Renyi-entropic bounds on quantum communication

TL;DR

This paper introduces new lower bounds on quantum communication complexity for state transformations and distributed function evaluation using Renyi entropy, encompassing and extending previous methods.

Contribution

It develops a Renyi entropy-based technique to establish lower bounds on quantum communication complexity, covering state transformations and distributed function evaluation.

Findings

Lower bounds for quantum state transformation complexity.

Bounds for distributed evaluation of functions like inner product.

Effective for problems with correlated inputs or promises.

Abstract

In this article we establish new bounds on the quantum communication complexity of distributed problems. Specifically, we consider the amount of communication that is required to transform a bipartite state into another, typically more entangled, state. We obtain lower bounds in this setting by studying the Renyi entropy of the marginal density matrices of the distributed system. The communication bounds on quantum state transformations also imply lower bounds for the model of communication complexity where the task consists of the the distributed evaluation of a function f(x,y). Our approach encapsulates several known lower bound methods that use the log-rank or the von Neumann entropy of the density matrices involved. The technique is also effective for proving lower bounds on problems involving a promise or for which the "hard" distributions of inputs are correlated. As examples, we show how to prove a nearly tight bound on the bounded-error quantum communication complexity of the inner product function in the presence of unlimited amounts of EPR-type entanglement and a similarly strong bound on the complexity of the shifted quadratic character problem.