Quantum contextuality: Difference between revisions

The need for contextuality was discussed informally in 1935 by [[Grete Hermann]],<ref>{{cite book |last1=Crull |first1=Elise |last2=Bacciagaluppi |first2=Guido |title=Grete Hermann - Between Physics and Philosophy |date=2016 |publisher=Springer |location=Netherlands |isbn=978-94-024-0968-0 |pages=154 |ref=46}}</ref> but it was more than 30 years later when [[Simon B. &nbsp;Kochen]] and [[Ernst Specker]], and separately [[John Stewart Bell|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 [[Hiddenhidden-variable theory|hidden variable theories]] cannot reproduce the empirical predictions of quantum mechanics.<ref>{{Cite journal |last=Carsten |first=Held |date=2000-09-11 |title=The Kochen–Specker Theorem |url=https://plato.stanford.edu/archives/spr2018/entries/kochen-specker/ |access-date=2018-11-17 |website=plato.stanford.eduStanford Encyclopedia of Philosophy}}</ref> 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 whichthat 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,<ref name=":1">S. Kochen and E. P. Specker, "The problem of hidden variables in quantum mechanics", ''Journal of Mathematics and Mechanics'' '''17''', 59–87 (1967).</ref> 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.<ref name=":2">Gleason, A. M, "Measures on the closed subspaces of a Hilbert space", ''Journal of Mathematics and Mechanics'' '''6''', 885–893 (1957).</ref>