Communication tasks with infinite quantum-classical separation

TL;DR

This paper demonstrates an infinite quantum-classical separation in communication tasks, showing quantum messages can exponentially reduce information revealed compared to classical messages, especially with entanglement assistance.

Contribution

It introduces a communication task illustrating infinite separation between quantum and classical resources, highlighting the power of entanglement in reducing communication complexity.

Findings

Quantum messages require nearly zero information to be revealed as n grows.

Entanglement allows constant classical communication regardless of input size.

Without entanglement, classical communication grows linearly with input size.

Abstract

Quantum resources can be more powerful than classical resources - a quantum computer can solve certain problems exponentially faster than a classical computer, and computing a function of two people's inputs can be done with exponentially less communication with quantum messages than with classical ones. Here we consider a task between two players, Alice and Bob where quantum resources are infinitely more powerful than classical ones. Alice is given a string of length n, and Bob's task is to exclude certain combinations of bits that Alice might have. If Alice must send classical messages, then she must reveal nearly n bits of information to Bob, but if she is allowed to send quantum bits, the amount of information she must reveal goes to zero with increasing n. Next, we consider a version of the task where the parties can only send classical messages but may have access to entanglement. When assisted by entanglement, Alice only needs to send a constant number of bits, while without entanglement, the number of bits Alice must send grows linearly with n. The task is related to the PBR theorem which arises in the context of the foundations of quantum theory.