The hierarchical structure of local unitary invariants

TL;DR

This paper explores the structure of local unitary invariants in multipartite quantum states, revealing a hierarchical organization that simplifies their study and relates them to cumulants and entanglement properties.

Contribution

It uncovers a hierarchical structure among local unitary invariants, showing they form families related by tracing out subsystems, and connects these to cumulants and separability.

Findings

Invariants form families related by tracing out subsystems.

A family of invariants in pure qubit systems is about half the orbit space dimension.

Members of these families are related to cumulants and multipartite separability.

Abstract

Local unitary invariants allow one to test whether multipartite states are equivalent up to local basis changes. Equivalently, they specify the geometry of the "orbit space" obtained by factoring out local unitary action from the state space. This space is of interest because of its intimate relationship to entanglement. Unfortunately, the dimension of the orbit space grows exponentially with the number of subsystems, and the number of invariants needed to characterise orbits grows at least as fast. This makes the study of entanglement via local unitary invariants seem very daunting. I point out here that there is a simplifying principle: Invariants fall into families related by the tracing-out of subsystems, and these families grow exponentially with the number of subsystems. In particular, in the case of pure qubit systems, there is a family whose size is about half the dimension of orbit space. These invariants are closely related to cumulants and to multipartite separability. Members of the family have been repeatedly discovered in the literature, but the fact that they are related to cumulants and constitute a family has apparently not been observed.