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We prove that there is a trade-off relation between the entanglement cost andthe number of rounds of communication, for two distant parties to accomplish abidirectional quantum information task by local operations and classicalcommunication (LOCC). We consider an implementation of a class of two-qubitcontrolled-unitary gate by LOCC assisted by shared entanglement, in aninformation theoretical scenario of asymptotically many input pairs andvanishingly small error. We prove the trade-off relation by showing that oneebit of entanglement per pair is necessary to be consumed for implementing theunitary by any two-round protocol, whereas the entanglement cost by afour-round protocol is strictly smaller than one ebit per pair.

A Four-Round LOCC Protocol Outperforms All Two-Round Protocols in Reducing the Entanglement Cost for A Distributed Quantum Information Processing