Quantifying contextuality.
Contextuality is central to both the foundations of quantum theory and to the novel information processing tasks. Despite some recent proposals, it still faces a fundamental problem: how to quantify its presence? In this work, we provide a universal framework for quantifying contextuality. We conduct two complementary approaches: (i) the bottom-up approach, where we introduce a communication game, which grasps the phenomenon of contextuality in a quantitative manner; (ii) the top-down approach, where we just postulate two measures, relative entropy of contextuality and contextuality cost, analogous to existent measures of nonlocality (a special case of contextuality). We then match the two approaches by showing that the measure emerging from the communication scenario turns out to be equal to the relative entropy of contextuality. Our framework allows for the quantitative, resource-type comparison of completely different games. We give analytical formulas for the proposed measures for some contextual systems, showing in particular that the Peres-Mermin game is by order of magnitude more contextual than that of Klyachko et al. Furthermore, we explore properties of these measures such as monotonicity or additivity.
Faculty of Physics, Adam Mickiewicz University, 61-614 Poznań, Poland.
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Grudka
National Quantum Information Center of Gdańsk, 81-824 Sopot, Poland and Institute of Informatics, University of Gdańsk, 80-952 Gdańsk, Poland.
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Horodecki
National Quantum Information Center of Gdańsk, 81-824 Sopot, Poland and Institute of Theoretical Physics and Astrophysics, University of Gdańsk, 80-952 Gdańsk, Poland.
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Horodecki
National Quantum Information Center of Gdańsk, 81-824 Sopot, Poland and Faculty of Applied Physics and Mathematics, Technical University of Gdańsk, 80-233 Gdańsk, Poland.
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Horodecki
National Quantum Information Center of Gdańsk, 81-824 Sopot, Poland and Institute of Theoretical Physics and Astrophysics, University of Gdańsk, 80-952 Gdańsk, Poland.
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Horodecki
National Quantum Information Center of Gdańsk, 81-824 Sopot, Poland and Institute of Theoretical Physics and Astrophysics, University of Gdańsk, 80-952 Gdańsk, Poland.
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Joshi
Faculty of Physics, Adam Mickiewicz University, 61-614 Poznań, Poland.
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Kłobus
Faculty of Physics, Adam Mickiewicz University, 61-614 Poznań, Poland.
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Wójcik