Doubly infinite separation of quantum information and communication

TL;DR

This paper demonstrates a fundamental separation between quantum information and communication complexities in certain tasks, showing that quantum communication can grow significantly even when quantum information remains minimal, revealing deep insights into quantum communication limits.

Contribution

It establishes the existence of tasks with a doubly infinite gap between quantum information and communication complexities, extending understanding of quantum communication efficiency.

Findings

Quantum information complexity tends to zero for certain tasks as input size increases.

Quantum communication complexity scales at least logarithmically with input size.

Lower bounds and gaps persist even with small error probabilities.

Abstract

We prove the existence of (one-way) communication tasks with a subconstant versus superconstant asymptotic gap, which we call "doubly infinite," between their quantum information and communication complexities. We do so by studying the exclusion game [C. Perry et al., Phys. Rev. Lett. 115, 030504 (2015)] for which there exist instances where the quantum information complexity tends to zero as the size of the input $n$ increases. By showing that the quantum communication complexity of these games scales at least logarithmically in $n$, we obtain our result. We further show that the established lower bounds and gaps still hold even if we allow a small probability of error. However in this case, the $n$-qubit quantum message of the zero-error strategy can be compressed polynomially.