Abstract
We introduce a hierarchy of semidefinite relaxations of the set of quantum correlations in generalised 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.
Highlights
The contextuality of quantum theory is a fundamental sign of its nonclassicality that has been investigated for several decades
Quantum contextuality can be witnessed in contextuality scenarios involving only operational equivalences between the preparations [along with the trivial measurement operational equivalence arising from Eq (9), which is necessarily satisfied by any quantum model for any contextuality scenario]
In this paper we introduce a semidefinite relaxation hierarchy for bounding the set of contextual quantum correlations and demonstrate its usefulness by applying it to solve several open problems in quantum contextuality
Summary
The contextuality of quantum theory is a fundamental sign of its nonclassicality that has been investigated for several decades. Quantum models that respect the operational equivalences may produce “contextual correlations” unobtainable by any such noncontextual model [3] This leads to a conceptually natural question: how can we determine if, for a given contextuality scenario, a given set of contextual correlations is compatible with quantum theory? We provide an answer to the question by introducing a hierarchy of semidefinite relaxations of the set of quantum correlations arising in contextuality scenarios involving arbitrary operational equivalences between preparations and measurements. This constitutes a sequence of increasingly precise necessary conditions that contextual correlations must satisfy in order to admit. We apply these concepts to the simplest preparation contextuality scenario [19], where we explicitly derive both the classical and quantum simulation costs of contextuality
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