Restricting the effects hides a nonphysical symmetry from every causal structure

In the quest to understand the fundamental rules of the universe, physicists often ask whether the mathematics we use to describe reality is the only possible way. One of the most successful frameworks is quantum theory, which relies on complex numbers to predict how particles behave. However, a simpler version of this theory exists, one that uses only real numbers. For decades, scientists believed that this simpler, real-number version was fundamentally weaker than the full complex version. They thought that in certain networked setups, where independent sources send information to a central observer, the complex theory could produce correlations that the real theory simply could not match. This difference was thought to be a hard boundary, a line in the sand separating what is physically possible from what is not.

The question of whether nature truly requires complex numbers has moved from abstract philosophy to experimental reality. Recent experiments have confirmed that the complex version of quantum theory does indeed outperform the real version in these specific network scenarios. This suggests that the universe is not just a real-number system in disguise; it possesses a richness that real numbers alone cannot capture. But this conclusion rests on a specific assumption: that the rules of the game allow for every possible measurement that the theory's states could logically support. If we were to restrict the types of measurements allowed, perhaps the gap between the real and complex worlds would disappear.

A new study challenges the idea that the difference between real and complex quantum theory is an unbreakable law of nature. The researchers constructed a specific, hypothetical version of quantum theory where the rules for what can be measured are deliberately limited. In this modified theory, the states of the system are exactly the same as in standard quantum mechanics, but the allowed measurements are a strict subset of what is usually permitted. The team then tested whether this restricted theory could still be distinguished from its own "real-number" version. They found that it could not. In every possible network arrangement, the restricted theory produced the exact same results as its real-number counterpart, even though the symmetry that separates them is mathematically "broken" in a way that usually signals a physical impossibility.

The core of this discovery lies in how the researchers defined the boundaries of their theory. In standard quantum mechanics, if a state exists, any measurement that yields a valid probability is considered allowed. This is known as the "no-restriction hypothesis." The new theory abandons this rule. It keeps all the standard states but removes many of the possible measurements. Specifically, it forbids any measurement that would reveal a contradiction if the system were subjected to a complex conjugation operation—a mathematical flip that turns complex numbers into their real counterparts. By removing these specific measurements, the theory effectively blinds itself to the very feature that usually distinguishes it from a real-number system.

The researchers demonstrated that this restricted theory behaves exactly like a real-number theory in every causal scenario, including the famous "bilocal" network where two independent sources feed a central party. In this setup, the full complex theory produces correlations that violate a limit known as the real bound. However, the restricted theory, despite being built on complex states, cannot produce these violations. It is as if the theory has voluntarily given up the ability to see the difference between itself and a simpler world. The symmetry that separates the two worlds is still there mathematically, but because the theory lacks the tools to detect it, the separation vanishes from an experimental perspective.

This finding suggests that the robustness of the difference between real and complex quantum theory is not a deep, unchangeable feature of the universe, but rather a consequence of the assumption that all logically possible measurements are physically allowed. The study shows that if you restrict the measurements, you can hide the non-physical nature of a symmetry. The symmetry itself, which involves flipping complex numbers, remains mathematically inconsistent with the theory's states when applied to entangled systems. Yet, because the theory does not include the specific measurements needed to expose this inconsistency, the theory remains internally consistent and experimentally indistinguishable from a real-number theory.

The researchers achieved this by carefully designing a simulation where each source in a network carries its own reference frame, rather than each individual particle. This subtle shift allows the mathematical operations that usually break the theory to be absorbed into the measurement process without causing errors. In a standard setup, applying a complex conjugation to part of an entangled pair would produce a result that is not a valid physical state. But in this restricted theory, the measurements are chosen such that they never interact with the parts of the system where this breakdown would be visible. The result is a theory that is mathematically complex but experimentally real.

The paper explicitly rules out the idea that the separation between real and complex theories is inevitable for any non-classical system. It shows that the separation depends on the specific choice of allowed measurements. If the set of measurements is too broad, the gap appears; if it is restricted in the right way, the gap disappears. This does not mean that our universe is necessarily a real-number system, but it does mean that the argument for why it must be complex cannot rely solely on the existence of a gap in network correlations. The gap exists only because we assume the theory allows every measurement that its states can support.

The study also clarifies the role of "weakly non-physical" symmetries. In physics, a symmetry is usually considered physical if it can be performed as an operation. If it cannot, it is often dismissed as impossible. However, this work identifies a middle ground where a symmetry is mathematically impossible to perform as a single operation, yet it does not lead to any logical contradictions or negative probabilities when applied within a closed circuit. This happens because the theory's measurements are restricted in a way that filters out the inconsistencies. The symmetry is "weakly non-physical," meaning it breaks the rules of the theory but never in a way that the theory can detect.

Ultimately, the research provides a precise criterion for when a theory and its symmetrized version will have the same correlations. It turns out that if the theory's measurements and operations are invariant under the symmetry acting independently on each source, then no gap will appear. This condition, which the author calls "sectorial closure," is sufficient to ensure that the theory behaves as if the symmetry were physical, even when it is not. The study constructs a theory that meets this condition and proves that it has no gap, demonstrating that the boundary between real and complex theories is not fixed by the symmetry itself, but by the structure of the theory's allowed measurements.

The implications are significant for how we understand the foundations of quantum mechanics. The difference between real and complex theories is not an absolute feature of the mathematical landscape but a property of the specific theory we inhabit. Our universe appears to allow every measurement that its states permit, which is why the gap between real and complex theories is visible to us. If the universe were different, and if it restricted its measurements in a specific way, that gap would vanish. The study does not claim that the real-number theory is the correct description of nature, but it does show that the argument for complex numbers cannot rest on the mere existence of a gap in network correlations. The gap is a feature of our specific theory, not a universal law.

In the end, the work reveals that the separation between real and complex quantum theories is sustained by a specific property of quantum theory: the assumption that every effect permitted by the states is physically realizable. When this assumption is relaxed, the symmetry that usually divides the two worlds becomes undetectable. The researchers have shown that by restricting the effects, one can build a theory that carries the same symmetry as complex quantum theory but behaves exactly like a real-number theory in every experiment. This finding shifts the focus from the symmetry itself to the structure of the theory, suggesting that the richness of the complex world is a consequence of the freedom to measure, rather than a fundamental necessity of the symmetry.