Locally Maximally Entangled States of Multipart Quantum Systems

TL;DR

This paper explores the mathematical structure and explicit constructions of locally maximally entangled states in multipart quantum systems, providing new insights into their classification, dimensions, and stabilizer groups.

Contribution

It offers a pedagogical overview, explicit constructions, and a general framework for understanding LME states, extending previous results on their existence and classification.

Findings

Constructed all LME states for tripartite systems with dimensions (2,A,B)

Provided a representation-theoretic construction for stabilizer LME states

Determined the dimension of the space of SLOCC equivalence classes for generic states

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 arXiv:1708.01645, which gives necessary and sufficient conditions for a system to admit LME states in terms of its subsystem dimensions $(d_1, d_2, \dots, d_n)$, and computes the dimension of the space ${\cal H}_{LME}/K$ of LME states up to local unitary transformations for all non-empty cases. In this paper, we provide a pedagogical overview and physical interpretation of the 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 also give the dimension of the space of SLOCC equivalence classes for states with "generic" entanglement for all multipart systems since this space is equivalent to ${\cal H}_{LME}/K$. Finally, we give the dimension of the stabilizer subgroup $S \subset SL(d_1, \mathbb{C}) \times \cdots \times SL(d_n, \mathbb{C})$ for a generic state in an arbitrary multipart system and identify all cases where this stabilizer is trivial.