Locality Implies Complex Numbers in Quantum Mechanics

Changliang Ren; Tianfeng Feng; Vlatko Vedral

arxiv: 2504.07808 · v2 · submitted 2025-04-10 · 🪐 quant-ph

Locality Implies Complex Numbers in Quantum Mechanics

Tianfeng Feng , Changliang Ren , Vlatko Vedral This is my paper

Pith reviewed 2026-05-22 20:28 UTC · model grok-4.3

classification 🪐 quant-ph

keywords quantum mechanicscomplex numbersreal numberslocalityentanglementindependent source assumptionnonlocal map

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The pith

Real-number quantum theories compatible with independent sources require hidden nonlocal maps to match complex quantum mechanics.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper investigates whether quantum mechanics can be written using only real numbers while keeping descriptions local. It demonstrates that any such real-number model, when it respects the assumption that separate sources produce independent states, must introduce a nonlocal mapping to reproduce the same results as standard complex quantum theory. This equivalence shows that complex numbers allow locality to be preserved in processes involving entanglement from independent sources. A reader would care because the work addresses whether complex numbers are optional mathematics or required for local quantum descriptions of multi-source entanglement.

Core claim

We show that the presented real-number quantum theories, compatible with the independent source assumption, require the inclusion of a nonlocal map. This means that if the independent source assumption holds, in these models, complex-number quantum theory is equivalent to a real-number quantum theory with hidden nonlocal degrees of freedom. Our results suggest that complex numbers may be indispensable for describing the process involving entanglement between two independent systems.

What carries the argument

The independent source assumption together with a required nonlocal map that converts real-number states into the predictions of complex quantum theory for entangled systems from separate sources.

If this is right

Complex-number quantum theory is equivalent to a real-number theory with hidden nonlocal degrees of freedom when the independent source assumption holds.
Real-number quantum theories must include a nonlocal map to be compatible with the independent source assumption.
Complex numbers become necessary to maintain locality when describing entanglement generated by two independent systems.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The result may limit the range of local real-number reformulations that can be applied to any multi-source quantum experiment.
Similar arguments could be tested in other quantum processes that combine states from separate sources, such as certain interference setups.
If the independent source assumption is relaxed, fully local real-number models might become viable again.

Load-bearing premise

The specific real-number quantum theories considered in the paper are representative of all possible real-number formulations that are compatible with the independent source assumption.

What would settle it

Construction of a real-number quantum model that respects the independent source assumption, reproduces all standard quantum predictions for entanglement between independent sources, and contains no nonlocal map.

read the original abstract

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper claims real QM needs hidden nonlocality under independent sources to match complex QM locally, but the abstract gives no math to check if this holds beyond the models they built.

read the letter

The main point is that the authors say real-number quantum theories, when they respect the independent source assumption for entangled systems, end up requiring nonlocal maps to reproduce standard correlations, while complex quantum mechanics handles it locally. This would make complex numbers necessary for a local description in that setting. The new angle is applying the real-versus-complex debate specifically to independent sources rather than single systems or other constraints. They apparently construct some real models that fit the assumption and then derive the need for those extra nonlocal elements, which ties the number field choice to multi-source entanglement in a direct way. That framing is a reasonable extension of prior work on real quantum mechanics. The paper does a decent job of picking a concrete scenario with two independent systems, which matches common experimental setups. The soft spots are clear from the abstract alone. No derivations, explicit Hilbert space constructions, or calculations appear, so there is no evidence to evaluate whether the nonlocality is forced or just an artifact of how the real models were chosen. The stress-test concern lands: showing the result for the specific real theories considered does not automatically rule out a more general real formulation that could keep everything local while preserving source independence. Without that generality, the claimed equivalence stays limited. There is also no visible discussion of how this relates to existing results on real quantum mechanics, which leaves the advance hard to gauge. This kind of paper is aimed at quantum foundations readers who track questions about complex numbers and locality. A specialist in that niche might find the idea worth following up if the full text supplies the missing technical steps. I would send it to peer review because the topic has enough substance to merit referee attention even if the current version needs substantial work to establish the claim properly.

Referee Report

2 major / 1 minor

Summary. The manuscript constructs specific real-number quantum theories compatible with the independent source assumption for entanglement between two independent systems. It shows that these models require inclusion of a nonlocal map to reproduce the predictions of complex-number quantum mechanics, establishing an equivalence between complex QM and real QM augmented by hidden nonlocal degrees of freedom. The results are presented as suggesting that complex numbers may be indispensable for a local description of such processes.

