Transformations in quantum networks via local operations assisted by finitely many rounds of classical communication

TL;DR

This paper investigates the capabilities and limitations of state transformations in quantum networks using local operations and classical communication, highlighting differences between network structures and the necessity of multiple communication rounds.

Contribution

It provides a systematic analysis of state transformation possibilities in bipartite entangled quantum networks, especially focusing on cycles and the role of communication rounds.

Findings

Transformations depend on network topology, with cycles limiting deterministic reachability.

A protocol with single-measurement steps can reach certain states in cycle networks.

Some transformations require three rounds of classical communication, the first known example of such complexity.

Abstract

Recent advances have led towards first prototypes of quantum networks in which entanglement is distributed by sources producing bipartite entangled states. This raises the question of which states can be generated in quantum networks based on bipartite sources using local operations and classical communication. In this work, we study state transformations under finite rounds of local operations and classical communication (LOCC) in networks based on maximally entangled two-qubit states. We first derive the symmetries for arbitrary network structures, as these determine which transformations are possible. Then, we show that contrary to tree graphs, for which it has already been shown that any state within the same entanglement class can be reached, there exist states which can be reached probabilistically but not deterministically if the network contains a cycle. Furthermore, we provide a systematic way to determine states which are not reachable in networks consisting of a cycle. Moreover, we provide a complete characterization of the states which can be reached in a cycle network with a protocol where each party measures only once, and each step of the protocol results in a deterministic transformation. Finally, we present an example which cannot be reached with such a simple protocol, and constitutes, up to our knowledge, the first example of a LOCC transformation among fully entangled states requiring three rounds of classical communication.