⚛️ quantum physics

Gap between quantum theory based on real and complex numbers is arbitrarily large

This paper demonstrates that in star-network quantum scenarios with $N+1$ parties, the performance gap between complex and real quantum theories grows linearly with the number of parties, proving that real Hilbert space formalisms become arbitrarily inadequate for describing complex quantum networks as the system scales.

Original authors: Shubhayan Sarkar, David Trillo, Marc Olivier Renou, Remigiusz Augusiak

Published 2026-07-22

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CC BY 4.0

Original authors: Shubhayan Sarkar, David Trillo, Marc Olivier Renou, Remigiusz Augusiak

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

Imagine the universe as a giant, cosmic puzzle where the pieces don't just sit next to each other; they talk to each other in ways that seem to break the rules of everyday logic. This is the world of quantum mechanics, the rulebook for how tiny particles like electrons and photons behave. For over a century, scientists have been arguing over the "language" this rulebook is written in. The standard version uses complex numbers, which sound like a math class nightmare but are actually a special tool that includes a mysterious ingredient called "imaginary" numbers. These aren't made-up; they are essential for describing how particles spin and dance. However, a bold alternative theory suggests we could get away with using only real numbers—the familiar 1, 2, 3, and so on—by simply doubling the size of our mathematical toolbox. For a long time, it seemed like these two languages might just be different ways of saying the exact same thing, like describing a sunset in English or Spanish. But here's the kicker: if the universe really needs complex numbers, then the "real number" version of physics is fundamentally broken, no matter how hard we try to fix it. The big question was: just how broken is it? Is the difference a tiny crack, or is it a canyon?

Enter a team of physicists who decided to put these two theories to the ultimate test in a cosmic game of "connect the dots." They set up a scenario called a star network, which is like a party where one central host (let's call her Eve) is connected to many guests (Alice, Bob, Charlie, and so on) via independent sources. In this game, the guests and the host perform measurements on shared quantum particles. The goal is to see if the correlations—the patterns in their results—can be explained by the "real number" theory or if they strictly require the "complex number" theory. The researchers constructed a special mathematical test, a Bell inequality, which acts like a scorecard. If the score stays within a certain limit, the "real number" theory holds up. If the score goes way over the limit, the complex theory wins.

What they found is nothing short of mind-bending. They discovered that as they added more guests to the party, the gap between what the complex theory predicts and what the real theory can possibly explain didn't just get a little bigger; it grew linearly with the number of parties. In plain English, if you have a few people, the difference might be small. But if you have a huge network of quantum systems, the "real number" theory becomes so inaccurate that it's practically useless. The gap becomes arbitrarily large, meaning there is no fundamental limit to how much the real theory fails compared to the complex one. It's as if trying to describe a symphony using only a single drumbeat; the more complex the music gets, the more obvious the failure becomes.

The team didn't just guess this; they proved it mathematically. They showed that in this star network, the complex theory can achieve a score of 1 on their test, while the real theory is capped at 1/(N - 1), where N is the number of external parties. As N grows, that fraction shrinks toward zero, creating a massive chasm between the two. They also checked if this result would hold up in a messy, noisy real-world experiment. Even when they allowed for small errors (denoted as ε) in the measurements, the gap remained robust, though the math showed that the error needs to be suppressed relative to the number of parties to see the full effect. Their analysis suggests that for a network with just N = 3 parties, the gap is already significant, and it only gets more dramatic as the network expands.

This work effectively shuts the door on the idea that complex numbers are just a convenient mathematical trick we could swap out for real numbers if we were clever enough. The paper demonstrates that imaginary numbers are a genuine resource in the quantum world, providing an advantage that grows without bound as systems get larger. While the authors note that creating such large, independent networks in a lab is challenging (because keeping sources truly independent is hard), their theoretical proof is solid. They have shown that the universe isn't just slightly different when you use real numbers; it's fundamentally incapable of describing complex quantum networks without the "imaginary" ingredient. The gap isn't a crack in the foundation; it's a widening abyss that proves the complex formalism of quantum mechanics isn't just a choice—it's a necessity.

Technical Summary: Gap between Quantum Theory Based on Real and Complex Numbers is Arbitrarily Large

Problem Statement
Quantum information theory is fundamentally formulated using complex Hilbert spaces (Complex Quantum Theory, CQT). While it has long been known that single-system quantum correlations can be simulated using real Hilbert spaces (Real Quantum Theory, RQT) by doubling the dimension, the necessity of complex numbers in multipartite scenarios has only recently been challenged. A previous study [8] demonstrated that in a specific three-party network (bilocality), RQT cannot reproduce all CQT predictions. However, a critical open question remained: Is there a fundamental, system-size-independent bound on the magnitude of the gap between CQT and RQT? If such a bound exists, RQT could serve as a valid approximation for large systems. Conversely, if the gap can grow arbitrarily large, the "unphysical" nature of complex numbers becomes a more profound resource.

