Locally Maximally Entangled States of Multipart Quantum Systems

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~[J. Bryan, Z. Reichstein and M. Van Raamsdonk, Existence of Locally Maximally Entangled Quantum States via Geometric Invariant Theory, Ann. Henri Poincaré 19 (2018), no. 8, 2491-2511. MR3830220], which gives necessary and sufficient conditions for a system to admit LME states in terms of its subsystem dimensions ( d 1 , d 2 , … , d n ) , and computes the dimension of the space S L M E / 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 S L M E / K correspond to natural families of SLOCC classes. Finally, we give the dimension of the stabilizer subgroup S ⊂ SL ( d 1 , C ) × ⋯ × SL ( d n , C ) for a generic state in an arbitrary multipart system and identify all cases where this stabilizer is trivial.

Abstract licence: CC BY

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