Entanglement cost of implementing controlled-unitary operations

TL;DR

This paper analyzes the minimum entanglement needed to implement two-qubit controlled-unitary operations deterministically using LOCC, revealing a gap between entanglement cost and entangling power due to input obliviousness.

Contribution

It establishes a lower bound of 1 ebit of entanglement for implementing controlled-unitaries with a specific LOCC protocol, highlighting a fundamental gap in quantum resource requirements.

Findings

Any controlled-unitary can be implemented with a three-turn LOCC protocol.

At least 1 ebit of entanglement is necessary for such implementation.

A gap exists between entanglement cost and entangling power due to input obliviousness.

Abstract

We investigate the minimum entanglement cost of the deterministic implementation of two-qubit controlled-unitary operations using local operations and classical communication (LOCC). We show that any such operation can be implemented by a three-turn LOCC protocol, which requires at least 1 ebit of entanglement when the resource is given by a bipartite entangled state with Schmidt number 2. Our result implies that there is a gap between the minimum entanglement cost and the entangling power of controlled-unitary operations. This gap arises due to the requirement of implementing the operations while oblivious to the identity of the inputs.