Characterising and bounding the set of quantum behaviours in contextuality scenarios

Abstract

The predictions of quantum theory resist generalised noncontextual explanations. In addition to the foundational relevance of this fact, the particular extent to which quantum theory violates noncontextuality limits available quantum advantage in communication and information processing. In the first part of this work, we formally define contextuality scenarios via prepare-and-measure experiments, along with the polytope of general contextual behaviours containing the set of quantum contextual behaviours. This framework allows us to recover several properties of set of quantum behaviours in these scenarios, including contextuality scenarios and associated noncontextuality inequalities that require for their violation the individual quantum preparation and measurement procedures to be mixed states and unsharp measurements. With the framework in place, we formulate novel semidefinite programming relaxations for bounding these sets of quantum contextual behaviours. Most significantly, to circumvent the inadequacy of pure states and projective measurements in contextuality scenarios, we present a novel unitary operator based semidefinite relaxation technique. We demonstrate the efficacy of these relaxations by obtaining tight upper bounds on the quantum violation of several noncontextuality inequalities and identifying novel maximally contextual quantum strategies. To further illustrate the versatility of these relaxations, we demonstrate monogamy of preparation contextuality in a tripartite setting, and present a secure semi-device independent quantum key distribution scheme powered by quantum advantage in parity oblivious random access codes.

Partial ignorance communication tasks in quantum theory

“…What we do get is an upper bound on the success metric -no quantum behavior can exceed this bound. Moreover, by considering monomials of length greater than one we obtain a converging hierarchy that converges on the set of quantum behaviors [11]. In practise we get very good results already on the first level.…”

“…There are many excellent sources on the theory of SDPs [19][20][21][22][23][24][25][26][27]. We will be using the unitary SDP hierarchy introduced in [11] largely due to the fact that it gives very good results already on the first level.…”

“…Suppose we have a quantum behavior p(k| i , M j ) = tr [ i M j (k)]. As was proven in [11], we can always find unitary matrices such that…”

Noise-robust preparation contextuality shared between any number of observers via unsharp measurements

“…Note added.-In the substantial time between the arXiv submission of this work and its publication, also other works concerning the sequential sharing of contextuality have appeared [28,37].…”

Quantum description of reality is empirically incomplete

“…Similar to the Navascués-Pironio-Acín hierarchy for nonlocal quantum correlations[17], and[18,19] for generalized contextuality scenarios,[20] for informationally restricted correlations.…”