State-Independent Proof of Kochen-Specker Theorem with 13 Rays

Abstract

Quantum contextuality, as proved by Kochen and Specker, and also by Bell, should manifest itself in any state in any system with more than two distinguishable states and recently has been experimentally verified on various physical systems. However for the simplest system capable of exhibiting contextuality, a qutrit, the quantum contextuality is verified only state-dependently in experiment because too many (at least 31) observables are involved in all the known state-independent tests. Here we report an experimentally testable inequality involving only 13 observables that is satisfied by all non-contextual realistic models while being violated by all qutrit states. Thus our inequality will practically facilitate a state-independent test of the quantum contextuality for an indivisible quantum system. We provide also a record-breaking state-independent proof of the Kochen-Specker theorem with 13 directions determined by 26 points on the surface of a three by three magic cube.It is believed, almost religiously, that every effect has its own cause and the same cause shall lead to the same effect. The predictions of quantum mechanics (QM) are however probabilistic and the effect that different outcomes appear in different runs of a measurement seem to have no definite cause, at least unexplainable using QM alone. Einstein, Podolsky, and Rosen [1] initiated a longlasting quest for a quantum reality by questioning the completeness of quantum mechanics. Hidden variable (HV) models are introduced in order to explain why a certain outcome appears in each run of a measurement, attempting to make QM complete. Years later Kochen, Specker [2], and Bell [3] discovered that quantum mechanics can be completed only by a hidden variable model that is contextual: the outcome of a measurement depends on which compatible observable might be measured alongside. Simply put, Kochen-Specker (KS) theorem states that non-contextual HV models cannot reproduce all the predictions of QM or quantum mechanics is contextual.In any non-contextual HV model all observables have definite values determined only by some HVs λ that are distributed according to a given probability distribution ̺ λ with normalization dλ̺ λ = 1. Two observables are compatible if they can be measured in a single experimental setup and a maximal set of mutually compatible observables defines a context. Non-contextuality is a typical classical property: the value of an observable revealed by a measurement is predetermined by HVs λ only regardless of which compatible observable might be measured alongside. Local realism is a form of non-contextuality enforced by the locality and thus Bell's inequalities [4] are a special form of KS inequalities [5][6][7][8][9], experimentally testable inequalities that are satisfied by all noncontextual HV models, some of which have been tested in recent measurements [10][11][12][13][14][15][16][17][18]. In general KS inequalities reveal the nonclassical nature of single systems demanding neither space-like separation nor entanglement, i.e....

Arbitrarily exhaustive hypergraph generation of 4-, 6-, 8-, 16-, and 32-dimensional quantum contextual sets

“…Yet, they entitle their paper "State-Independent Proof of Kochen-Specker Theorem with 13 Rays" and on p. 3 they claim to have nevertheless "proved the original KS theorem" [67]. How come, when the Lemma XII.1 proves the contrary?…”

“…In 2012, Yu and Oh published a paper [67] in which they introduced a set with 13 vertices which they call a 13-ray set-13-vertex set in our notation. The set is displayed in Fig.…”

“…19 [67], have 49,51,57,192, and 25 vertices, respectively (and 36, 37, 40, 118, and 16 edges, respectively). In Fig.…”

State-independent contextuality sets for a qutrit

“…The rays form a set to test contextualtiy in a state-independent way. Later on Peres reported a 33-ray proof [2], Conway and Kochen reported a 31-ray proof [3], followed by Yu and Oh [4] who reduced the required number of observables to a recordbreaking 13 rays and gave the state-independent proof of the KS theorem. In three dimensions, it has been proven that 13 is the minimal number of observables by Cabello [5].…”

Exclusivity principle forbids sets of correlations larger than the quantum set

“…This third approach is based on two observations: on one hand, that quantum contextual correlations, i.e., quantum correlations for compatible (but not necessarily spacelike compatible) measurements provide a natural generalization of quantum nonlocal correlations that leaves room for a wider range of experimental scenarios, including systems that cannot be separated into parts or represented as tensor product of smaller spaces [18,[24][25][26][27][28] and for systems prepared in arbitrary quantum states [19,[25][26][27][29][30][31][32]. The second observation comes from the graph approach to quantum correlations introduced in Ref.…”