Methodology
The authors address this by generalizing the network scenario to a star network involving $N+1$ parties: $N$ external parties ( $A_i$ ) and a central party ( $E$ , Eve). The network consists of $N$ independent sources, each distributing a bipartite quantum state between an external party and Eve.

Operational Setup: External parties perform one of three measurements ( $x_i \in \{0,1,2\}$ ) with binary outcomes. Eve performs a single measurement with $2^N$ outcomes.
Conditional Bell Functional: To rigorously compare CQT and RQT, the authors construct a "conditional" Bell inequality. This involves two functionals:
1. $\hat{I}_l$ : A functional derived from self-testing protocols [27, 28] designed to certify that the network achieves maximal quantum violation. In CQT, the maximum value is $2(N-1)$ .
2. $J_N$ : A functional based on Mermin's inequality [29], normalized to the range $[-1, 1]$ .
The Witness: The authors define a gap witness by maximizing $J_N$ subject to the operational condition that $\hat{I}_l$ achieves its maximal quantum value for all outcomes $l$ . This "conditioning" ensures that the comparison is made between the ideal CQT state/measurements and the best possible RQT approximation of that specific configuration.
Analytical Approach: The authors employ self-testing techniques to characterize the states and measurements in RQT under the constraint of maximal $\hat{I}_l$ violation. They analyze the tensor-product structure and the constraints imposed by real-valued observables.

Key Contributions and Results

Arbitrarily Large Gap: The paper proves that the ratio of the maximal value of $J_N$ in CQT to that in RQT grows linearly with the number of parties $N$ .
- In CQT, with the ideal GHZ state and specific observables (including Pauli $Y$ ), the value of $J_N$ is $\beta_{\text{CQT}} = 1$ .
- In RQT, under the constraint that $\hat{I}_l$ is maximally violated, the authors prove (Theorem 1) that the maximal value of $J_N$ is bounded by $\beta_{\text{RQT}} \leq 1/(N-1)$ .
- Consequently, the gap ratio $\beta_{\text{CQT}}/\beta_{\text{RQT}} \geq N-1$ . As $N \to \infty$ , this gap becomes arbitrarily large, demonstrating that no fundamental bound independent of system size exists.
Robustness to Noise: The authors extend their results to noisy scenarios (Theorem 2). They show that even if the observed violation of $\hat{I}_l$ deviates from the ideal by a small error $\epsilon_N$ , and Eve's outcome probabilities deviate slightly, the upper bound for $J_N$ in RQT remains suppressed. Specifically, $\beta_{\text{RQT}} \leq \frac{1}{N-1} + O(N^6\sqrt{\epsilon_N})$ . This indicates that while the error tolerance requires suppression relative to $N$ , the gap persists in the presence of small experimental imperfections.
Numerical Verification: For the specific case of $N=3$ , the authors performed numerical analysis using semidefinite programming. The results indicate that the actual gap in RQT is significantly larger than the analytical lower bound derived in the paper, suggesting the analytical bounds are conservative.

Significance
The paper claims to resolve a foundational question in quantum mechanics: Complex numbers are not merely a mathematical convenience but a necessary resource that provides an unbounded advantage over real numbers in describing quantum networks.

Foundational Impact: The results imply that as quantum networks scale, the real-number formalism becomes "exceedingly worse" at describing the system's correlations compared to the complex formalism. This challenges the notion that RQT is a sufficient approximation for large-scale quantum systems.
Experimental Implications: The authors suggest that larger star networks are better suited for closing the detection loophole in experiments distinguishing CQT from RQT. A larger gap reduces the detector efficiency required to experimentally verify the discrepancy between the theories.
Resource Theory: The work identifies "imaginarity" (the presence of complex numbers) as a resource capable of providing unbounded advantage in multipartite scenarios, distinct from the advantages provided by entanglement alone in standard Bell scenarios.

The authors maintain a modest tone regarding future work, noting that while they have demonstrated the existence of an unbounded gap, realizing this with a fixed number of parties remains an open question. They also acknowledge the experimental challenges of enforcing source independence but argue that such independence is a fundamental assumption of physics.

Technical Summary: Gap between Quantum Theory Based on Real and Complex Numbers is Arbitrarily Large

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