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arXiv:1507.03142 · Physical Review A

Quantum theory allows for absolute maximal contextuality

Bárbara Amaral, Marcelo Terra Cunha, Adán Cabello

Published Dec 16, 2015• 23 citations
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Abstract

Contextuality is believed to be a fundamental property of quantum theory. On investigating how large it can be, instances are shown wherein quantum theory achieves the maximum conceivable value of a certain contextuality quantifier.
DOI
10.1103/physreva.92.062125

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Abstract This paper investigates various properties that may by possessed by quantum states, which are believed to be specifically “quantum” (entanglement, nonlocality, steerability, negative conditional entropy, non-zero quantum discord, non-zero quantum super discord and contextuality) and their opposites. It also considers their “absolute” counterparts in the following sense: a given state has a given property absolutely if after an arbitrary unitary transformation it still possesses it. The known relations between the listed properties and between their absolute counterparts are summarized. It is proven that the only two-qubit state that has zero quantum discord absolutely is the maximally mixed state. Finally, related conceptual issues concerning the terms “classical” and “quantum” are discussed.
The operational characterization of contextuality is the cornerstone in the development of resource theory of contextuality. Here, we introduce contextual robustness as an operational measure of contextuality on empirical models. We first show that it satisfies all the properties a proper contextual measure should satisfy. Besides, we derive the linear programing of contextual robustness and prove that contextual robustness exactly characterizes the maximal violation of certain types of Bell inequalities. Moreover, we show that contextual robustness quantifies the accessible advantage of empiric models in measurement-based quantum computation, and provide a tighter upper bound on accessible advantage in certain cases.
We introduce a hierarchy of semidefinite relaxations of the set of quantum correlations in generalized contextuality scenarios.This constitutes a simple and versatile tool for bounding the magnitude of quantum contextuality.To illustrate its utility, we use it to determine the maximal quantum violation of several noncontextuality inequalities whose maximum violations were previously unknown.We then go further and use it to prove that certain preparation-contextual correlations cannot be explained with pure states, thereby showing that mixed states are an indispensable resource for contextuality.In the second part of the paper, we turn our attention to the simulation of preparation-contextual correlations in general operational theories.We introduce the information cost of simulating preparation contextuality, which quantifies the additional, otherwise forbidden, information required to simulate contextual correlations in either classical or quantum models.In both cases, we show that the simulation cost can be efficiently bounded using a variant of our hierarchy of semidefinite relaxations, and we calculate it exactly in the simplest contextuality scenario of parity-oblivious multiplexing.