Significance. If the equivalence holds for the constructed models, the work supplies a concrete illustration linking the choice of number field in QM to locality and source independence. The explicit model construction allows direct verification of the nonlocal requirement and contributes to foundational discussions on the mathematical structure of quantum theory.

major comments (2)

[§3] §3 (construction of the real-number quantum theories): the demonstration that these specific models require a nonlocal map is shown, but the manuscript does not establish that every conceivable real-number Hilbert-space formulation satisfying the independent source assumption must introduce an equivalent nonlocal map. This generality gap is load-bearing for the suggestion that complex numbers are indispensable rather than merely one way to achieve locality.
[§5] §5 (conclusions and title): the claim that 'locality implies complex numbers' is framed more broadly than the results, which apply only to the presented models under the assumption; without a proof ruling out other local real formulations, the central implication remains conditional on model choice.

minor comments (1)

[Abstract] The abstract could more explicitly qualify that the equivalence is shown for the constructed models rather than all real formulations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments. We address the major comments point by point below, acknowledging the scope limitations of our results and indicating revisions to clarify the claims.

read point-by-point responses

Referee: [§3] §3 (construction of the real-number quantum theories): the demonstration that these specific models require a nonlocal map is shown, but the manuscript does not establish that every conceivable real-number Hilbert-space formulation satisfying the independent source assumption must introduce an equivalent nonlocal map. This generality gap is load-bearing for the suggestion that complex numbers are indispensable rather than merely one way to achieve locality.

Authors: We agree that the demonstration applies specifically to the real-number models constructed in §3. These models are built to satisfy the independent source assumption while using real Hilbert spaces, and we explicitly show that reproducing the predictions of complex QM requires a nonlocal map. The construction is intended to capture the essential structural features needed for independent sources in real-number theories. We do not provide a general no-go theorem excluding all other possible real formulations. To address this, we will revise §3 to include an explicit statement of the models' scope and note that the results apply to this class of theories. revision: yes
Referee: [§5] §5 (conclusions and title): the claim that 'locality implies complex numbers' is framed more broadly than the results, which apply only to the presented models under the assumption; without a proof ruling out other local real formulations, the central implication remains conditional on model choice.

Authors: The referee is correct that the title and some phrasing in §5 could suggest a broader implication than the model-specific results support. The abstract qualifies the findings as applying to 'the presented real-number quantum theories' and uses suggestive language ('may be indispensable'). Nevertheless, to prevent misinterpretation, we will revise the title and §5 to more precisely indicate that the equivalence and necessity of complex numbers hold for the constructed models under the independent source assumption. Proposed title revision: 'Locality and Independent Sources Require Complex Numbers in Real-Number Quantum Models'. Corresponding adjustments will be made to the conclusions. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected; derivation relies on explicit model construction

full rationale

The paper constructs specific real-number quantum theories that are compatible with the independent source assumption and then demonstrates that these models require a nonlocal map to reproduce the correlations of complex quantum theory. This establishes an equivalence only within the presented models rather than by redefining the inputs or fitting parameters to force the outcome. No self-citations, ansatzes smuggled via prior work, or uniqueness theorems imported from the authors appear in the abstract or description, and the central step is an explicit comparison between the constructed real models and standard complex QM. The derivation remains self-contained through direct construction and analysis of the given models.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the independent source assumption and the equivalence constructed for the presented real-number theories; hidden nonlocal degrees of freedom are introduced to achieve the match with complex theory.

axioms (1)

domain assumption Independent source assumption holds for the systems considered
Invoked in the abstract as the condition under which real-number theories require nonlocal maps.

invented entities (1)

hidden nonlocal degrees of freedom no independent evidence
purpose: To allow real-number quantum theory to reproduce complex quantum predictions while satisfying the independent source assumption
Postulated in the abstract to establish the equivalence; no independent evidence or falsifiable prediction is mentioned.

pith-pipeline@v0.9.0 · 5578 in / 1300 out tokens · 45392 ms · 2026-05-22T20:28:50.219893+00:00 · methodology

discussion (0)

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Quantum mechanics over real numbers fully reproduces standard quantum theory
quant-ph 2026-04 unverdicted novelty 7.0

A real-valued quantum framework based on ka space and symplectic tensor product is isomorphic to standard complex QM via explicit bijection and reproduces all predictions including maximal CHSH violation of 6√2.

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages · cited by 1 Pith paper · 2 internal anchors

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M. McKague, M. Mosca, and N. Gisin, Simulating quantum sy stems using real Hilbert spaces, Phys. Rev. Lett. 102, 02050 5 (2009)

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