Comment on "Quantum theory based on real numbers cannot be experimentally falsified": On the compatibility of physical principles with information theory for fermions
The manuscript [arXiv:2603.19208] proposes a physically motivated postulate to select the appropriate formulation of quantum theory over real Hilbert spaces, ruling out the theory considered in [Nature 600, 625-629 (2021)] in favour of the alternative theory which reproduces the predictions of standard quantum information theory (QIT). Here, we first make the claim that a general physical postulate should in particular be satisfied by Fermionic Information Theory (FIT), the standard framework describing information encoded in the presence or absence of identical fermions. We then show that this postulate proposed by [arXiv:2603.19208] fails in FIT, hence is not a general physical postulate according to our claim. More broadly, our results highlight the importance of confronting proposed foundational principles with fermionic information theories, a point that also deserves further examination in recent related works such as [arXiv:2503.17307] and [arXiv:2504.02808].
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Comment on arXiv:2603.19208. 5+5 pages. Feel free to discuss
The manuscript [arXiv:2603.19208] proposes a physically motivated postulate to select the appropriate formulation of quantum theory over real Hilbert spaces, ruling out the theory considered in [Nature 600, 625-629 (2021)] in favour of the alternative theory which reproduces the predictions of standard quantum information theory (QIT). Here, we first make the claim that a general physical postulate should in particular be satisfied by Fermionic Information Theory (FIT), the standard framework describing information encoded in the presence or absence of identical fermions. We then show that this postulate proposed by [arXiv:2603.19208] fails in FIT, hence is not a general physical postulate according to our claim. More broadly, our results highlight the importance of confronting proposed foundational principles with fermionic information theories, a point that also deserves further examination in recent related works such as [arXiv:2503.17307] and [arXiv:2504.02808].
Complex numbers play a crucial role in quantum mechanics. However, their necessity remains debated: whether they are fundamental or merely convenient. Recently, it was shown that any real-number quantum theory satisfying certain postulates can be falsified with multipartite experiments. In this Letter we show that a physically motivated postulate about composite quantum systems allows us to construct quantum mechanics based on real numbers that reproduces predictions for all multipartite quantum experiments. Thus, we argue that real-valued quantum mechanics cannot be falsified, and therefore the use of complex numbers is a matter of convenience.
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) 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 γ. 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 violation of 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 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.
In a previous work, we showed that real quantum theory (RQT) cannot be experimentally distinguished from ordinary quantum theory (QT). In this paper, we investigate whether this indistinguishability survives two natural extensions of the setting: allowing separated measurements to be represented by commuting rather than tensor-product observables, and imposing superselection rules, as in fermionic information theory. We show that it does in both cases.
Our first result is a positive answer to the Tsirelson problem in RQT: commuting and tensor-product measurements generate the same finite-dimensional correlations. This is surprisingly not a trivial consequence of the standard complex proof, as it does not carry over to the reals. The proof obstruction becomes particularly relevant in the presence of superselection rules, which is why we first establish the equivalence in RQT before extending the framework to superselected RQT.
Our second result is a local, real quantum formulation of superselected theories. We extend the RQT framework to accommodate a representation of superselection over the reals and show, in particular, that it admits a fermionic version whose predictions are experimentally indistinguishable from those of fermionic information theory. More generally, the construction extends to all superselected QTs.
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.
Modern theoretical physics faces an abundance of mathematically consistent frameworks and a scarcity of decisive empirical constraints. This work proposes a principled response to that situation by identifying stability under intervention as a primary selection criterion for viable physical theories. Adopting an explicitly operational perspective, the paper formulates stability as the non-amplification of operational distinguishability under physical processes, a requirement motivated by the finite precision and reproducibility of experimental practice. When combined with compositional consistency, purification, and reversible symmetry, this principle strongly constrains admissible statistical frameworks. Within this setting, quantum theory emerges not as a fundamental ontology or a historically contingent choice, but as the uniquely stable non-classical fixed point compatible with reproducible physical control by finite agents embedded within the systems they study. The defining structural features of quantum mechanics, including convex state spaces, affine dynamics, state–effect duality, and a Born-type probability rule, arise as consequences rather than postulates. The framework further suggests that spacetime locality and geometry need not be assumed a priori, but can be understood as constraints on the propagation of distinguishability in composite systems. Potential deviations from quantum theory are predicted to manifest as failures of stability, such as breakdown of dilation, violation of distinguishability monotonicity, or loss of local tomography. This repository contains the final submission-ready manuscript together with its LaTeX source and bibliography, deposited on Zenodo prior to journal submission.This version (v1.1) incorporates clarifications on scope, logical dependencies between principles, worked examples within the Jordan-algebra classification, strengthened hedging of speculative material, and refined statements of falsifiability, in response to detailed referee feedback.
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.