Entanglement-assisted zero-error source-channel coding

TL;DR

This paper investigates how quantum entanglement can enhance zero-error source-channel coding, demonstrating that entanglement can significantly improve coding rates and providing bounds related to graph parameters like the theta number.

Contribution

It extends previous results by establishing bounds on entanglement-assisted source coding rates and showing unbounded improvements enabled by entanglement.

Findings

Entanglement can unboundedly improve classical coding rates.

The theta number bounds the entangled Witsenhausen rate.

A lower bound on entanglement-assisted source-codes using Szegedy's number.

Abstract

We study the use of quantum entanglement in the zero-error source-channel coding problem. Here, Alice and Bob are connected by a noisy classical one-way channel, and are given correlated inputs from a random source. Their goal is for Bob to learn Alice's input while using the channel as little as possible. In the zero-error regime, the optimal rates of source codes and channel codes are given by graph parameters known as the Witsenhausen rate and Shannon capacity, respectively. The Lov\'asz theta number, a graph parameter defined by a semidefinite program, gives the best efficiently-computable upper bound on the Shannon capacity and it also upper bounds its entanglement-assisted counterpart. At the same time it was recently shown that the Shannon capacity can be increased if Alice and Bob may use entanglement. Here we partially extend these results to the source-coding problem and to the more general source-channel coding problem. We prove a lower bound on the rate of entanglement-assisted source-codes in terms Szegedy's number (a strengthening of the theta number). This result implies that the theta number lower bounds the entangled variant of the Witsenhausen rate. We also show that entanglement can allow for an unbounded improvement of the asymptotic rate of both classical source codes and classical source-channel codes. Our separation results use low-degree polynomials due to Barrington, Beigel and Rudich, Hadamard matrices due to Xia and Liu and a new application of remote state preparation.