Communication Complexity of One-Shot Remote State Preparation

TL;DR

This paper provides a detailed information-theoretic analysis of the communication complexity involved in approximate remote state preparation, revealing bounds and differences between worst-case and average-case scenarios.

Contribution

It offers tight characterizations of the worst-case and average-case communication complexities of RSP using non-asymptotic information theory, and establishes a lower bound on classical communication in LOCC protocols.

Findings

Average-case communication complexity can be significantly lower than worst-case.

Tight bounds are derived for both worst-case and average-case RSP complexities.

n bits cannot be communicated with less than n transmitted bits in LOCC protocols.

Abstract

Quantum teleportation uses prior shared entanglement and classical communication to send an unknown quantum state from one party to another. Remote state preparation (RSP) is a similar distributed task in which the sender knows the entire classical description of the state to be sent. (This may also be viewed as the task of non-oblivious compression of a single sample from an ensemble of quantum states.) We study the communication complexity of approximate remote state preparation, in which the goal is to prepare an approximation of the desired quantum state. Jain [Quant. Inf. & Comp., 2006] showed that the worst-case communication complexity of approximate RSP can be bounded from above in terms of the maximum possible information in an encoding. He also showed that this quantity is a lower bound for communication complexity of (exact) remote state preparation. In this work, we tightly characterize the worst-case and average-case communication complexity of remote state preparation in terms of non-asymptotic information-theoretic quantities. We also show that the average-case communication complexity of RSP can be much smaller than the worst-case one. In the process, we show that n bits cannot be communicated with less than n transmitted bits in LOCC protocols. This strengthens a result due to Nayak and Salzman [J. ACM, 2006] and may be of independent interest.