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

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.