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arXiv:1201.0374

Minimal proofs of state-independent contextuality

Adán Cabello

Published Jan 1, 2012• 3 citations
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Abstract

It has been recently shown that state-independent contextuality (SIC) is a fundamental resource linked with a type of nonlocality which cannot be improved by nonsignaling resources. Therefore, it is of fundamental importance to identify the simplest sets of quantum observables needed to prove SIC. We show that d+10 rank-1 projectors are sufficient to prove SIC for any physical system in dimension d>2. This result outperforms both the best Kochen-Specker proofs and the results presented by Yu and Oh in arXiv:1112.5513v1.

DOI
10.48550/arxiv.1201.0374

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It has been recently shown that state-independent contextuality (SIC) is a fundamental resource linked with a type of nonlocality which cannot be improved by nonsignaling resources. Therefore, it is of fundamental importance to identify the simplest sets of quantum observables needed to prove SIC. We show that d+10 rank-1 projectors are sufficient to prove SIC for any physical system in dimension d>2. This result outperforms both the best Kochen-Specker proofs and the results presented by Yu and Oh in arXiv:1112.5513v1.
The strength of quantum contextuality is closely related to quantum computation power. Yu-Oh set is the minimal quantum system with state-independent contextuality(SIC). However, its strength of the contextuality has not been taken into account. In this paper, we present a general method to determine whether there is a quantum state with strong contextuality in the quantum system composed of rank-one projectors. Based on this method, we conclude that Yu-Oh set does not have quantum states with strong contextuality. This indicates that strong contextuality and SIC are mutually independent.
Recently, Yu and Oh [Phys. Rev. Lett. 108, 030402 (2012)] have conjectured that the simplest set of vectors needed to prove state-independent contextuality on a qutrit requires 13 vectors. Here we first prove that a necessary and sufficient condition for a set of vectors in any dimension d≥3d \ge 3d≥3 to prove contextuality for a system prepared in a maximally mixed state is that the graph GCG_CGC​ in which vertices represent vectors and edges link orthogonal ones has chromatic number χ(GG)χ(G_G)χ(GG​) larger than ddd. Then, we prove Yu and Oh's conjecture. Finally, we prove that any set satisfying χ(GG)>dχ(G_G)>dχ(GG​)>d assisted with ddd-party maximum entanglement generates nonlocality which cannot be improved without violating the no-signalling principle. This shows that any set satisfying χ(GG)>dχ(G_G)>dχ(GG​)>d is a valuable resource for quantum information processing.
Abstract Virtually all of the analysis of quantum contextuality is restricted to the case where events are represented by rank-one projectors. This restriction is arbitrary and not motivated by physical considerations. We show here that loosening the rank constraint opens a new realm of quantum contextuality and we demonstrate that state-independent contextuality (SIC) can even require projectors of nonunit rank. This enables the possibility of SIC with less than 13 projectors, which is the established minimum for the case of rank one. We prove that for any rank, at least 9 projectors are required. Furthermore, in an exhaustive numerical search we find that 13 projectors are also minimal for the cases where all projectors are uniformly of rank two or uniformly of rank three.
Contextuality is a key distinguishing feature between classical and quantum physics. It expresses a fundamental obstruction to describing quantum theory using classical concepts. In turn, understood as a resource for quantum computation, it is expected to hold the key to quantum advantage. Yet, despite its long recognised importance in quantum foundations and, more recently, in quantum computation, the structural essence of contextuality has remained somewhat elusive - different frameworks address different aspects of the phenomenon, yet their precise relationship often remains unclear. This issue already looms large at the level of the Bell-Kochen-Specker theorem: while traditional proofs proceed by showing the nonexistence of valuations, the notion of state-independent contextuality in the marginal approach allows to prove contextuality from seemingly weaker assumptions. In the light of this, and at the absence of a unified mathematical framework for Kochen-Specker contextuality, the original algebraic approach has been widely abandoned, in favour of the study of contextual correlations. Here, we reinstate the algebraic perspective on contextuality. Concretely, by building on the novel concept of context connections, we reformulate the algebraic relations between observables originally postulated by Kochen and Specker, and we explicitly demonstrate their consistency with the notion of state-independent contextuality. In the present paper, we focus on the new conceptual ideas and discuss them in the concrete setting of spin-1 observables, specifically those in the example of [S. Yu and C.H. Oh, Phys. Rev. Lett., 108, 030402 (2012)]; in a companion paper, we generalise these ideas, obtain a complete characterisation of Kochen-Specker contextuality and provide a detailed comparison with the related notions of contextuality in the marginal and graph-theoretic approach.
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. 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 noncontextual realistic models while being violated by all qutrit states. Thus our inequality facilitates a state-independent test of the quantum contextuality for an indivisible quantum system. We also provide a record-breaking state-independent proof of the Kochen-Specker theorem with 13 directions determined by 26 points on the surface of a magic cube.