Symmetric mixed states of $n$ qubits: local unitary stabilizers and entanglement classes

TL;DR

This paper classifies symmetric mixed states of n qubits based on their local unitary stabilizers and entanglement classes, using polynomial representations and SO(3) symmetry to organize the states.

Contribution

It introduces a classification of symmetric mixed states by their stabilizer Lie algebras and entanglement types, extending understanding of their structure and symmetries.

Findings

Six classes of stabilizer Lie algebras identified

Entanglement classes classified for all but the zero algebra

Application of SO(3) representation theory to symmetric states

Abstract

We classify, up to local unitary equivalence, local unitary stabilizer Lie algebras for symmetric mixed states into six classes. These include the stabilizer types of the Werner states, the GHZ state and its generalizations, and Dicke states. For all but the zero algebra, we classify entanglement types (local unitary equivalence classes) of symmetric mixed states that have those stabilizers. We make use of the identification of symmetric density matrices with polynomials in three variables with real coefficients and apply the representation theory of SO(3) on this space of polynomials.