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arXiv:2011.04048 · New Journal of Physics

State-independent quantum contextuality with projectors of nonunit rank

Zhen‐Peng Xu, Xiao‐Dong Yu, Matthias Kleinmann

Published Feb 16, 2021• 6 citations
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Abstract

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.

DOI
10.1088/1367-2630/abe6e3

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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.
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.
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.
A set of quantum measurements exhibits quantum contextuality when any consistent value assignment to the measurement outcomes leads to a contradiction with quantum theory. In the original Kochen-Specker-type of argument the measurement projectors are assumed to be rays, that is, of unit rank. Only recently a contextuality scenario has been identified where state-independent contextuality requires measurements with projectors of rank two. Using the disjunctive graph product, we provide a systematic method to construct contextuality scenarios which require non-unit rank. We construct explicit examples requiring ranks greater than rank one up to rank five.
Fully revealing the mathmatical structure of quantum contextuality is a significant task, while some known contextuality theories are only applicable for rank-1 projectors. That is because they adopt the observable-based definitions. This paper overcomes the challenges faced by some known contextuality theories by establishing an event-based contextuality theory with exclusive partial Boolean algebra, which is used to describe the contextual systems with local consistency and exclusivity principle. Our theory provides a precise mathematical framework for quantum contextuality, which can handle the scenarios composed of general projectors, and introduces a more complete contextuality hierarchy. We conclude that the Kochen-Specker contextuality is equivalent to the state-independent strong contextuality for finite dimensional quantum systems. Therefore, when considering both the strength and proportion of contextual quantum states, Kochen-Specker contextuality is the strongest.
Abstract Quantum contextuality represents a fundamental form of nonclassicality in quantum mechanics. To provide a more complete characterization of nonclassical properties in quantum systems, we adopt a logical perspective and propose a mathematical framework based on exclusive partial Boolean algebras. This framework enables a unified description of contextuality and nonclassicality across finite general, quantum, and classical systems. We establish a unified and minimal classical counterpart for any finite general system. Within this framework, we formalize major categories of quantum contextuality, demonstrating that: 12 projectors suffice to generate Kochen–Specker scenarios; 10 projectors suffice to witness state-independent contextuality; and 3 observables suffice to witness quantum contextuality. Finally, we prove that contextuality is a sufficient but not necessary condition for nonclassicality.