⚛️ quantum physics

Quantum theory based on real numbers cannot be experimentally falsified

This paper demonstrates that real quantum theory is experimentally indistinguishable from standard complex quantum theory in any finite network or sequential multipartite scenario, provided that source independence is defined operationally rather than by untestable mathematical assumptions, thereby proving that real quantum theory cannot be falsified as long as standard quantum theory holds.

Original authors: Timothée Hoffreumon, Mischa P. Woods

Published 2026-03-20

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Timothée Hoffreumon, Mischa P. Woods

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Question: Does the Universe Need "Imaginary" Numbers?

For decades, physicists have wondered if the standard version of quantum mechanics (the theory that explains how atoms and particles work) needs complex numbers to be true.

Standard Quantum Theory (QT): Uses complex numbers (numbers involving $i$ , the square root of -1). Think of these as having a "real" part and an "imaginary" part.
Real Quantum Theory (RQT): A simplified version that tries to describe the universe using only real numbers, throwing away the "imaginary" part.

For a long time, scientists thought they could prove RQT was wrong. In 2021, a famous experiment (by Renou et al.) seemed to show that RQT couldn't explain certain results in a network of three people sharing particles. It looked like the universe was screaming, "I need complex numbers!"

This new paper says: "Hold on. You didn't actually prove that."

The authors, Hoffreumon and Woods, argue that the 2021 experiment relied on a hidden assumption that cannot be tested in a lab. When you remove that assumption, Real Quantum Theory becomes indistinguishable from Standard Quantum Theory. You can't tell them apart by doing experiments.

The Analogy: The "Ghost" in the Machine

To understand why, let's use an analogy involving maps and territories.

1. The Two Maps

Imagine you are trying to navigate a city.

Map A (Complex/QT): A detailed map that shows streets, parks, and also invisible "ghost tunnels" that only exist in the math.
Map B (Real/RQT): A simplified map that only shows the streets and parks. It claims the ghost tunnels don't exist.

For a long time, we thought Map B was wrong because it couldn't explain why you could get from Point A to Point B faster than the streets allowed. We thought the "ghost tunnels" (complex numbers) were real physical shortcuts.

2. The "Independence" Trap

The 2021 experiment tried to prove Map B was wrong by setting up a scenario with two independent sources (like two separate delivery trucks dropping off packages).

The Assumption: The experimenters assumed that if the trucks are "independent," their packages must be mathematically "separate" (a product state).
The Problem: The authors of this new paper say this assumption is like assuming that because two trucks are parked in different garages, their cargo must be completely unrelated.

In the world of Real Quantum Theory, two trucks can be parked in different garages (operationally independent) and their cargo can still be mysteriously linked in a way that only the Complex Map can see, but the Real Map treats as "normal."

3. The "Blindfold" Effect

Here is the kicker: The Real Map is blind to certain links.

Imagine you have two boxes.

In the Complex World, you can see that Box A and Box B are secretly tied together with a red string.
In the Real World, the red string is invisible. To the Real World observer, the boxes look completely separate.

The 2021 experiment tried to pull on the string to prove it existed. But the authors of this paper say: "If you are using a Real World observer, they can't see the string. If you force the Real World observer to assume the boxes are separate just because they look separate, you are cheating."

The authors prove that any result you get with the Complex Map can be perfectly mimicked by the Real Map, as long as you accept that the Real Map has a blindfold on. The Real Map can't see the "entanglement" (the red string), so it just assumes the boxes are separate. But because it can't see the string, it predicts the exact same outcomes as the Complex Map.

The Core Discovery: "Operational Independence" vs. "Mathematical Independence"

The paper makes a crucial distinction between two types of "independence":

Mathematical Independence (Product-State): The math says the two things are totally separate. (This is what the 2021 experiment assumed).
Operational Independence: You look at the data, and the two things act like they are separate. There are no weird correlations in the results.

The Big Reveal:
In the Complex World, these two things are the same. If they act separate, they are separate.
In the Real World, they are different. You can have things that act separate (Operational Independence) but are actually mathematically linked (not Product-State).

The 2021 experiment forced the Real World to act like the Complex World by assuming "Mathematical Independence." But since we can't test the math directly—only the results (Operational Independence)—that assumption was untestable.

The Conclusion: The Universe is a "Blind" Realist

So, what does this mean for us?

We can't prove Complex numbers are necessary. As long as our experiments agree with Standard Quantum Theory, a "Real Number" version of the universe could be hiding underneath, just with a different internal structure that we can't see.
The "Hidden Correlations": In the Real Number version of the universe, everything might be secretly connected in a dense web of correlations. But because the "Real" math is less powerful at detecting these connections (it's less "tomographic"), we can't see them. They are there, but they are invisible to our local measurements.
The Verdict: You can't falsify Real Quantum Theory. It's like trying to prove a magician is using a hidden wire when the magician is wearing a cloak that hides the wire. If the trick works, you can't tell if the wire is there or not.

Summary in One Sentence

The authors show that you can't prove the universe needs "imaginary" numbers because a "real number" version of the universe can mimic every single experiment we've ever done, provided we accept that the real version has a built-in blindfold that hides its own secret connections.

1. Problem Statement

The central question addressed is whether Complex Quantum Theory (QT) is experimentally distinguishable from Real Quantum Theory (RQT).

