Conditions for a Class of Entanglement Transformations

TLDR

The Letter provides necessary and sufficient conditions for transforming a shared pure state |ψ⟩ into another state |φ⟩ using local operations and classical communication. These conditions establish a partial ordering of entangled states, link entanglement to the algebraic theory of majorization, and demonstrate that bipartite quantum systems possess distinct types of entanglement.

Abstract

Suppose Alice and Bob jointly possess a pure state, $|\ensuremath{\psi}〉$. Using local operations on their respective systems and classical communication it may be possible for Alice and Bob to transform $|\ensuremath{\psi}〉$ into another joint state $|\ensuremath{\varphi}〉$. This Letter gives necessary and sufficient conditions for this process of entanglement transformation to be possible. These conditions reveal a partial ordering on the entangled states and connect quantum entanglement to the algebraic theory of majorization. As a consequence, we find that there exist essentially different types of entanglement for bipartite quantum systems.