random variables, defined as a set \mathcal{R} = \{R_q^c : q \in Q, q \prec c, c \in C\} in which each random variable R_q^c is labeled by its
content q the property it measures, and its
context c the set of recorded circumstances under which it is recorded (including but not limited to which other random variables it is recorded together with); q \prec c stands for "q is measured in c". The variables within a context are jointly distributed, but variables from different contexts are
stochastically unrelated, defined on different sample spaces. A
(probabilistic) coupling of the system \mathcal{R} is defined as a system S in which all variables are jointly distributed and, in any context c, R^c = \{R_q^c : q \in Q, q \prec c\} and S^c = \{S_q^c : q \in Q, q \prec c\} are identically distributed. The system is considered noncontextual if it has a coupling S such that the probabilities \Pr[S_q^c = S_q^{c'}] are maximal possible for all contexts c, c' and contents q such that q \prec c, c'. If such a coupling does not exist, the system is contextual. For the important class of
cyclic systems of dichotomous (\pm1) random variables, \mathcal{C}_n = \big\{(R_1^1, R_2^1), (R_2^2, R_3^2), \ldots, (R_n^n, R_1^n)\big\} (n \geq 2), it has been shown that such a system is noncontextual if and only if D (\mathcal{C}_n) \leq \Delta(\mathcal{C}_n), where \Delta(\mathcal{C}_n) = (n - 2) + |R_1^1 - R_1^n| + |R_2^1 - R_2^2| + \ldots + |R_n^{n-1} - R_n^n|, and D(\mathcal{C}_n) = \max\big(\lambda_1 \langle R_1^1 R_2^1\rangle + \lambda_2 \langle R_2^2 R_3^2\rangle + \ldots + \lambda_n \langle R_n^n R_1^n\rangle\big), with the maximum taken over all \lambda_i = \pm1 whose product is -1. If R_q^c and R_q^{c'}, measuring the same content in different context, are always identically distributed, the system is called
consistently connected (satisfying "no-disturbance" or "no-signaling" principle). Except for certain logical issues, That nonlocality is a special case of contextuality follows in CbD from the fact that being jointly distributed for random variables is equivalent to being measurable functions of one and the same random variable (this generalizes
Arthur Fine's analysis of
Bell's theorem). CbD essentially coincides with the probabilistic part of Abramsky's sheaf-theoretic approach if the system is
strongly consistently connected, which means that the joint distributions of \{R_{q_1}^c, \ldots, R_{q_k}^c\} and \{R_{q_1}^{c'}, \ldots, R_{q_k}^{c'}\} coincide whenever q_1, \ldots, q_k are measured in contexts c, c'. However, unlike most approaches to contextuality, CbD allows for
inconsistent connectedness, with R_q^c and R_q^{c'} differently distributed. This makes CbD applicable to physics experiments in which no-disturbance condition is violated, as well as to human behavior where this condition is violated as a rule. In particular, Victor Cervantes, Ehtibar Dzhafarov, and colleagues have demonstrated that random variables describing certain paradigms of simple decision making form contextual systems, whereas many other decision-making systems are noncontextual once their inconsistent connectedness is properly taken into account. With respect to measurements, it removes the assumption of determinism of value assignments that is present in standard definitions of contextuality. This breaks the interpretation of nonlocality as a special case of contextuality, and does not treat irreducible randomness as nonclassical. Nevertheless, it recovers the usual notion of contextuality when outcome determinism is imposed. Spekkens' contextuality can be motivated using Leibniz's law of the
identity of indiscernibles. The law applied to physical systems in this framework mirrors the entended definition of noncontextuality. This was further explored by Simmons
et al, who demonstrated that other notions of contextuality could also be motivated by Leibnizian principles, and could be thought of as tools enabling ontological conclusions from operational statistics.
Extracontextuality and extravalence Given a pure quantum state |\psi \rangle, Born's rule tells that the probability to obtain another state | \phi \rangle in a measurement is | \langle \phi | \psi \rangle|^2. However, such a number does not define a full probability distribution, i.e. values over a set of mutually exclusive events, summing up to 1. In order to obtain such a set one needs to specify a context, that is a complete set of commuting operators (CSCO), or equivalently a set of N orthogonal projectors | \phi_n \rangle \langle \phi_n | that sum to identity, where N is the dimension of the Hilbert space. Then one has \sum_n | \langle \phi_n | \psi \rangle|^2 = 1 as expected. In that sense, one can tell that a state vector | \psi \rangle alone is predictively incomplete, as long a context has not been specified. The actual physical state, now defined by | \phi_n \rangle within a specified context, has been called a modality by Auffèves and Grangier Since it is clear that | \psi \rangle alone does not define a modality, what is its status ? If N \geq 3, one sees easily that | \psi \rangle is associated with an equivalence class of modalities, belonging to different contexts, but connected between themselves with certainty, even if the different CSCO observables do not commute. This equivalence class is called an extravalence class, and the associated transfer of certainty between contexts is called extracontextuality. As a simple example, the usual singlet state for two spins 1/2 can be found in the (non commuting) CSCOs associated with the measurement of the total spin (with S=0, \; m=0), or with a Bell measurement, and actually it appears in infinitely many different CSCOs - but obviously not in all possible ones. The concepts of extravalence and extracontextuality are very useful to spell out the role of contextuality in quantum mechanics, that is not non-contextual (like classical physical would be), but not either fully contextual, since modalities belonging to incompatible (non-commuting) contexts may be connected with certainty. Starting now from extracontextuality as a postulate, the fact that certainty can be transferred between contexts, and is then associated with a given projector, is the very basis of the hypotheses of Gleason's theorem, and thus of Born's rule. Also, associating a state vector with an extravalence class clarifies its status as a mathematical tool to calculate probabilities connecting modalities, which correspond to the actual observed physical events or results. This point of view is quite useful, and it can be used everywhere in quantum mechanics.
Other frameworks and extensions A form of contextuality that may present in the dynamics of a quantum system was introduced by Shane Mansfield and
Elham Kashefi, and has been shown to relate to computational
quantum advantages. As a notion of contextuality that applies to transformations it is inequivalent to that of Spekkens. Examples explored to date rely on additional memory constraints which have a more computational than foundational motivation. Contextuality may be traded-off against Landauer erasure to obtain equivalent advantages. == Fine's theorem ==