[Paper Review] Quantum physics needs complex numbers

Marc-Olivier Renou, David Trillo+6 more|arXiv (Cornell University)|Jan 26, 2021

Quantum Mechanics and Applications18 citations

TL;DR

This paper proves that complex numbers are fundamentally necessary in quantum physics by demonstrating that complex quantum mechanics makes predictions distinct from real quantum mechanics in network scenarios with independent sources. The key result is a Bell-type experiment where complex quantum theory predicts correlations that cannot be reproduced by any real quantum model, thereby establishing complex numbers as indispensable for a complete quantum description of nature.

ABSTRACT

Complex numbers, i.e., numbers with a and an imaginary part, are essential for mathematical analysis, while their role in other subjects, such as electromagnetism or special relativity, is far less fundamental. Quantum is the only physical theory where these numbers seem to play an indispensible role, as the theory is explicitly formulated in terms of operators acting on complex Hilbert spaces. The occurrence of complex numbers within the quantum formalism has nonetheless puzzled countless physicists, including the fathers of the theory, for whom a version of quantum physics, where states and observables are represented by operators, seemed much more natural. In fact, previous works showed that such real quantum physics can reproduce the outcomes of any multipartite experiment, as long as the parts share arbitrary quantum states. Thus, are complex numbers really needed for a quantum description of nature? Here, we show this to be case by proving that and complex quantum make different predictions in network scenarios comprising independent quantum state sources. This allows us to devise a Bell-type quantum experiment whose input-output correlations cannot be approximated by any quantum model. The successful realization of such an experiment would disprove quantum physics, in the same way as standard Bell experiments disproved local physics.

Motivation & Objective

• To determine whether complex numbers are fundamentally required in quantum physics or if real quantum mechanics could reproduce all quantum phenomena.
• To investigate whether real quantum theories—where states and observables are represented by real operators—can account for all quantum correlations in network configurations.
• To identify experimental scenarios where complex quantum mechanics makes predictions that cannot be approximated by any real quantum model.
• To establish a Bell-type experiment framework that can empirically test the necessity of complex numbers in quantum theory.

Proposed method

• The study analyzes quantum networks with multiple independent quantum state sources, where each source prepares entangled states independently.
• It formulates the correlations expected in such networks under both complex and real quantum mechanics using operator formalism on Hilbert spaces.
• The method involves comparing the set of possible input-output correlations predicted by complex quantum mechanics versus real quantum mechanics in the same network structure.
• It identifies specific network configurations where the correlation sets differ, implying that complex numbers are not merely a calculational tool but physically essential.
• The approach uses a generalization of Bell-type inequalities to detect the non-replicability of complex quantum correlations by real quantum models.

Experimental results

Research questions

• RQ1 Can real quantum mechanics reproduce all predictions of complex quantum mechanics in quantum networks with independent sources?
• RQ2 Are there experimental configurations where complex quantum mechanics leads to correlations that cannot be approximated by any real quantum model?
• RQ3 Does the use of complex numbers in quantum theory represent a fundamental physical requirement rather than a mathematical convenience?
• RQ4 Can a Bell-type experiment be designed to empirically distinguish complex quantum mechanics from real quantum mechanics?

Key findings

• Complex quantum mechanics makes predictions in certain network scenarios that cannot be reproduced by any real quantum model, proving that complex numbers are physically indispensable.
• The paper identifies specific network configurations where the set of achievable correlations under complex quantum mechanics strictly exceeds that of real quantum mechanics.
• The theoretical framework allows for a Bell-type experiment whose outcome cannot be explained by any real quantum theory, thus providing a testable distinction.
• The results show that complex numbers are not just a calculational tool but are required to describe the full range of quantum phenomena, especially in non-local network settings.