Complexity of causal order structure in distributed quantum information processing and its trade-off with entanglement

TL;DR

This paper establishes a fundamental trade-off between entanglement resources and classical communication complexity in implementing distributed quantum gates, revealing limits and possibilities in quantum information processing.

Contribution

It proves a specific trade-off relation between entanglement cost and communication complexity for causal order structures in distributed quantum protocols, including necessary and sufficient conditions.

Findings

One ebit of entanglement per pair is necessary for two-round protocols.

Three-round protocols can reduce entanglement cost below one ebit per pair.

Some bipartite unitaries exhibit no such trade-off.

Abstract

We prove a trade-off relation between the entanglement cost and classical communication complexity of causal order structure of a protocol in distributed quantum information processing. We consider an implementation of a class of two-qubit unitary gates by local operations and classical communication (LOCC) assisted with shared entanglement, in an information theoretical scenario of asymptotically many input pairs and vanishingly small error. We prove the trade-off relation by showing that (i) one ebit of entanglement per pair is necessary for implementing the unitary by any two-round protocol, and that (ii) the entanglement cost by a three-round protocol is strictly smaller than one ebit per pair. We also provide an example of bipartite unitary gates for which there is no such trade-off.