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

Abstract

<jats:p>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.</jats:p>

Even and Odd Schrödinger Cat States in the Probability Representation of Quantum Mechanics

“…Tomogram of cat states (24) contains the interference contribution expressed in terms of integrals (23), with the parameters The Wigner functions of cat states W ± (q, p) are given by Fourier integrals (22) of the cat-state tomograms. For μ = cos θ and ν = sin θ, the calculated tomograms are optical tomograms measured for photon states by homodyne detectors [86].…”

Probability Representation of Quantum States

“…The four quantizers are of the form The quantizers and dequantizers satisfy the relations The density matrix can be presented in the form where the probabilities , are determined by the dequantizers and density matrix as follows: The density matrix is linear combination of quantizers. The standard parameterization of qudit density matrices by Bloch sphere parameters [ 65 ] and relation of it with discussed probability representation of qubit states was studied in Reference [ 28 ].…”

“…The main aim of the paper is to review the methods of obtaining the probability representation of quantum states and to formulate the rules of constructing the probability distribution representations of quantum system states on examples of the quantum oscillator, free particle and qubit. The idea of this construction is to find the invertible map of density operators onto probability distributions applying the Born’s rule [ 22 , 23 ] and the method of quantizer–dequantizer operators [ 24 , 25 , 26 ] (also see References [ 27 , 28 ]). The development and review of this method of quantizer–dequantizer, which is important for studying the systems with continuous variables is presented in Reference [ 29 ].…”

PT -Symmetric Qubit-System States in the Probability Representation of Quantum Mechanics

“…Our goal in this section is to demonstrate the extension of the approach to other states; for this, we construct the probability representation of eigenvectors for qudit system on the example of the qutrit state. As it was shown in [51,53], an arbitrary N×N matrix ρ, such that ρ = ρ † , Tr ρ = 1, with nonzero eigenvalues, has matrix elements which can be parameterized as follows:…”