A link between symmetries of critical states and the structure of SLOCC classes in multipartite systems

TL;DR

This paper explores the structure of SLOCC classes in multipartite quantum states, providing criteria for the existence of strictly semistable states and classifying orbit types in tripartite qubit systems.

Contribution

It introduces a new criterion for identifying strictly semistable states based on orbit dimensions and classifies all SLOCC classes in tripartite qubit systems.

Findings

Existence of strictly semistable states linked to orbit dimension differences.

Complete classification of SLOCC classes in tripartite qubit systems.

Identification of representatives for strictly semistable classes.

Abstract

Central in entanglement theory is the characterization of local transformations among pure multipartite states. As a first step towards such a characterization, one needs to identify those states which can be transformed into each other via local operations with a non-vanishing probability. The classes obtained in this way are called SLOCC classes. They can be categorized into three disjoint types: the null-cone, the polystable states and strictly semistable states. Whereas the former two are well characterized, not much is known about strictly semistable states. We derive a criterion for the existence of the latter. In particular, we show that there exists a strictly semistable state if and only if there exist two polystable states whose orbits have different dimensions. We illustrate the usefulness of this criterion by applying it to tripartite states where one of the systems is a qubit. Moreover, we scrutinize all SLOCC classes of these systems and derive a complete characterization of the corresponding orbit types. We present representatives of strictly semistable classes and show to which polystable state they converge via local regular operators.