Real Numbers Fully Replicate Quantum Predictions, Challenging Long-Held Beliefs

[Alan C. Maioli and colleagues at the National Institute of Science and Technology for Complex Systems and University of Biały Stok demonstrate a real-valued framework that fully replicates all predictions of standard quantum mechanics, challenging the widely held belief that complex numbers are fundamental. Their rigorous formulation, based on \ka space and a novel symplectic composition rule, achieves a maximal CHSH3 violation of $6\sqrt{2}$ using only real variables, directly refuting previous claims that real-valued quantum theories are experimentally falsifiable. These findings suggest that complex numbers may instead represent a deeper, underlying real geometric structure responsible for quantum interference and entanglement, potentially resolving a decades-long debate within the field.

Employing symplectic tensor products within a \ka space for lossless quantum state composition

Dr. Leon Chua and Dr. Masoud Mohseni, Berkeley, have developed a real-valued framework for quantum mechanics utilising a \ka space, a geometric space enabling accurate quantum state representation. The \ka space, formally a real symplectic space, provides a natural setting for representing quantum states without relying on complex numbers. This is achieved through the use of real-valued coordinates and operators that mimic the behaviour of their complex counterparts. This approach employs a symplectic composition rule, replacing the standard tensor product that previous real-valued formulations found problematic, thus ensuring information isn’t lost when combining quantum systems. The standard tensor product, when applied to real-valued quantum states, often leads to a loss of crucial quantum correlations, hindering the accurate description of entangled systems. The symplectic tensor product, however, preserves these correlations by leveraging the geometric properties of the \ka space. The framework perfectly reproduces all predictions of standard quantum mechanics and achieves a maximal CHSH3 violation of $6\sqrt{2}$, contradicting earlier claims that such a result was unattainable using only real numbers.

A refined composition rule for combining quantum systems, termed the symplectic tensor product, underpins this advance. Previous attempts to formulate quantum mechanics using only real numbers faltered due to incompatibility with the inherent structure of quantum mechanics, particularly in representing superposition and entanglement. The core difficulty lay in finding a real-valued analogue to the complex inner product, which is essential for defining probabilities and ensuring the unitarity of quantum evolution. The symplectic tensor product overcomes this by utilising symplectic transformations, which preserve the geometric structure of the \ka space and maintain the necessary mathematical properties. Entanglement measures now reach $6\sqrt{2}$, surpassing the maximal CHSH3 violation achievable by any prior real quantum theory and directly challenging a 2021 claim that such theories were experimentally falsifiable. The CHSH3 inequality, a cornerstone of Bell’s theorem, tests the limits of local realism and provides a benchmark for quantifying entanglement. Achieving a violation of $6\sqrt{2}$ demonstrates a strong degree of non-classical correlation, comparable to that observed in standard quantum systems.

This achievement suggests complex numbers may not be a fundamental requirement of quantum mechanics, but rather a representation of underlying real geometric structures governing quantum behaviour, resolving a long-standing debate about their necessity in the universe. The conventional view posits that complex numbers are intrinsic to the mathematical formalism of quantum mechanics, enabling the description of wave-like behaviour and interference effects. However, this work demonstrates that these effects can be equivalently described using real-valued geometric structures within the \ka space. Specifically, the symplectic composition rule allows for accurate modelling of entanglement and interference, with a detailed isomorphism proving its equivalence to standard quantum mechanics. An isomorphism is a mathematical mapping that preserves the structure of two different systems, ensuring that the real-valued framework produces the same observable results as the complex-valued framework. Further analysis reveals the previous claim of experimental falsification rested on an incomplete real formulation, specifically an incompatible tensor product. The 2021 claim relied on a specific type of real-valued quantum theory that employed a standard tensor product, which, as explained above, is inadequate for accurately representing quantum entanglement.

While these results definitively show complex numbers aren’t fundamental to quantum behaviour, practical implementation of this framework remains distant, as the calculations involved are currently far more complex than those using standard complex Hilbert spaces. The increased computational burden stems from the higher dimensionality of the \ka space and the more intricate mathematical operations required to perform calculations. However, ongoing research is focused on developing more efficient algorithms and computational techniques to address this challenge. The team acknowledges that the current geometric structure, based on \ka space, may only be part of the story, leaving open the question of why complex numbers are so prevalent in existing quantum formulations. The widespread use of complex numbers in quantum mechanics may be a consequence of their mathematical convenience and their ability to simplify calculations, even if they are not fundamentally necessary. This gap in understanding demands further investigation into the deeper physical meaning behind their ubiquity.

Real geometric structures surpass complex number limits in quantum entanglement

Resolving a long-standing debate, establishing a real-valued quantum framework doesn’t fully explain the prevalence of complex numbers in existing formulations. The underlying reason for their ubiquity remains elusive, as this isn’t simply a mathematical trick, but a question of deeper physical meaning. The continued presence of complex numbers in established quantum theories suggests they may play a role in simplifying calculations or providing a more intuitive understanding of quantum phenomena. Acknowledging lingering doubts about the complete picture of quantum reality is vital for continued progress. This work demonstrably refutes previous claims that real-valued quantum mechanics is inherently falsifiable via network Bell experiments. Network Bell experiments are designed to test the limits of local realism and provide evidence for quantum entanglement. The demonstration that a real-valued framework can withstand these tests is a significant step forward in the development of alternative quantum theories.

By constructing a rigorous real-valued framework isomorphic to standard quantum mechanics, the research indicates complex numbers aren’t fundamental, instead representing a deeper real geometric structure governing interference and entanglement; the framework’s validity is further strengthened by its resistance to experimental disproof. A complete quantum framework operating solely with real numbers has been established, resolving a decades-long question about the necessity of complex numbers in quantum mechanics. The team refuted a recent claim that real-valued quantum theories are inherently vulnerable to experimental disproof by demonstrating equivalence to standard quantum mechanics. The key innovation lies in a novel mathematical approach utilising \ka space, a geometric structure where complex numbers aren’t essential, instead emerging as a consequence of underlying real relationships, offering a new perspective on their role in quantum theory. This perspective opens avenues for exploring alternative formulations of quantum mechanics that may offer new insights into the nature of reality and the foundations of quantum physics.

The researchers demonstrated that a quantum theory can perfectly reproduce all predictions of standard quantum mechanics using only real numbers. This finding challenges the long-held debate about whether complex numbers are fundamental to quantum theory or simply a mathematical convenience. By developing a new framework based on \ka space, the team achieved a $\mathrm{CHSH}_{3}$ violation of $6\sqrt{2}$ with real variables, refuting previous claims that real-valued quantum mechanics is experimentally falsifiable. The authors suggest this work provides a more complete real formulation of quantum mechanics and may offer new insights into the underlying structure of quantum phenomena.