Quantum contextuality is a feature of the phenomenology of quantum mechanics whereby measurements of quantum observables cannot simply be thought of as revealing pre-existing values. Any attempt to do so in a realistic hidden-variable theory leads to values that are dependent upon the choice of the other (compatible) observables which are simultaneously measured (the measurement context). More formally, the measurement result (assumed pre-existing) of a quantum observable is dependent upon which other commuting observables are within the same measurement set.
Contextuality was first demonstrated to be a feature of quantum phenomenology by the Bell–Kochen–Specker theorem. The study of contextuality has developed into a major topic of interest in quantum foundations as the phenomenon crystallises certain non-classical and counter-intuitive aspects of quantum theory. A number of powerful mathematical frameworks have been developed to study and better understand contextuality, from the perspective of sheaf theory, graph theory, hypergraphs, algebraic topology, and probabilistic couplings.
Nonlocality, in the sense of Bell's theorem, may be viewed as a special case of the more general phenomenon of contextuality, in which measurement contexts contain measurements that are distributed over spacelike separated regions. This follows from Fine's theorem.
Quantum contextuality has been identified as a source of quantum computational speedups and quantum advantage in quantum computing. Contemporary research has increasingly focused on exploring its utility as a computational resource.
Kochen and Specker
The need for contextuality was discussed informally in 1935 by Grete Hermann, but it was more than 30 years later when Simon B. Kochen and Ernst Specker, and separately John Bell, constructed proofs that any realistic hidden-variable theory able to explain the phenomenology of quantum mechanics is contextual for systems of Hilbert space dimension three and greater. The Kochen–Specker theorem proves that realistic noncontextual hidden-variable theories cannot reproduce the empirical predictions of quantum mechanics. Such a theory would suppose the following.
All quantum-mechanical observables may be simultaneously assigned definite values (this is the realism postulate, which is false in standard quantum mechanics, since there are observables that are indefinite in every given quantum state). These global value assignments may deterministically depend on some "hidden" classical variable, which in turn may vary stochastically for some classical reason (as in statistical mechanics). The measured assignments of observables may therefore finally stochastically change. This stochasticity is, however, epistemic and not ontic, as in the standard formulation of quantum mechanics. Value assignments pre-exist and are independent of the choice of any other observables, which, in standard quantum mechanics, are described as commuting with the measured observable, and they are also measured. Some functional constraints on the assignments of values for compatible observables are assumed (e.g., they are additive and multiplicative, there are, however, several versions of this functional requirement).
In addition, Kochen and Specker constructed an explicitly noncontextual hidden-variable model for the two-dimensional qubit case in their paper on the subject,S. Kochen and E. P. Specker, "The problem of hidden variables in quantum mechanics", Journal of Mathematics and Mechanics 17, 59–87 (1967). thereby completing the characterisation of the dimensionality of quantum systems that can demonstrate contextual behaviour. Bell's proof invoked a weaker version of Gleason's theorem, reinterpreting the theorem to show that quantum contextuality exists only in Hilbert space dimension greater than two.Gleason, A. M, "Measures on the closed subspaces of a Hilbert space", Journal of Mathematics and Mechanics 6, 885–893 (1957).
Frameworks for contextuality
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, a representative example of which is the Greenberger–Horne–Zeilinger proof of nonlocality.
Graph and hypergraph frameworks Adán Cabello, Simone Severini, and Andreas Winter introduced a general graph-theoretic framework for studying contextuality of different physical theories.A. Cabello, S. Severini, A. Winter, Graph-Theoretic Approach to Quantum Correlations", Physical Review Letters 112 (2014) 040401. 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.
Contextuality-by-default (CbD) framework In the CbD approach, 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 in which each random variable is labeled by its content the property it measures, and its context the set of recorded circumstances under which it is recorded (including but not limited to which other random variables it is recorded together with); stands for " is measured in ". 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 is defined as a system in which all variables are jointly distributed and, in any context , and are identically distributed. The system is considered noncontextual if it has a coupling such that the probabilities are maximal possible for all contexts and contents such that . If such a coupling does not exist, the system is contextual. For the important class of cyclic systems of dichotomous () random variables, (), it has been shown that such a system is noncontextual if and only if
where
and
with the maximum taken over all whose product is . If and , 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, in this case CbD specializes to traditional treatments of contextuality in quantum physics. In particular, for consistently connected cyclic systems the noncontextuality criterion above reduces to which includes the Bell/CHSH inequality (), KCBS inequality (), and other famous inequalities. 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 and coincide whenever are measured in contexts . However, unlike most approaches to contextuality, CbD allows for inconsistent connectedness, with and 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.
Tezzin and Amaral have made important contributions to this area, in particular showing that disturbing systems cannot be Kochen-Specker contextual, as well innovative usages of graphical structures in exploring contextuality.
Operational framework An extended notion of contextuality due to Robert Spekkens applies to preparations and transformations as well as to measurements, within a general framework of operational physical theories. 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,A.W. Simmons, Joel J. Wallman, H. Pashayan, S. D. Bartlett, T. Rudolph, "Contextuality under weak assumptions", New J. Phys. 19 033030, (2017). 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