Majorana representation for mixed states

TL;DR

This paper extends the Majorana stellar representation from pure to mixed spin states, establishing a bijective correspondence with polynomial spaces and constellation classes, and demonstrating its rotational and partial trace properties.

Contribution

It introduces a generalized Majorana representation for mixed states, linking density matrices to polynomial and constellation frameworks with consistent transformation behaviors.

Findings

Representation is well-behaved under rotations.

Reduced density matrices inherit constellation classes.

Expresses density matrix concepts via polynomials.

Abstract

We generalize the Majorana stellar representation of spin-$s$ pure states to mixed states, and in general to any hermitian operator, defining a bijective correspondence between three spaces: the spin density-matrices, a projective space of homogeneous polynomials of four variables, and a set of equivalence classes of points (constellations) on spheres of different radii. The representation behaves well under rotations by construction, and also under partial traces where the reduced density matrices inherit their constellation classes from the original state $\rho$. We express several concepts and operations related to density matrices in terms of the corresponding polynomials, such as the anticoherence criterion and the tensor representation of spin-$s$ states described in [1].