Quantum contextuality
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 the  Fine–Abramsky–Brandenburger 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. In addition, Kochen and Specker constructed an explicitly noncontextual hidden variable model for the two-dimensional qubit case in their paper on the subject, 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. 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". Analogies may be drawn with impossible figures like the Penrose staircase, which in a formal sense may  also be said to exhibit a kind of contextuality. This framework gives rise in a natural way to a qualitative hierarchy 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. 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 formula_1 in which each random variable formula_2 is labeled by its "content" formula_3, the property it measures, and its "context" formula_4, the set of recorded circumstances under which it is recorded (including but not limited to which other random variables it is recorded together with); formula_5 stands for "formula_3 is measured in formula_4". 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 formula_8 is defined as a system formula_9 in which all variables are jointly distributed and, in any context formula_4, formula_11 and formula_12 are identically distributed. The system  is considered noncontextual if it has a coupling formula_9 such that the probabilities formula_14 are maximal possible for all contexts formula_15 and contents formula_3 such that formula_17. If such a coupling does not exist, the system is contextual. For the important class of "cyclic systems" of dichotomous (formula_18) random variables,  formula_19 (formula_20), it has been shown that such a system is noncontextual if and  only if formula_21 where formula_22 and formula_23 with the maximum taken over all formula_24 whose product is formula_25. If formula_2 and formula_27, 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 formula_28which includes the Bell/CHSH inequality (formula_29), KCBS inequality (formula_30), 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 formula_31 and formula_32 coincide whenever formula_33 are measured in contexts formula_15. However, unlike most approaches to contextuality, CbD allows for "inconsistent connectedness," with formula_2 and formula_27 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, Vctor 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. 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", 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. Fine–Abramsky–Brandenburger theorem. The Kochen–Specker theorem proves that quantum mechanics is incompatible with realistic noncontextual hidden variable models. On the other hand Bell's theorem proves that quantum mechanics is incompatible with factorisable hidden variable models in an experiment in which measurements are performed at distinct spacelike separated locations. Arthur Fine showed that in the experimental scenario in which the famous CHSH inequalities and proof of nonlocality apply, a factorisable hidden variable model exists if and only if an noncontextual hidden variable model exists. This equivalence was proven to hold more generally in  any experimental scenario by Samson Abramsky and Adam Brandenburger. It is for this reason that we may consider nonlocality to be a special case of contextuality. Measures of contextuality. Contextual fraction. A number of methods exist for quantifying contextuality. One approach is by measuring the degree to which some particular noncontextuality inequality is violated, e.g. the KCBS inequality, the Yu–Oh inequality, or some Bell inequality. A more general measure of contextuality is the contextual fraction. Given a set of measurement statistics "e", consisting of a probability distribution over joint outcomes for each measurement context, we may consider factoring "e  into a noncontextual part e" and some remainder "e"', formula_37 The maximum value of λ over all such decompositions is the noncontextual fraction of "e" denoted NCF("e"), while the remainder CF("e")=(1-NCF("e")) is the contextual fraction of "e". The idea is that we look for a noncontextual explanation for the highest possible fraction of the data, and what is left over is the irreducibly contextual part. Indeed for any such decomposition that maximises λ the leftover "e'" is known to be strongly contextual. This measure of contextuality takes values in the interval [0,1], where 0 corresponds to noncontextuality and 1  corresponds to strong contextuality. The contextual fraction may be computed using linear programming. It has also been proved that CF("e") is an upper bound on the extent to which "e" violates "any" normalised noncontextuality inequality. Here normalisation means that violations are expressed as fractions of the algebraic maximum violation of the inequality. Moreover, the dual linear program to that which maximises λ computes a noncontextual inequality for which this violation is attained. In this sense the contextual fraction is a more neutral measure of contextuality, since it optimises over all possible noncontextual inequalities