Symmetric states: local unitary equivalence via stabilizers

TL;DR

This paper classifies symmetric quantum states based on their local unitary stabilizer groups, providing a comprehensive understanding of their equivalence classes, especially highlighting the role of continuous and discrete stabilizers.

Contribution

It introduces a classification scheme for symmetric states using their stabilizer subgroups, including a complete classification for states with continuous stabilizers.

Findings

States with continuous stabilizers have an exhaustive classification.

Finite stabilizers are isomorphic to subgroups of SO(3).

Examples of states with discrete stabilizers are provided.

Abstract

We classify local unitary equivalence classes of symmetric states via a classification of their local unitary stabilizer subgroups. For states whose local unitary stabilizer groups have a positive number of continuous degrees of freedom, the classification is exhaustive. We show that local unitary stabilizer groups with no continuous degrees of freedom are isomorphic to finite subgroups of the rotation group SO(3), and give examples of states with discrete stabilizers.