Quantum theory based on real numbers can be experimentally falsified (2101.10873v2)

Abstract: While complex numbers are essential in mathematics, they are not needed to describe physical experiments, expressed in terms of probabilities, hence real numbers. Physics however aims to explain, rather than describe, experiments through theories. While most theories of physics are based on real numbers, quantum theory was the first to be formulated in terms of operators acting on complex Hilbert spaces. This has puzzled countless physicists, including the fathers of the theory, for whom a real version of quantum theory, in terms of real operators, seemed much more natural. In fact, previous works showed that such "real quantum theory" can reproduce the outcomes of any multipartite experiment, as long as the parts share arbitrary real quantum states. Thus, are complex numbers really needed in the quantum formalism? Here, we show this to be case by proving that real and complex quantum theory make different predictions in network scenarios comprising independent states and measurements. This allows us to devise a Bell-like experiment whose successful realization would disprove real quantum theory, in the same way as standard Bell experiments disproved local physics.

• The paper demonstrates that a network-based experimental design can differentiate the predictions of real-number and complex quantum theories.
• It utilizes entanglement swapping and Bell-type inequalities to reveal that real quantum models fail to achieve the same quantum correlations as complex formulations.
• The findings imply that current quantum technologies can test and potentially refute real-number-based quantum mechanics, emphasizing the necessity of complex numbers.

An Analysis of "Quantum theory based on real numbers can be experimentally falsified"

The paper "Quantum theory based on real numbers can be experimentally falsified" confronts a fundamental aspect of quantum mechanics: the intrinsic necessity and role of complex numbers within its formalism. While quantum theory typically utilizes complex Hilbert spaces, the paper investigates whether a real-number-based quantum theory can account for all physical phenomena explained by complex quantum theory.

Summary of Key Concepts

Quantum mechanics, in its standard form, employs complex Hilbert spaces to describe quantum states and their evolutions. A longstanding question in quantum physics is whether the use of complex numbers is indispensable or if real numbers could suffice for the same theoretical predictions. The paper introduces an experimental framework that seeks to differentiate between the predictions of real and complex quantum theories.

The central argument presented is that while real quantum theory can replicate the outcomes of specific quantum experiments, it falls short in certain network scenarios. In these scenarios, independent quantum states and measurements are introduced, enabling experimental designs that could ultimately refute the feasibility of real quantum mechanics altogether.

Theoretical Framework and Results

The authors propose a network-based experimental setup reminiscent of a Bell test but extended to involve multiple independent sources and more complex configurations. The core result of the paper is that real and complex quantum theories yield different predictions for such network configurations, particularly those involving entanglement swapping and independent sources.

The analysis includes a complex quantum scenario where two sources produce pairs of entangled particles. These particles are then distributed among three observers—Alice, Bob, and Charlie. Bob performs a Bell-state measurement that entangles Alice's and Charlie's particles. The resultant statistics can be represented using complex quantum states and measurements, whereas real quantum theory fails to capture the same correlations.

Furthermore, the paper identifies experimental conditions and Bell-type inequalities that can help distinguish between real and complex quantum predictions. A particular measure, denoted by $\mathscr{T}$, is defined to quantify the extent of quantum correlations. It is shown that for complex quantum systems, $\mathscr{T}$ can reach higher values than any real quantum counterpart could achieve.

Implications and Future Directions

This work has significant implications for both theoretical physics and potential experimental validations. It proposes that real quantum theory can be tested and potentially falsified using current or near-future quantum technologies, specifically through scenarios involving entanglement swapping and network non-locality.

From a foundational perspective, these results underscore the necessity of complex numbers in quantum mechanics, moving beyond a convenient mathematical abstraction to an essential feature of the physical world. The findings suggest that the exclusion of complex numbers from quantum theory fundamentally limits its descriptive power concerning observable phenomena in multipartite quantum systems.

In experimental physics, the paper opens a pathway to designing experiments, akin to Bell tests, that can rigorously test and potentially refute real-number-based quantum formulations. This exploration into foundational quantum mechanics could augment our understanding of quantum correlations and the mathematical structures necessary to describe reality accurately.

Conclusion

The paper "Quantum theory based on real numbers can be experimentally falsified" presents compelling arguments and scenarios for the critical role of complex numbers in quantum mechanics. By devising specific network architectures and leveraging independent states and measurements, it sets the stage for possible experimental refutation of real quantum mechanics. This work not only pushes the boundaries of theoretical physics but also invites further experimental investigations into the true nature of quantum systems.