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Contextuality is a fundamental feature of quantum theory and is necessary for quantum computation and communication. Serious steps have therefore been taken towards a formal framework for contextuality as an operational resource. However, the most important component for a resource theory - a concrete, explicit form for the free operations of contextuality - was still missing. Here we provide such a component by introducing noncontextual wirings: a physically-motivated class of contextuality-free operations with a friendly parametrization. We characterize them completely for the general case of black-box measurement devices with arbitrarily many inputs and outputs. As applications, we show that the relative entropy of contextuality is a contextuality monotone and that maximally contextual boxes that serve as contextuality bits exist for a broad class of scenarios. Our results complete a unified resource-theoretic framework for contextuality and Bell nonlocality.
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Quantum measurements are noncontextual, with outcomes independent of which other commuting observables are measured at the same time, when consistently analyzed using principles of Hilbert space quantum mechanics rather than classical hidden variables.
We discuss chromatic constructions on orthogonality hypergraphs which are classical set representable or have a faithful orthogonal representation. The latter ones have a quantum mechanical realization in terms of intertwined contexts or maximal observables. Structure reconstruction of these hypergraphs from their table of two-valued states is possible for a class of hypergraphs, namely perfectly separable hypergraphs. Some examples from exempt categories that either cannot be reconstructed by two-valued states or whose set of two-valued states does not yield a coloring are presented.
One implication of Bells theorem is that there cannot in general be hidden variable models for quantum mechanics that both are noncontextual and retain the structure of a classical probability space. Thus, some hidden variable programs aim to retain noncontextuality at the cost of using a generalization of the Kolmogorov probability axioms. We generalize a theorem of Feintzeig (2015) to show that such programs are committed to the existence of a finite null cover for some quantum mechanical experiments, i.e., a finite collection of probability zero events whose disjunction exhausts the space of experimental possibilities.
It has been recently shown, that some of the tripartite boxes admitting bilocal decomposition, lead to non-locality under wiring operation applied to two of the subsystems [R. Gallego et al. Physical Review Letters 109, 070401 (2012)]. In the following, we study this phenomenon quantitatively. Basing on the known classes of boxes closed under wirings, we introduced multipartite monotones which are counterparts of bipartite ones - the non-locality cost and robustness of non-locality. We then provide analytical lower bounds on both the monotones in terms of the Maximal Non-locality which can be obtained by Wirings (MWN). We prove also upper bounds for the MWN of a given box, based on the weight of boxes signaling in a particular direction, that appear in its bilocal decomposition. We study different classes of partially local boxes and find MWN for each class, using Linear Programming. We identify also the wirings which lead to MWN and exhibit that some of them can serve as a witness of certain classes. We conclude with example of partially local boxes being analogue of quantum states that allow to distribute entanglement in separable manner.
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