The hierarchical structure of local unitary invariants

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.