Malevich’s Suprematist Composition Picture for Spin States

Vladimir I Man’Ko,Liubov A Markovich

doi:10.3390/e21090870

Abstract

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 random vector variables and 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.

Highlights

In modern science quantum systems are powerful resource for information processing
The single qubit is identified with the set of three probability distributions of spin projections on three perpendicular directions in space
We proposed to go further and introduce polygons, built on simplexes, formed by the probabilities, that parametrize the density matrix, to describe the higher dimension quantum systems

Summary

Introduction

In modern science quantum systems are powerful resource for information processing. Many physical properties and phenomena are difficult to understand, but the geometric interpretations of quantum mechanical systems deliver an elegant way of understanding and “feeling” them. That is why the geometrical picture of physical theories draw attention in a wide range of fields from classical and quantum mechanics to the general relativity (cf [1]). The practical implementation of large scale quantum communication networks and key distribution [2], quantum cryptography [3,4,5], quantum random number generation [6] or a quantum computer, is the main goal of quantum information science. That is a lot due to the fact, that for the practical use, multipartite high fidelity entangled quantum states are needed, that are challenging to control technologically. The quantum computer with two 32-state qudits, would be Entropy 2019, 21, 870; doi:10.3390/e21090870 www.mdpi.com/journal/entropy

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