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.
Highlights
The Leibnizian methodological principle of “ontological identity of empirical indiscernibles” [CvLNA56] creates a bridge across the schism dividing the “empiricist”, and “realist” viewpoints on physics [Spe19]
We sought to characterise the behaviours appearing in generalised contextuality scenarios with operational equivalences of preparations and measurements
11We note that in Ref. [SBA17] the authors study quantum key distribution based on the monogamy of the Klyachko–Can–Binicioglu–Shumovsky-type contextuality, and in Ref. [TF15] the authors propose a quantum key distribution protocol based on generalised measurement contextuality
Summary
The Leibnizian methodological principle of “ontological identity of empirical indiscernibles” [CvLNA56] creates a bridge across the schism dividing the “empiricist”, and “realist” viewpoints on physics [Spe19]. In analogy with quantum non-locality [BCP+14], quantum theory surpasses these noncontextual limitations, but not necessarily to the maximum mathematically possible extent [BBM+15] Bounding this violation provides foundational insights, such as establishing how distinct operationally indistinguishable entities must be in order to explain quantum predictions [CS20, Mar20], and finds technological applications in capping quantum advantage. Our most significant contribution constitutes a semidefinite relaxation technique which employs moment matrices indexed exclusively by monomials of unitary operators, which may be of independent interest We benchmark these relaxations by recovering maximal quantum violations of several noncontextuality inequalities, and identifying optimal quantum protocols with respect to several noncontextuality inequalities in a diverse selection of contextuality scenarios. To exhibit the relevance of these relaxations to real-world applications, we present a secure, semi-device-independent one-way quantum key distribution scheme powered by preparation contextuality
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