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arXiv:2101.10873 · Nature

Quantum theory based on real numbers can be experimentally falsified

Marc-Olivier Renou, David Trillo, Mirjam Weilenmann, Lê Phuc Thinh, Armin Tavakoli, Nicolas Gisin, Antonio Acín, Miguel Navascués

Published Dec 15, 2021• 215 citations
arXivHTMLPDF

Abstract

Abstract Although complex numbers are essential in mathematics, they are not needed to describe physical experiments, as those are expressed in terms of probabilities, hence real numbers. Physics, however, aims to explain, rather than describe, experiments through theories. Although most theories of physics are based on real numbers, quantum theory was the first to be formulated in terms of operators acting on complex Hilbert spaces 1,2 . This has puzzled countless physicists, including the fathers of the theory, for whom a real version of quantum theory, in terms of real operators, seemed much more natural 3 . In fact, previous studies have shown that such a ‘real quantum theory’ can reproduce the outcomes of any multipartite experiment, as long as the parts share arbitrary real quantum states 4 . Here we investigate whether complex numbers are actually needed in the quantum formalism. We show this to be case by proving that real and complex Hilbert-space formulations of quantum theory make different predictions in network scenarios comprising independent states and measurements. This allows us to devise a Bell-like experiment, the successful realization of which would disprove real quantum theory, in the same way as standard Bell experiments disproved local physics.

DOI
10.1038/s41586-021-04160-4

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Abstract Although complex numbers are essential in mathematics, they are not needed to describe physical experiments, as those are expressed in terms of probabilities, hence real numbers. Physics, however, aims to explain, rather than describe, experiments through theories. Although most theories of physics are based on real numbers, quantum theory was the first to be formulated in terms of operators acting on complex Hilbert spaces 1,2 . This has puzzled countless physicists, including the fathers of the theory, for whom a real version of quantum theory, in terms of real operators, seemed much more natural 3 . In fact, previous studies have shown that such a ‘real quantum theory’ can reproduce the outcomes of any multipartite experiment, as long as the parts share arbitrary real quantum states 4 . Here we investigate whether complex numbers are actually needed in the quantum formalism. We show this to be case by proving that real and complex Hilbert-space formulations of quantum theory make different predictions in network scenarios comprising independent states and measurements. This allows us to devise a Bell-like experiment, the successful realization of which would disprove real quantum theory, in the same way as standard Bell experiments disproved local physics.
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.
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.
The role of complex quantities in quantum theory has been puzzling physicists since the beginnings. It is, thus, natural to ask whether, in order to describe our experiments, the mathematical structure of the complex Hilbert spaces it is built on is really necessary. Recently, it was shown that this structure is inevitable in network scenarios with independent sources. More precisely, Real Quantum Theory cannot explain the predictions of (Complex) Quantum Theory [Renou et al., Nature (London) 600, 625 (2021).NATUAS0028-083610.1038/s41586-021-04160-4]. Here, we revisit the independence assumption underlying this work. We show that assuming partial independence is sufficient for showing the inadequacy of Real Quantum Theory. We derive a tradeoff between source independence and the Bell value achievable in Real Quantum Theory, which also lower bounds the source correlations required to explain previous experiments by means of real quantum systems. We further show that 1 bit of entanglement is necessary and sufficient for recovering the complex quantum correlations by means of Real Quantum Theory in the scenario from [Renou et al., Nature (London) 600, 625 (2021).NATUAS0028-083610.1038/s41586-021-04160-4]. Finally, building on [McKague et al., Phys. Rev. Lett. 102, 2009PRLTAO0031-900710.1103/PhysRevLett.102.020505], we provide a construction to simulate any complex quantum setup with m independent sources by means of Real Quantum Theory by allowing the sources to share an m real-qubit entangled state in the first round of the experiment.
In our previous paper on the matter, we proved that real quantum theory (RQT) cannot be experimentally falsified without the experiment falsifying quantum theory (QT) as well. Our result is in direct contradiction to the central claim of an article by Renou et al., published in Nature [600, 625--629 (2021)], reporting a bilocal network experiment that could falsify real quantum theory without falsifying quantum theory. The purpose of this note is to address several seemingly natural attempts to recover the experimental conclusion of Renou et al. We first explain how these results, although based on nonlocality experiments, are far from a Bell-like level of proof. Then we explain why the proposed experiment requires an experimentally untestable premise about independent preparations necessarily being represented as product states. Finally, we discuss some concerns that lie outside the scope of both results.
Quantum theory is commonly formulated in complex Hilbert spaces. However, the question of whether complex numbers need to be given a fundamental role in the theory has been debated since its pioneering days. Recently it has been shown that tests in the spirit of a Bell inequality can reveal quantum predictions in entanglement swapping scenarios that cannot be modeled by the natural real-number analog of standard quantum theory. Here, we tailor such tests for implementation in state-of-the-art photonic systems. We experimentally demonstrate quantum correlations in a network of three parties and two independent EPR sources that violate the constraints of real quantum theory by over 4.5 standard deviations, hence disproving real quantum theory as a universal physical theory.
Standard quantum mechanics employs complex Hilbert spaces, but whether complex numbers are fundamental or merely convenient has long been debated. For decades, real-valued equivalents were considered mathematically possible but cumbersome. However, a highly cited 2021 result claimed that any quantum theory based on real numbers is experimentally falsifiable via network Bell experiments. Yet, it remains an open question whether this falsification applies to all real-valued theories. Here we show that this conclusion rests on an incomplete real formulation, and we present a rigorous real-valued framework that perfectly reproduces all predictions of standard quantum mechanics. We demonstrate that the standard real tensor product (⊗R\otimes_{\mathbb{R}}⊗R​) used in previous no-go theorems is algebraically incompatible with the rich structure of conventional quantum mechanics. We present a real framework based on Kähler space and prove that it is exactly isomorphic to established quantum mechanics via an explicit bijection γγγ. The isomorphism extends to composite systems through a symplectic composition rule $\otimes^{\ks}$ that replaces the Kronecker product. Consequently, our formulation achieves the maximal CHSH3\mathrm{CHSH}_{3}CHSH3​ violation of 626\sqrt{2}62​ using purely real variables, demonstrating that the no-go theorem is specific to a particular real representation of states and operators and to the composition rule ⊗R\otimes_\mathbb{R}⊗R​ built upon it, neither of which extends to the present Kähler framework. These results demonstrate that complex numbers are not fundamentally required by nature; rather, they encode a deeper real geometric structure that governs quantum interference and entanglement, settling this long debate.