Quantum theory based on real numbers cannot be experimentally falsified

TL;DR

This paper demonstrates that real quantum theory cannot be experimentally distinguished from standard quantum theory when source independence is defined operationally, challenging previous claims of potential falsification.

Contribution

It proves that, under operational source independence, all correlations achievable in quantum theory are also achievable in real quantum theory, making them empirically indistinguishable.

Findings

Real quantum theory reproduces all quantum correlations under operational independence.

Previous claims of falsification relied on untestable assumptions.

Quantum and real quantum theories are empirically indistinguishable under certain conditions.

Abstract

Whether the complex numbers of standard quantum theory are experimentally indispensable has remained open for decades. Real quantum theory (RQT), obtained by replacing complex amplitudes with real ones while retaining the usual Kronecker-product composition rule, reproduces all single-party and bipartite Bell correlations of quantum theory (QT), but its lack of local tomography suggested that the two theories might diverge in more general local experiments. This possibility appeared to be confirmed by Renou et al., who argued that a bilocal network experiment can falsify RQT without falsifying QT. Here we show that this conclusion relies on an experimentally untestable assumption. The key distinction is between product-state independence, which constrains the mathematical form of source states, and operational independence, which is defined entirely by the absence of observable cross-source correlations. We prove that, once source independence is imposed operationally, every finite network correlation achievable in QT is also achievable in RQT with the same locality structure of the measurements. We then extend this equivalence to arbitrary finite sequential multipartite protocols involving channels and measurements with prescribed locality structure. Thus, as long as no violation of QT is observed, RQT cannot be experimentally falsified. Our results restore the empirical indistinguishability of QT and RQT, while showing that they support markedly different pictures of the correlation structure underlying the same observed world.