The Round Complexity of Local Operations and Classical Communication (LOCC) in Random-Party Entanglement Distillation

TL;DR

This paper investigates the number of communication rounds needed in LOCC protocols for entanglement distillation, revealing that the required rounds can vary significantly based on the entanglement measure and state, with some protocols needing unbounded rounds.

Contribution

It demonstrates that LOCC round complexity depends on the entanglement measure and provides explicit bounds, including constructions requiring unbounded rounds, advancing understanding of quantum communication complexity.

Findings

Round complexity varies with entanglement measure.

Some protocols require unbounded rounds to implement.

The original W-state protocol is near-optimal in round complexity.

Abstract

A powerful operational paradigm for distributed quantum information processing involves manipulating pre-shared entanglement by local operations and classical communication (LOCC). The LOCC round complexity of a given task describes how many rounds of classical communication are needed to complete the task. Despite some results separating one-round versus two-round protocols, very little is known about higher round complexities. In this paper, we revisit the task of one-shot random-party entanglement distillation as a way to highlight some interesting features of LOCC round complexity. We first show that for random-party distillation in three qubits, the number of communication rounds needed in an optimal protocol depends on the entanglement measure used; for the same fixed state some entanglement measures need only two rounds to maximize whereas others need an unbounded number of rounds. In doing so, we construct a family of LOCC instruments that require an unbounded number of rounds to implement. We then prove explicit tight lower bounds on the LOCC round number as a function of distillation success probability. Our calculations show that the original W-state random distillation protocol by Fortescue and Lo is essentially optimal in terms of round complexity.