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arXiv:1712.01927 · Journal of Physics Conference Series

Quantum suprematism picture of Triada of Malevich’s squares for spin states and the parametric oscillator evolution in the probability representation of quantum mechanics

V. N. Chernega, O. V. Man’ko, V. I. Man’ko

Published Aug 1, 2018• 17 citations
arXivHTMLPDF

Abstract

Review of tomographic probability representation of quantum states is presented both for oscillator systems with continious variables and spin–systems with discrete variables. New entropy–information inequalities are obtained for Franck–Condon factors. Density matrices of qudit states are expressed in terms of probabilities of artificial qubits as well as the quantum suprematism approach to geometry of these states using the triadas of Malevich squares is developed. Examples of qubits, qutrits and ququarts are considered.

DOI
10.1088/1742-6596/1071/1/012008

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Review of tomographic probability representation of quantum states is presented both for oscillator systems with continious variables and spin–systems with discrete variables. New entropy–information inequalities are obtained for Franck–Condon factors. Density matrices of qudit states are expressed in terms of probabilities of artificial qubits as well as the quantum suprematism approach to geometry of these states using the triadas of Malevich squares is developed. Examples of qubits, qutrits and ququarts are considered.
Abstract We develop the tomographic-probability picture where quantum states of qudits (spin- j systems, N -level atoms) are described by means of sets of fair classical probability distributions associated with properties of sets of dichotomic classical random variables. We formulate the superposition principle of pure quantum states as nonlinear addition rule of standard probabilities. The invertible map of the density matrices of qudit states onto the probability distributions is illustrated by their geometric representation in terms of triangles and squares called Malevich’s squares. We discuss an analogy of the squares with the quantum carpet picture. New entropic inequalities obtained due to the introduced map of the density matrices onto Malevich’s squares are studied. The correlations in systems without subsystems (hidden correlations) are discussed.
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.
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.
Semigroups of stochastic and bistochastic matrices constructed by means of spin tomograms or tomographic probabilities and their relations to the problem of Bell's inequalities and entanglement are reviewed. The probability determining the quantum state of spins and the probability densities determining the quantum states of particles with continuous variables are considered. Entropies for semigroups of stochastic and bisctochastic matrices are studied, in view of both the Shannon information entropy and its generalization like Rényi entropy. Qubit portraits of qudit states are discussed in the connection with the problem of Bell's inequality violation for entangled states.
A new geometric representation of qubit and qutrit states based on probability simplexes is used to describe the separability and entanglement properties of density matrices of two qubits. The Peres-Horodecki positive partial transpose (ppt) -criterion and the concurrence inequalities are formulated as the conditions that the introduced probability distributions must satisfy to present entanglement. A four-level system, where one or two states are inaccessible, is considered as an example of applying the elaborated probability approach in an explicit form. The areas of three Triadas of Malevich's squares for entangled states of two qubits are defined through the qutrit state, and the critical values of the sum of their areas are calculated. We always find an interval for the sum of the square areas, which provides the possibility for an experimental checkup of the entanglement of the system in terms of the probabilities.