rather than checking the statistics against one  inequality in particular. Measures of (non)contextuality within the Contextuality-by-Default (CbD) framework. Several measures of the degree of contextuality in contextual systems were proposed within the CbD framework, but only one of them, denoted CNT, has been shown to naturally extend into a measure of noncontextuality in noncontextual systems, NCNT. This is important, because at least in the non-physical applications of CbD contextuality and noncontextuality are of equal interest. Both CNT and NCNT are defined as the formula_38-distance between a probability vector formula_39 representing a system and the surface of the "noncontextuality polytope" formula_40 representing all possible  noncontextual systems with the same single-variable marginals. For cyclic systems  of dichotomous random variables, it is shown that if the system is contextual (i.e., formula_41), formula_42 and if it is noncontextual ( formula_43), formula_44 where formula_45 is the formula_38-distance from the vector formula_47 to the surface of the box circumscribing the noncontextuality polytope. More generally, NCNT and CNT are computed by means of linear programming. The same is true for other CbD-based measures of contextuality. One of them, denoted CNT, uses the notion of a "quasi-coupling", that differs from a coupling in that the probabilities in the  joint distribution of its values are replaced with arbitrary reals (allowed to be negative but summing to 1). The class of quasi-couplings formula_48 maximizing the probabilities formula_14 is always nonempty, and the minimal total variation of the signed measure in this class is a natural measure of contextuality. Contextuality as a resource for quantum computing. Recently, quantum contextuality has been investigated as a source of quantum advantage and computational speedups in quantum computing. Magic state distillation. Magic state distillation is a scheme for quantum computing in which quantum circuits constructed only of Clifford operators, which by themselves are fault -tolerant but efficiently classically simulable, are injected with certain "magic" states that promote the computational power to universal fault-tolerant quantum computing. In 2014, Mark Howard, "et al." showed that contextuality characterises magic states for qudits of odd prime dimension and for qubits with real wavefunctions. Extensions to the qubit case have been investigated by Juani Bermejo-Vega "et al." This line of research builds on earlier work by Ernesto Galvão, which showed that Wigner function negativity is necessary for a state to be "magic"; it later emerged that Wigner negativity and contextuality are in a sense equivalent notions of  nonclassicality. Measurement-based quantum computing. Measurement-based quantum computation (MBQC) is a model for quantum computing in which a classical control computer interacts with a quantum system by specifying measurements to be performed and receiving measurement outcomes in return. The measurement statistics for the quantum system may or may not exhibit contextuality. A variety of results have shown that the presence of contextuality enhances the computational power of an MBQC. In particular, researchers have considered an artificial situation in which the power of the classical control computer is restricted to only being able to compute linear Boolean functions, i.e. to  solve problems in the Parity L complexity class ⊕L. For interactions with multi-qubit quantum systems a natural assumption is that each step of the interaction consists of a binary choice of measurement which in turn returns a binary outcome. An MBQC of this restricted kind is known as an "l2"-MBQC. Anders and Browne. In 2009, Janet Anders and Dan Browne showed that two specific examples of nonlocality and contextuality were sufficient to compute a non-linear function. This in turn could be used to boost computational power to that of a universal classical computer, i.e. to solve problems in  the complexity class P. This is sometimes referred to as measurement-based classical computation. The specific examples made use of the Greenberger–Horne–Zeilinger nonlocality proof and the supra-quantum Popescu–Rohrlich box. Raussendorf. In 2013, Robert Raussendorf showed more generally that access to "strongly contextual" measurement statistics is necessary and sufficient for an "l2"-MBQC to compute a non-linear function. He also showed that to compute non-linear Boolean functions with sufficiently high probability requires contextuality. Abramsky, Barbosa and Mansfield. A further generalisation and refinement of these results due to Samson Abramsky, Rui Soares Barbosa and Shane Mansfield appeared  in 2017, proving a precise quantifiable relationship between the probability of successfully computing any given non-linear function and the degree of contextuality present in the "l2"-MBQC as measured by the contextual fraction. Specifically, formula_50where formula_51 are the probability of success, the contextual fraction of the measurement statistics "e", and a measure of the non-linearity of the function to be computed formula_52, respectively.