Classical systems can be contextual too: Analogue of the Mermin–Peres square

Abstract

Contextuality lays at the heart of quantum mechanics. In the prevailing opinion it is considered as a signature of "quantumness" that classical theories lack. However, this assertion is only partially justified. Although contextuality is certainly true of quantum mechanics, it cannot be taken by itself as discriminating against classical theories. Here we consider a representative example of contextual behaviour, the so-called Mermin-Peres square, and present a discrete toy model of a bipartite system which reproduces the pattern of quantum predictions that leads to contradiction with the assumption of non-contextuality. This illustrates that quantum-like contextual effects have their analogues within classical models with epistemic constraints such as limited information gain and measurement disturbance. way out of this conundrum and, in particular, what form acceptable hidden variable models could take to that effect. Certainly, at this point one should seriously reflect on the John Bell's dictum "... what is proved by impossibility proofs is lack of imagination" [9]. It might suggest that perhaps revision of the relation between the concept of observable and measurement is required for better understanding of the theory. In this article we attempt to explore this possibility and show that careful distinction between these concepts opens a way for methodological discussion of contextual effects in classical systems too.Difficulty in making sense of contextuality in classical terms often prompts to consider it as a signature distinguishing between quantum and classical realms. Indeed, the possibility of contextual hidden variable models aiming at reconstruction of quantum predictions is hardly explored. On the other hand, the hypothesis of non-contextual hidden variable models has been thoroughly investigated [5] and proved to be directly testable [10,11]. In particular, many state-independent quantum-contextuality experiments have been recently performed e.g. with trapped ions [12], photons [13,14] and magnetic-resonance systems [15]. Essentially all of them boil down to checking of the pattern captured in the so called Mermin-Peres square [16,17]. Certainly, these results provide compelling evidence for contextual behaviour in these experimental setups, thus pushing the project of hidden variable models to the less explored contextual camp. In this work we present a simple probabilistic model which reproduces the pattern of quantum observables considered in the Mermin-Peres square and demonstrates quantum-like contextuality in a classical bipartite system. One immediate feature of the presented model is state-independent violation of contextuality inequality [10] in accord with quantum mechanical predictions.Many results suggest that quantum states can be understood as states of knowledge. Strong evidence in favour of this view is given, in particular, by concrete models providing classical analogues of various phenomena typically associated with strictly quantum mechanical characteristics [18,19,20,21,22,23,2...

Nonlocality from Local Contextuality

“…In addition, our experiment solves a problem that previous experiments testing the Peres-Mermin inequality [27][28][29] have. While the results of all these experiments can be simulated with classical models [43][44][45], our experiment rules out all these models, since no contextual but local hidden variable model can explain the observed correlations. In this sense, our experiment constitutes a crucial development of the experiments in Refs.…”

Local model of a qubit in the interferometric setup

“…Indeed, recent research show that many phenomena typically associated with strictly quantum mechanical effects have analogues in classical models with epistemic restrictions [21][22][23][24][25][26][27][28][29]. Most notable in this respect is the Spekkens' toy model [24] which reproduces a surprisingly large array of quantum phenomena in a simple discrete system constrained by the so-called 'knowledge balance principle' (it also reproduces certain aspects of the interferometric phenomena [30]).…”

“…[ˆ]  see equations (29) and (31). In terms of classes this induces outcome dependent 'projection', i.e.…”

Revisiting the admissibility of non-contextual hidden variable models in quantum mechanics

“…Contextuality has thus been identified as a fundamental non-classical feature of the quantum world and experiments have also been proposed and conducted to this effect [10,11]. Contextuality, on the one hand has led to investigations on the foundational aspects of QM including attempts to prove the completeness of QM [12,13], and on the other hand has been harnessed for computation and cryptography [14,15]. While there have been generalisations, in this letter, we restrict ourselves to the standard notion of non-contextuality as used by KS [6].…”