Entanglement requirements for implementing bipartite unitary operations

Abstract: We prove, using a new method based on map-state duality, lower bounds on entanglement resources needed to deterministically implement a bipartite unitary using separable (SEP) operations, which include LOCC (local operations and classical communication) as a particular case. It is known that the Schmidt rank of an entangled pure state resource cannot be less than the Schmidt rank of the unitary. We prove that if these ranks are equal the resource must be uniformly (maximally) entangled: equal nonzero Schmidt coefficients. Higher rank resources can have less entanglement: we have found numerical examples of Schmidt rank 2 unitaries which can be deterministically implemented, by either SEP or LOCC, using an entangled resource of two qutrits with less than one ebit of entanglement.