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Malevich’s Suprematist Composition Picture for Spin States

by 1,2,† and 3,4,*,†
1
P.N. Lebedev Physical Institute, Russian Academy of Sciences, Leninskii Prospect 53, Moscow 119991, Russia
2
Moscow Institute of Physics and Technology, Institutskii Per. 9, Dolgoprudny, Moscow Region 141700, Russia
3
Russian Quantum Center, 100 Novaya St., Skolkovo, Moscow 143025, Russia
4
V. A. Trapeznikov Institute of Control Sciences, Moscow, Profsoyuznaya 65, Moscow 117997, Russia
*
Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Entropy 2019, 21(9), 870; https://doi.org/10.3390/e21090870
Received: 29 July 2019 / Accepted: 28 August 2019 / Published: 6 September 2019
(This article belongs to the Section Quantum Information)
This paper proposes an alternative geometric representation of single qudit states based on probability simplexes to describe the quantum properties of noncomposite systems. In contrast to the known high dimension pictures, we present the planar picture of quantum states, using the elementary geometry. The approach is based on, so called, Malevich square representation of the single qubit state. It is shown that the quantum statistics of the single qudit with some spin j and observables are formally equivalent to statistics of the classical system with N 2 - 1 random vector variables and N 2 - 1 classical probability distributions, obeying special constrains, found in this study. We present a universal inequality, that describes the single qudits state quantumness. The inequality provides a possibility to experimentally check up entanglement of the system in terms of the classical probabilities. The simulation study for the single qutrit and ququad systems, using the Metropolis Monte-Carlo method, is obtained. The geometrical representation of the single qudit states, presented in the paper, is useful in providing a visualization of quantum states and illustrating their difference from the classical ones.
Keywords: geometry of quantum states; qudit; noncomposite quantum systems; uncertainty relations geometry of quantum states; qudit; noncomposite quantum systems; uncertainty relations
MDPI and ACS Style

Man’ko, V.I.; Markovich, L.A. Malevich’s Suprematist Composition Picture for Spin States. Entropy 2019, 21, 870.

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