Strict advantage of complex quantum theory in a communication task

TL;DR

This paper demonstrates that complex quantum theory offers a strict operational advantage over real quantum and classical theories in a specific communication task, highlighting the fundamental importance of complex amplitudes.

Contribution

It identifies a communication task where complex quantum theory outperforms both real quantum and classical approaches, proving the necessity of complex numbers for optimal quantum strategies.

Findings

Complex quantum theory has lower communication cost than real quantum and classical theories.

The advantage is certified through geometric properties of quantum state ensembles.

This establishes a fundamental operational benefit of complex over real quantum theory.

Abstract

Standard formulations of quantum theory are based on complex numbers: Quantum states can be in superpositions, with weights given by complex probability amplitudes. Motivated by quantum theory promising a range of practical advantages over classical for a multitude of tasks, we investigate how the presence of complex amplitudes in quantum theory can yield operational advantages over counterpart real formulations. We identify a straightforward communication task for which complex quantum theory exhibits a provably lower communication cost than not just any classical approach, but also any approach based on real quantum theory. We certify the necessity of complex quantum theory for optimal approaches to the task through geometric properties of quantum state ensembles that witness the presence of basis-independent complexity. This substantiates a strict operational advantage of complex quantum theory. We discuss the relevance of this finding for quantum advantages in stochastic simulation.