Multi-copy and stochastic transformation of multipartite pure states

TL;DR

This paper explores the transformation and classification of multipartite pure states under various operational environments, revealing equivalences and structures that extend the concept of maximally entangled states in quantum information.

Contribution

It introduces the equivalence of multi-copy LOCC and SLOCC environments for state transformation and classifies multipartite states using tensor rank and partial order.

Findings

Transformability under LOCC/SLOCC implies transformability under MCLOCC/MCSLOCC.

Multipartite states can be classified into inequivalent sets and orbits based on transformation properties.

GHZ states can generate all states with smaller or equal tensor rank under SLOCC.

Abstract

Characterizing the transformation and classification of multipartite entangled states is a basic problem in quantum information. We study the problem under two most common environments, local operations and classical communications (LOCC), stochastic LOCC and two more general environments, multi-copy LOCC (MCLOCC) and multi-copy SLOCC (MCSLOCC). We show that two transformable multipartite states under LOCC or SLOCC are also transformable under MCLOCC and MCSLOCC. What's more, these two environments are equivalent in the sense that two transformable states under MCLOCC are also transformable under MCSLOCC, and vice versa. Based on these environments we classify the multipartite pure states into a few inequivalent sets and orbits, between which we build the partial order to decide their transformation. In particular, we investigate the structure of SLOCC-equivalent states in terms of tensor rank, which is known as the generalized Schmidt rank. Given the tensor rank, we show that GHZ states can be used to generate all states with a smaller or equivalent tensor rank under SLOCC, and all reduced separable states with a cardinality smaller or equivalent than the tensor rank under LOCC. Using these concepts, we extended the concept of "maximally entangled state" in the multi-partite system.