SU(2) Symmetry of Qubit States and Heisenberg–Weyl Symmetry of Systems with Continuous Variables in the Probability Representation of Quantum Mechanics

Autor: Peter Adam, Vladimir A. Andreev, Margarita A. Man’ko, Vladimir I. Man’ko, Matyas Mechler
Jazyk: angličtina
Rok vydání: 2020
Předmět:
Zdroj: Symmetry, Vol 12, Iss 7, p 1099 (2020)
Druh dokumentu: article
ISSN: 12071099
2073-8994
DOI: 10.3390/sym12071099
Popis: In view of the probabilistic quantizer–dequantizer operators introduced, the qubit states (spin-1/2 particle states, two-level atom states) realizing the irreducible representation of the S U ( 2 ) symmetry group are identified with probability distributions (including the conditional ones) of classical-like dichotomic random variables. The dichotomic random variables are spin-1/2 particle projections m = ± 1 / 2 onto three perpendicular directions in the space. The invertible maps of qubit density operators onto fair probability distributions are constructed. In the suggested probability representation of quantum states, the Schrödinger and von Neumann equations for the state vectors and density operators are presented in explicit forms of the linear classical-like kinetic equations for the probability distributions of random variables. The star-product and quantizer–dequantizer formalisms are used to study the qubit properties; such formalisms are discussed for photon tomographic probability distribution and its correspondence to the Heisenberg–Weyl symmetry properties.
Databáze: Directory of Open Access Journals
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