Researchers at Inria Saclay are approaching the long-standing question of whether quantum phenomena require complex numbers with a physically motivated principle, rather than a purely mathematical postulate. While two real-valued formulations of quantum theory have emerged, one version preserving the standard mathematical rule for combining systems has been theoretically and experimentally ruled out for failing to reproduce quantum experiments. This leaves an alternative that necessitates a rethinking of fundamental quantum information concepts by modifying how independent quantum systems combine. As Pedro Barrios Hita and colleagues explore, reframing the debate with an additional assumption offers a compelling path forward: introducing an assumption, based on a physically motivated principle, about how independent quantum systems combine may allow for a viable real-valued quantum theory.
Real-Valued Quantum Theory: Two Compositional Approaches
The pursuit of a quantum theory formulated entirely with real numbers, rather than the conventionally used complex numbers, has revealed a surprising divergence in approaches, with one definitively ruled out by recent experiments. For decades, physicists have debated whether complex numbers are fundamental to describing nature, or merely a mathematical convenience in quantum mechanics. Two distinct formulations emerged attempting to bypass the need for them; one preserved the standard mathematical rule for combining quantum systems, while the other modified it. Research published in Physics 19, 85 demonstrated that the approach maintaining the standard combination rule ultimately fails to reproduce the results of more complex quantum experiments. This failure stems from a fundamental issue with representing quantum information using only real numbers without introducing inconsistencies.
The initial strategy involved doubling the dimensions of the Hilbert space, the mathematical space describing a quantum system, to accommodate the real components of what would normally be complex amplitudes. This is understood by associating each quantum system with an additional two-level system, known as the flag qubit. The problem arises when combining these systems; the resulting space requires twice as many real dimensions as its complex counterpart. The first approach attempted to address this by allowing arbitrary real-valued representations of experiments, potentially disconnecting them from standard quantum theory. While initially plausible, and even able to reproduce Bell-type experiments, it proved unsustainable under more rigorous testing.
This alternative modifies the composition rule for combining quantum systems, effectively sharing the “flag qubit” globally across all subsystems. Researchers explain that by exploiting a physically motivated principle, rather than a mathematical postulate, they offer a new perspective on how a real-valued quantum theory can be constructed. This approach successfully reproduces all predictions of standard quantum mechanics, but at the cost of altering our understanding of how quantum information is structured. The team, working at the Centre for Theoretical Physics and Computer Science Laboratory, École Polytechnique, along with Pedro Barrios Hita at the German Aerospace Center and his collaborators, have reframed the debate by focusing on the physical constraints governing the combination of independent quantum systems.
Barrios Hita and colleagues have taken a step in a new direction, beginning with a physically motivated principle: a local operation acting on one subsystem should not affect another independently prepared subsystem. This leads to a modified composition rule that restores a redundancy inherent in standard quantum theory; a composite system can have multiple mathematically distinct descriptions representing the same physical situation. Complex numbers are not fundamentally necessary to reproduce observable quantum phenomena once the tensor-product rule is relaxed, and Barrios Hita and collaborators offer a physically motivated justification for the alternate rule that they use.
Flag Qubit Strategy for Real Number Representation
One strategy attempted to preserve the standard mathematical rules governing how independent quantum systems combine, while the other modified those rules to accommodate a real-number framework. However, the first of these, which doubled the dimensionality of quantum systems by associating each with an additional “flag qubit,” has now been definitively ruled out by experimental results. A study published in Physics 19, 85 demonstrated that more-general experiments involving independent systems cannot be reproduced by any real-valued formulation that preserves the standard tensor-product structure. Barrios Hita and colleagues have already shifted the focus from purely mathematical postulates to physically motivated principles. The core challenge lies in representing quantum information without the inherent properties of complex numbers; the flag qubit strategy addresses this by effectively encoding the missing information from the imaginary component within an auxiliary two-level system.
The difficulty arises when combining these independent systems. The standard tensor product, which dictates how dimensions multiply in combined systems, results in a composite space with an excess of real dimensions. The second approach, which shares a single flag qubit globally across all subsystems, circumvents this issue, successfully reproducing the predictions of standard quantum mechanics. However, this comes at a cost, abandoning the conventional understanding of how independently prepared systems interact, a structure that underpins much of our current understanding of quantum information.
Locally Independent Systems Define Modified Tensor Product
Fatemeh Moradi-Kalarde and Marc-Olivier Renou, researchers at Inria Saclay, are currently refining a novel approach to formulating quantum theory without relying on complex numbers, a long-standing challenge in physics. Their work diverges from previous attempts by prioritizing a physically motivated principle over purely mathematical postulates, a step Barrios Hita and colleagues already took, yielding promising results in the quest for a real-valued quantum framework. The team’s focus isn’t simply to eliminate complex numbers, but to understand if their presence is truly fundamental, or an artifact of the mathematical tools used to describe quantum phenomena. Two distinct real-valued formulations have previously emerged, each tackling the problem differently. Beyond the specific construction, the work emphasizes the importance of grounding quantum theory in physical principles, rather than abstract mathematical axioms.
Redundancy in Real Framework Mirrors Complex Quantum Theory
While complex numbers are ubiquitous in the standard mathematical description of quantum mechanics, their fundamental necessity has long been debated, prompting exploration of alternative formulations. Researchers at Inria Saclay and École Polytechnique have now demonstrated a pathway where a modified approach to combining quantum systems, guided by physical reasoning, effectively replicates all observable quantum phenomena. The resulting framework introduces a redundancy mirroring that found in standard quantum theory; because multiplying a quantum state by a complex phase has no physical consequence, multiple mathematical descriptions can represent the same physical situation. The team’s construction recovers an analogous redundancy in the real-valued setting, effectively demonstrating that complex numbers aren’t fundamentally required to describe quantum phenomena if the standard tensor-product rule is relaxed. This isn’t simply a mathematical exercise; it’s a reframing of the debate, centering on the physical principles that should dictate how independent quantum systems combine.
