The asymptotic spectrum of LOCC transformations

TL;DR

This paper characterizes the asymptotic spectrum of LOCC transformations for pure quantum states, providing explicit formulas for entanglement conversion rates based on the converse error exponent, especially in bipartite systems.

Contribution

It introduces a full characterization of the asymptotic spectrum of LOCC conversion, enabling explicit rate formulas for bipartite pure states, extending previous entanglement concentration results.

Findings

Explicit formula for the asymptotic conversion rate in bipartite states.

Complete description of the spectrum in the bipartite case.

The derived rate bounds are tight in the bipartite scenario.

Abstract

We study exact, non-deterministic conversion of multipartite pure quantum states into one-another via local operations and classical communication (LOCC) and asymptotic entanglement transformation under such channels. In particular, we consider the maximal number of copies of any given target state that can be extracted exactly from many copies of any given initial state as a function of the exponential decay in success probability, known as the converese error exponent. We give a formula for the optimal rate presented as an infimum over the asymptotic spectrum of LOCC conversion. A full understanding of exact asymptotic extraction rates between pure states in the converse regime thus depends on a full understanding of this spectrum. We present a characterisation of spectral points and use it to describe the spectrum in the bipartite case. This leads to a full description of the spectrum and thus an explicit formula for the asymptotic extraction rate between pure bipartite states, given a converse error exponent. This extends the result on entanglement concentration in [Hayashi et al, 2003], where the target state is fixed as the Bell state. In the limit of vanishing converse error exponent the rate formula provides an upper bound on the exact asymptotic extraction rate between two states, when the probability of success goes to 1. In the bipartite case we prove that this bound holds with equality.