Simple Communication Complexity Separation from Quantum State Antidistinguishability

TL;DR

This paper introduces a simple quantum communication task demonstrating an exponential separation between quantum and classical one-way communication complexities using antidistinguishable quantum states, with implications for understanding quantum advantages.

Contribution

The paper presents a new quantum communication task based on antidistinguishable states that achieves a significant separation from classical communication complexity, with a simple and self-contained proof.

Findings

Quantum states with antidistinguishability properties enable efficient quantum communication protocols.

Classical communication complexity for the task scales linearly with the dimension, while quantum protocols are logarithmic.

The separation is robust to certain errors but disappears with two-way classical communication.

Abstract

A set of $n$ pure quantum states is called antidististinguishable if there exists an $n$-outcome measurement that never outputs the outcome `$k$' on the $k$-th quantum state. We describe sets of quantum states for which any subset of three states is antidistinguishable and use this to produce a two-player communication task that can be solved with $\log d$ qubits, but requires one-way communication of at least $\log (4/3) (d-1) - 1 \approx 0.415 (d-1) - 1$ classical bits. The advantages of the approach are that the proof is simple and self-contained -- not needing, for example, to rely on hard-to-establish prior results in combinatorics -- and that with slight modifications, non-trivial bounds can be established in any dimension $\geq 3$. The task can be framed in terms of the separated parties solving a relation, and the separation is also robust to multiplicative error in the output probabilities. We show, however, that for this particular task, the separation disappears if two-way classical communication is allowed. Finally, we state a conjecture regarding antidistinguishability of sets of states, and provide some supporting numerical evidence. If the conjecture holds, then there is a two-player communication task that can be solved with $\log d$ qubits, but requires one-way communication of $\Omega (d \log d)$ classical bits.