Locally maximally entangled states of multipart quantum systems

Abstract

For a multipart quantum system, a locally maximally entangled (LME) state is one where each elementary subsystem is maximally entangled with its complement. This paper is a sequel to [BRV], which gives necessary and sufficient conditions for a system to admit LME states in terms of its subsystem dimensions (d1, d2, . . . , dn), and computes the dimension of the space SLME/K of LME states up to local unitary transformations for all non-empty cases. Here we provide a pedagogical overview and physical interpretation of the underlying mathematics that leads to these results and give a large class of explicit constructions for LME states. In particular, we construct all LME states for tripartite systems with subsystem dimensions (2,A,B) and give a general representation-theoretic construction for a special class of stabilizer LME states. The latter construction provides a common framework for many known LME states. Our results have direct implications for the problem of characterizing SLOCC equivalence classes of quantum states, since points in SLME/K correspond to natural families of SLOCC classes. Finally, we give the dimension of the stabilizer subgroup S ⊂ SL(d1,C) × × SL(dn,C) for a generic state in an arbitrary multipart system and identify all cases where this stabilizer is trivial.