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

Peter Adam,Vladimir Andreev,Vladimir Man’Ko,Margarita Man’Ko,Matyas Mechler

doi:10.3390/sym12071099

Abstract

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.

Highlights

In conventional formulation of quantum mechanics, the states of quantum systems are identified either with wave functions [1,2] and state vectors [3] or with density matrices [4] and density operators [5]
We presented the probability representation of qubit states introducing the quantizer–dequantizer operator formalism in an explicit form and calculating the structure constants of the associative product of the symbols of spin operators, as well as the Lie algebra structure constants of the systems associated with SU (2) symmetry
We obtained in an explicit form the structure constants of the associative algebra and Lie and Jordan algebras using the quantizer–dequantizer operators determining symplectic tomograms of quantum states for systems with continuous degrees of freedom

Summary

Introduction

In conventional formulation of quantum mechanics, the states of quantum systems are identified either with wave functions [1,2] and state vectors [3] (in the case of pure states) or with density matrices [4] and density operators [5] (in the case of mixed states). We introduce the approach to Lie algebras and corresponding symmetries of systems with the star-product structure constants associated with quantization procedure providing the description of states by probability distributions. Xn are considered as fixed parameters, while the indices j are determined according the physical properties of the system under consideration (for example, in Section 9, x = x1 , x2 , x2 and j = 1) This property means that, in this case, the function f ρ ( x ) can be interpreted as a conditional probability distribution, and we introduce the notation f ρ ( x ) = P x 1 , x 2 , . We consider particular examples of the above approach for qubit states and photon states, where the number of random variable is not larger than three

Structure Constants of Lie Algebras and Their Relation to Associative Product

The von Neumann Evolution Equation for the Density Matrix ρ in Vector Form

The Probability Representation of the von Neumann Equation

Transforms of Quantizer and Dequantizer Operators

Symplectic Tomographic Probability Distribution

10. Conclusions