Geometrical Domain of Spin 1/2 Probability Mass Function

TL;DR

This paper explores the geometric domain where the probability mass function of a spin 1/2 quantum system is positive, revealing it as an octahedron within the Bloch sphere and linking it to quantum characteristic functions.

Contribution

It identifies the domain of positive quasi distributions for spin 1/2 systems as an octahedron inscribed in the Bloch sphere, connecting geometry with quantum probability functions.

Findings

Positive regions of Wigner and Margenau-Hill distributions form an octahedron.

Quantum characteristic functions admit a probability mass function within this octahedral domain.

Classical variates are independent only when the Bloch vector is on an axis.

Abstract

The quantum analogue of the classical characteristic function for a spin 1/2 assembly is considered and the probability mass function of the random vector associated with the assembly is derived. It is seen that the positive regions of Wigner and Margenau-Hill quasi distributions for the three components of spin, correspond to a trivariate probability mass function. We identify the domain of these positive regions as an Octahedron inscribed in the Bloch sphere with its vertices on the surface of the sphere. It is in this domain that a quantum characteristic function characterizing the quasi distribution, admits a probability mass function in IR^3 . It is also observed that the classical variates X1, X2, X3 corresponding to the 3 spin operators \sigma_1,\sigma_2,\sigma_3 in the domain, are independent iff the Bloch vector lies on any one of the axes.