Transformations among Pure Multipartite Entangled States via Local Operations are Almost Never Possible

Nolan R Wallach,Barbara Kraus,David Sauerwein,Gilad Gour

doi:10.1103/physrevx.8.031020

Nolan R Wallach, Barbara Kraus + Show 2 more

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https://doi.org/10.1103/physrevx.8.031020

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Abstract

Local operations assisted by classical communication (LOCC) constitute the free operations in entanglement theory. Hence, the determination of LOCC transformations is crucial for the understanding of entanglement. We characterize here almost all LOCC transformations among pure multipartite multilevel states. Combined with the analogous results for qubit states shown by Gour \emph{et al.} [J. Math. Phys. 58, 092204 (2017)], this gives a characterization of almost all local transformations among multipartite pure states. We show that nontrivial LOCC transformations among generic, fully entangled, pure states are almost never possible. Thus, almost all multipartite states are isolated. They can neither be deterministically obtained from local-unitary-inequivalent (LU-inequivalent) states via local operations, nor can they be deterministically transformed to pure, fully entangled LU-inequivalent states. In order to derive this result, we prove a more general statement, namely, that, generically, a state possesses no nontrivial local symmetry. We discuss further consequences of this result for the characterization of optimal, probabilistic single copy and probabilistic multi-copy LOCC transformations and the characterization of LU-equivalence classes of multipartite pure states.

Highlights

Entanglement lies at the heart of quantum theory and is the essential resource for many striking applications of quantum information science [1,2,3,4,5,6]
We show that a generic state jψi can be deterministically transformed to a fully entangled state jφi via Local operations assisted by classical communication (LOCC) if and only if jφi 1⁄4 u1 ⊗ ... ⊗ unjψi, where ui is unitary; that is, only if jψi and jφi are local unitary (LU) equivalent
In order to explain them, we briefly review the connection between the local symmetries of multipartite states and their transformation properties under LOCC and separable operations (SEP)

Summary

Introduction

Entanglement lies at the heart of quantum theory and is the essential resource for many striking applications of quantum information science [1,2,3,4,5,6]. The entanglement properties of multipartite states are, fundamental to important concepts in condensed matter physics [7]. This relevance of entanglement in various fields of science has motivated great research efforts to gain a better understanding of these intriguing quantum correlations. Local operations assisted by classical communication (LOCC) play an essential role in the theoretical and experimental investigation of quantum correlations. The parties are free to communicate classically with

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