Quantum contextuality for rational vectors

Zhenghao Liu

Ida Cortez, Camilo Morales, Manuel L. Reyes

This paper is a contribution to the algebraic study of contextuality in quantum theory. As an algebraic analogue of Kochen and Specker’s no-hidden-variables result, we investigate rational subrings over which the partial ring of [Formula: see text] symmetric matrices ([Formula: see text]) admits no morphism to a nonzero commutative ring, which we view as an “algebraic hidden state”. For [Formula: see text], the minimal such ring is shown to be [Formula: see text], while for [Formula: see text] the minimal subring is [Formula: see text] itself. The proofs rely on the construction of new sets of integer vectors in dimensions 3 and 6 that have no Kochen–Specker coloring.

Jochen Szangolies

Chang He, Yongjun Wang, Baoshan Wang +2 more

Quantum contextuality is a fundamental nonclassical property of quantum systems, regarded as a key resource that demonstrates the computational and informational advantages of quantum over classical systems. Our present work aims to construct Hardy's paradoxes, a set of possibilistic conditions witnessing contextuality, for Yu-Oh set, which is the state-independent contextual quantum system with the least number of vectors. To achieve the aim, we systematically enumerate all logically contextual pure states on Yu-Oh set, and theoretically prove that no mixed states in this scenario are logically contextual. Based on the identified logically contextual quantum states, we construct 12 Hardy's paradoxes with identical success probability SP=11.1%. Furthermore, we present corresponding observables to experimentally witness these Hardy's paradoxes.

Jaskaran Singh, Rajendra Singh Bhati, Arvind

The two-state vector formalism (TSVF) proposed by Aharonov, Bergmann, and Lebowitz [A. Aharonov, P. G. Bergmann, and J. L. Lebowitz, Phys. Rev. 134, B1410 (1964)] allows a counterfactual assignment of probabilities of outcomes of contemplated (but unperformed) measurements on quantum systems. The probabilities assigned by the Aharonov-Bergmann-Lebowitz (ABL) rule and the associated weak values have been used to provide insights into quantum situations and to unearth underlying quantum contextuality. We apply the principle of exclusivity on ABL probabilities which are assigned to mutually orthogonal projectors to define paradoxical and nonparadoxical scenarios. Given a pre- and a postselected pair of states, we consider the nonparadoxical sector with a view to explore the demonstration of quantum contextuality. For the Klyachko-Can-Binicio\ifmmode \breve{g}\else \u{g}\fi{}lu-Shumovsky scenario, we numerically show that the ABL probabilities of the TSVF in the nonparadoxical sector do not offer any contextual advantage. Our approach can be easily generalized to other contextual scenarios as well. We thus argue that several previous proofs of the emergence of contextuality in pre- and postselected scenarios are only possible if the principle of exclusivity is violated and are therefore classified as paradoxical.

Adán Cabello

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 $d \ge 3$ to prove contextuality for a system prepared in a maximally mixed state is that the graph $G_C$ in which vertices represent vectors and edges link orthogonal ones has chromatic number $χ(G_G)$ larger than $d$. Then, we prove Yu and Oh's conjecture. Finally, we prove that any set satisfying $χ(G_G)>d$ assisted with $d$-party maximum entanglement generates nonlocality which cannot be improved without violating the no-signalling principle. This shows that any set satisfying $χ(G_G)>d$ is a valuable resource for quantum information processing.