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Entropy 2018, 20(9), 692; https://doi.org/10.3390/e20090692

New Entropic Inequalities and Hidden Correlations in Quantum Suprematism Picture of Qudit States

1
Lebedev Physical Institute, Russian Academy of Sciences, Leninskii Prospect 53, Moscow 119991, Russia
2
Moscow Institute of Physics and Technology (State University), Institutskii per. 9, Dolgoprudnyi, Moscow Region 141700, Russia
3
Department of Physics, Tomsk State University, Lenin Avenue 36, Tomsk 634050, Russia
We dedicate this article to the memory of Professor E. C. G. Sudarshan, the Great Scientist and our friend and colleague with whom we were happy to collaborate during many years.
*
Author to whom correspondence should be addressed.
Received: 17 August 2018 / Revised: 5 September 2018 / Accepted: 7 September 2018 / Published: 11 September 2018
(This article belongs to the Special Issue Quantum Probability and Randomness)
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Abstract

We study an analog of Bayes’ formula and the nonnegativity property of mutual information for systems with one random variable. For single-qudit states, we present new entropic inequalities in the form of the subadditivity and condition corresponding to hidden correlations in quantum systems. We present qubit states in the quantum suprematism picture, where these states are identified with three probability distributions, describing the states of three classical coins, and illustrate the states by Triada of Malevich’s squares with areas satisfying the quantum constraints. We consider arbitrary quantum states belonging to N-dimensional Hilbert space as ( N 2 1 ) fair probability distributions describing the states of ( N 2 1 ) classical coins. We illustrate the geometrical properties of the qudit states by a set of Triadas of Malevich’s squares. We obtain new entropic inequalities for matrix elements of an arbitrary density N×N-matrix of qudit systems using the constructed maps of the density matrix on a set of the probability distributions. In addition, to construct the bijective map of the qudit state onto the set of probabilities describing the positions of classical coins, we show that there exists a bijective map of any quantum observable onto the set of dihotomic classical random variables with statistics determined by the above classical probabilities. Finally, we discuss the physical meaning and possibility to check derived inequalities in the experiments with superconducting circuits based on Josephson junction devices. View Full-Text
Keywords: entropy; correlations; qubits; probability representation; Bayes’ formula entropy; correlations; qubits; probability representation; Bayes’ formula
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Man’ko, M.A.; Man’ko, V.I. New Entropic Inequalities and Hidden Correlations in Quantum Suprematism Picture of Qudit States. Entropy 2018, 20, 692.

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