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arXiv:2607.28668

Comment on "Quantum mechanics based on real numbers: a consistent description"

S C Tiwari

Published Jul 25, 2026• 0 citations
arXivHTMLPDF

Abstract

In this comment on arXiv:2503.17307 (PRL, 136, 240202, 2026), a novel idea using flag states to transform complex state vector and operators to real ones is argued to effectively lead to a quantum mechanics that is complex in the disguised form of real. The claim that real-valued quantum mechanics cannot be falsified is conceptually flawed. A new criterion based on the geometric phases is suggested to compare real and complex versions of quantum mechanics.

DOI
10.48550/arxiv.2607.28668

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In this comment on arXiv:2503.17307 (PRL, 136, 240202, 2026), a novel idea using flag states to transform complex state vector and operators to real ones is argued to effectively lead to a quantum mechanics that is complex in the disguised form of real. The claim that real-valued quantum mechanics cannot be falsified is conceptually flawed. A new criterion based on the geometric phases is suggested to compare real and complex versions of quantum mechanics.
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.
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 an alternative theory which reproduces the predictions of standard complex quantum information theory. Here, we argue that, to qualify as a general physical principle, this postulate should also hold in fermionic quantum information theory, the standard framework describing information encoded in the presence or absence of identical fermions. We show, however, that it fails in this framework. More broadly, our result highlights the importance of confronting proposed foundational principles with fermionic quantum information theory, a point that also deserves further examination in recent related works such as [arXiv:2503.17307] and [arXiv:2504.02808].
Barrios Hita et al. [Phys. Rev. Lett. 136, 240202 (2026)] proposed a quotient-space formulation of quantum mechanics over the real numbers and concluded that complex numbers are only a convenience. A preceding analysis [arXiv:2607.05865] showed that, when the construction is empirically equivalent to complex quantum mechanics, its real flag is protected by a hidden complex structure J^\hat{J}J^ and its composition rule is the balanced tensor product over that structure. Here we ask what would follow from the stronger operational reading that the flag is an accessible real degree of freedom. The answer is a direct clash with overlap-determinability (OD), introduced in [arXiv:2601.14638], and with standard no-go theorems. A readable or coherently controllable flag fixes a representative of each ray, and hence supplies precisely the phase convention that OD identifies as the missing resource for universal superposition of unknown states. Once promoted to a generic phase-access primitive, this resource enables probabilistic cloning of a linearly dependent set, steering-based signaling, and logarithmic-query unstructured search. Thus the quotient construction has a sharp follow-up interpretation: as a protected gauge representation it is standard complex quantum mechanics in real notation; as an accessible real theory it leaves the quantum operational framework.
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.
According to quantum theory, pure physical states correspond to equivalence classes of state vectors, where any two members of one class differ by a complex factor. The point is that such a factor does not change the probability for the occurrence of any measurement result as computed within the formalism of quantum mechanics. In the formalism to be presented here, the state vector does not only determine the probabilities for the occurrence of measurement results, rather it determines the result itself. The information which guides the selection process is obtained from the state vector in a way that is not invariant against multiplication by a complex factor. It can be seen as a kind of global phase of the state vectors. This global phase then is a perfectly hidden variable. For this presentation of quantum mechanics we have a remarkable parallelism to classical statistical mechanics: If we would know exactly the initial state of a system, we could compute the outcomes for all experiments to be done later on this system. In the quantum case the initial state is to be represented by its state vector with specified phase, e.g. as a concrete complex-valued wave function.
It is argued that many real-valued constructions of quantum mechanics are only \emph{nominally} real in that the operators and states are restricted to impose a \emph{complex structure} on the Hilbert space. That is, complex algebra between pairs of elements representing complex numbers is preserved in these formulations. It is therefore mistaken to think of these constructions as being `real' as they are actually a representation of complex linear algebra. It is then shown that under the assumptions of state normalisation and strict energy conservation, this complex structure (at the very least) is required to allow for time evolution in quantum mechanics. This is only a \emph{minimum} requirement, as our arguments do not preclude the formulation of quantum mechanics in terms of hyper-complex entities, such as quaternions.