Entanglement requirements for implementing bipartite unitary operations

TL;DR

This paper establishes lower bounds on the entanglement resources required for deterministic implementation of bipartite unitaries, revealing that maximal entanglement is necessary when resource and unitary Schmidt ranks match.

Contribution

It introduces a new method based on map-state duality to derive entanglement bounds and provides examples where less entanglement suffices for certain unitaries.

Findings

Maximal entanglement is required when resource and unitary Schmidt ranks are equal.

Examples of Schmidt rank 2 unitaries implemented with less than one ebit of entanglement.

New bounds on entanglement resources for bipartite unitary implementation.

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.