Entangled pure state transformations via local operations assisted by finitely many rounds of classical communication

TL;DR

This paper characterizes which pure quantum states can be transformed into others using finite-round local operations and classical communication, revealing that most states cannot be reached from others in high-dimensional multipartite systems.

Contribution

It provides a measure-theoretic characterization of state convertibility under finite-round LOCC and identifies classes of states with deterministic transformation protocols.

Findings

Set of reachable states is measure zero for n>3 qubits.

Maximally entangled set is of full measure under practical LOCC scenarios.

Existence of probabilistic steps distinguishes multipartite from bipartite LOCC transformations.

Abstract

We consider generic pure $n$-qubit states and a general class of pure states of arbitrary dimensions and arbitrarily many subsystems. We characterize those states which can be reached from some other state via Local Operations assisted by finitely many rounds of Classical Communication ($LOCC_{\mathbb{N}}$). For $n$ qubits with $n>3$ we show that this set of states is of measure zero, which implies that the maximally entangled set is generically of full measure if restricted to the practical scenario of $LOCC_{\mathbb{N}}$. Moreover, we identify a class of states for which any $LOCC_{\mathbb{N}}$ protocol can be realized via a concatenation of deterministic steps. We show, however, that in general there exist state transformations which require a probabilistic step within the protocol, which highlights the difference between bipartite and multipartite LOCC.