The Generation Cost of Bipartite Quantum States under LOCC

TL;DR

This paper characterizes the minimal quantum communication cost to generate bipartite states under LOCC, showing it equals the logarithm of the state's Schmidt number, and compares it with classical-communication-restricted schemes.

Contribution

It provides a complete characterization of the generation cost for bipartite states under LOCC, linking it to the Schmidt number, and analyzes the role of classical communication.

Findings

Generation cost equals the logarithm of the Schmidt number.

Classical communication can sometimes be completely unhelpful.

Comparison between LOCC and classical-only schemes reveals the role of classical communication.

Abstract

We consider a realistic setting of quantum tasks that generate shared bipartite quantum states. Suppose \alice and \bob are located at different places and need to produce a target shared quantum state $\rho$. In order to save quantum communication, they can choose to share a proper smaller quantum state $\sigma$ first, and then turn $\sigma$ to $\rho$ by performing only local quantum operations and classical communications (LOCC). We hope $\sigma$ is the optimal such that the quantum communication needed is as little as possible, which is called the generation cost of $\rho$. In this paper, for an arbitrary bipartite $\rho$, we characterize its generation cost completely by proving that it is exactly equivalent to the logarithm of the Schmidt number of $\rho$. Similar quantum schemes where classical communication is not allowed have actually been considered. By comparing the two settings, we are able to look into the role that classical communication plays in these fundamental tasks, where we exhibit some instances in which classical communication is not helpful completely.