Quantum contextuality

Quantum contextuality is a feature of quantum mechanics whereby measurements of quantum observables cannot simply be thought of as revealing pre-existing values. Any attempt to do so leads to values that are dependent upon which other measurements are being performed (the measurement context). More formally, the measurement result 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 mechanics 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, and algebraic topology.

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

Main article: Kochen–Specker theorem

Simon B. Kochen and Ernst Specker, and separately John Bell, constructed proofs that quantum mechanics is contextual for systems of Hilbert space dimension three and greater. The Kochen–Specker theorem proves that noncontextual hidden variable theories cannot reproduce the empirical predictions of quantum mechanics. Such a theory would suppose the following.

1. All quantum-mechanical observables may be assigned definite values. These global value assignments may be 'hidden' in the sense that the state of a quantum system could be described by a probabilistic mixture of assignments.
2. Value assignments are independent of which other commuting observables are also measured.

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.[1]

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. 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 though 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.

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.

Other frameworks and extensions

• A framework due to Ehtibar Dzhafarov, Janne Kujala and Victor Cervantes known as contextuality-by-default has recently inspired researchers to apply the concept of contextuality to human psychology and sociology. This has met with mixed results.
• 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–Abramsky–Brandenburger theorem

The Kochen–Specker theorem proves that quantum mechanics is incompatible with 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.

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^NC and some remainder e',$$e=\lambda e^{NC}+(1-\lambda )e'\,.$$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 was also proved by Abramsky, Barbosa and Mansfield 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 reliable measure of contextuality, since it optimises over all possible noncontextual inequalities rather than checking the statistics against one inequality in particular.

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 injection

Magic state injection 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 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 which 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.

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,$$(1-p_{s})\geq (1-CF).\nu (f)$$where $p_{s},CF,\nu (f)\in [0,1]$ are the probability of success, the contextual fraction of the quantum resource, and a measure of the non-linearity of the function to be computed $f$, respectively.

Further examples

• The Abramsky–Barbosa–Mansfield inequality was also shown to relate quantum advantage in non-local games to the degree of contextuality required by the strategy and an appropriate measure of the difficulty of the game.
• Similarly the inequality arises in a transformation-based model of quantum computation analogous to l2-MBQC where it relates the degree of sequential contextuality present in the dynamics of the quantum system to the probability of success and the degree of non-linearity of the target function.
• Preparation contextuality has been shown to enable quantum advantages in cryptographic random-access codes and in state-discrimination tasks.
• In classical simulations of quantum systems, contextuality has been shown to incur memory costs.

See also

• Kochen–Specker theorem
• Mermin–Peres square
• KCBS pentagram
• Quantum nonlocality
• Quantum foundations