Local Transformations Requiring Infinite Rounds of Classical Communication

TL;DR

This paper investigates the complexity of implementing quantum tasks with local operations and classical communication, revealing that increasing communication rounds enhances capabilities and some transformations require infinitely many rounds.

Contribution

It demonstrates that LOCC operations become strictly more powerful with more rounds and that some entanglement transformations need infinite rounds, highlighting the discontinuity in entanglement distillation.

Findings

More rounds of communication enable more LOCC operations.

Certain entanglement transformations require infinite rounds.

Distillation efficiency can change abruptly with entanglement amount.

Abstract

In this paper, we study the number of rounds of communication needed to implement certain tasks by local quantum operations and classical communication (LOCC). We find that the class of LOCC operations becomes strictly more powerful as more rounds of classical communication are permitted. Specifically, for every $n$, there always exists an $n$ round protocol that is impossible to implement in $n-2$ rounds. Furthermore, we show that certain entanglement transformations are possible if and only if the protocol uses an infinite (unbounded) number of rounds. Interestingly, the number of rounds required to deterministically distill bipartite entanglement from a single multipartite state can be strongly discontinuous with respect to the amount of entanglement distilled.