Context: RQT is a "foil theory" where complex amplitudes are replaced by real numbers, while retaining the standard Born rule, Lüders update, and Kronecker product ( $\otimes_K$ ) for system composition.
Historical Context: It has long been known that RQT can reproduce all single-party and bipartite Bell correlations of QT. However, RQT lacks local tomography (the state of a composite system cannot be fully reconstructed from local statistics alone).
The Conflict: In 2021, Renou et al. claimed that RQT could be experimentally falsified using a bilocal network (entanglement swapping) scenario. They derived a Bell-like inequality where the maximum value achievable in RQT (assuming independent sources are represented as product states) is strictly lower than in QT. Experiments have since violated this bound, seemingly confirming the falsification of RQT.
The Flaw Identified: The authors argue that Renou et al.'s conclusion relies on an untestable assumption: that "independent sources" in RQT must be mathematically represented as product states ( $\rho_X \otimes_K \rho_Y$ ). The paper posits that this conflates product-state independence (a mathematical constraint) with operational independence (the absence of observable correlations).

2. Methodology and Theoretical Framework

The authors employ a rigorous mathematical framework involving three theories:

QT: Standard complex Hilbert space theory.
RQT: Real Hilbert space theory with Kronecker product composition.
RNQT (Real-Number Quantum Theory): A sub-theory of RQT introduced in the authors' previous work, which is isomorphic to QT but uses a different composition rule ( $\otimes_R$ ) to handle the complex structure within real matrices.

Key Conceptual Distinction

Product-State Independence: The joint state of independent sources is a tensor product $\rho_{XY} = \rho_X \otimes_K \rho_Y$ . This is a model-dependent mathematical assumption.
Operational Independence: The joint probability distribution of outcomes for local measurements factorizes: $p(a,b) = p(a)p(b)$ . This is an experimentally verifiable condition.
The Crucial Insight: In QT, these two notions coincide. In RQT, they do not. There exist RQT states that are not product states (they are entangled in the real formalism) but are operationally independent (they produce no observable correlations). These states are "locally indistinguishable" from product states.

Technical Tools

Standard Mapping ( $\Gamma$ ): A mapping from complex matrices to real matrices of double dimension ( $d \to 2d$ ) that preserves algebraic structure.
$n$ -fold Standard Mapping ( $\Gamma^{(n)}$ ): An extension of the standard mapping to multipartite systems that distributes the "phase-carrying" fiducial system across all parties, ensuring locality is preserved.
R-Product ( $\otimes_R$ ): A composition rule for real matrices defined such that $\Gamma^{(n)}(A \otimes_K B) \cong \Gamma^{(n)}(A) \otimes_R \Gamma^{(n)}(B)$ . This rule allows the embedding of QT models into RQT.
Contamination Property: A technical lemma showing that if a state is in the "QT-image" (an R-product state), it can "contaminate" Kronecker products of effects, effectively converting them into R-products within the trace calculation, thereby preserving statistics.

3. Key Contributions and Results

Theorem 1: Network Equivalence

Statement: For any finite network of independent sources and locally measuring parties, if the sources are required only to be operationally independent (not necessarily product states), then every outcome distribution predicted by a QT model can also be predicted by an equivalent RQT model.

Mechanism: The authors prove that the set of RQT states that are operationally independent includes the image of QT product states under the mapping $\Gamma$ . These RQT states are non-product (entangled in the real formalism) but yield identical local statistics to QT product states.
Implication: The "falsification" of RQT by Renou et al. relied on restricting the source states to the subset of product states. Once the restriction is relaxed to the experimentally justified operational independence, RQT can reproduce all network correlations (including bilocal, triangle, and star networks) achievable by QT.

Theorem 2: Sequential Protocol Equivalence

Statement: The equivalence extends to arbitrary finite sequential multipartite protocols involving channels and measurements with prescribed locality structures.

Scope: This covers adaptive strategies, multi-round experiments, and protocols like LOCC (Local Operations and Classical Communication).
Mechanism: The authors construct a component-wise mapping where every QT channel and measurement is mapped to a corresponding RQT channel and measurement. The mapping preserves the locality structure (tensor product factorization) and the outcome statistics at every stage.
Significance: This demonstrates that no sequence of local operations and measurements can distinguish QT from RQT, provided the observed statistics are consistent with QT.

Interpretation of "Falsification"

The paper clarifies that while RQT cannot be falsified by experiments consistent with QT, the reverse is not true. It is theoretically possible to observe statistics consistent with RQT but inconsistent with QT (e.g., observing two systems that are locally indistinguishable but globally distinguishable). However, since no such violation has been observed, RQT remains empirically indistinguishable from QT.

4. Significance and Implications

Restoration of Empirical Indistinguishability: The paper restores the status of RQT as a viable empirical alternative to QT. The previous claim of falsification was an artifact of imposing a mathematical constraint (product states) that is not experimentally verifiable in a network setting.
Ontological Distinction: While the theories are empirically indistinguishable, they offer radically different pictures of reality:
- QT: Independent systems are genuinely uncorrelated unless interacting.
- RQT: Independent systems may possess "hidden" correlations (entanglement of representation) that are inaccessible to local tomography. The universe described by RQT has a denser correlation structure that is largely invisible to local observers.
Security Implications: The authors note a potential security concern. Cryptographic protocols proven secure in QT might be vulnerable in an RQT universe if the hidden correlations can be exploited via global operations, even if local statistics appear secure.
Methodological Lesson: The paper highlights the danger of testing "foil theories" without ensuring that the operational definitions of concepts (like "independence") are identical across theories. A theory cannot be falsified if the experimental constraints imposed on it are stronger than what the experiment actually tests.

Conclusion

Hoffreumon and Woods demonstrate that Real Quantum Theory cannot be experimentally falsified as long as the observed data is consistent with standard Quantum Theory. The apparent distinction found in bilocal networks arises from an untestable assumption about the mathematical form of source independence. By adopting the operationally motivated definition of independence, RQT is shown to be fully capable of reproducing the predictions of QT in all network and sequential scenarios. The distinction between the two theories is therefore purely interpretational, relating to the underlying correlation structure of the world rather than observable phenomena.