Quantum Physics Needs Imaginary Numbers

Multiple experiments have shown that the most straightforward versions of quantum physics equations that don't use imaginary numbers (i.e. quantities with values equal to the square root of negative one called "i"), don't work. This confirms the theoretical expectation.

Another way of putting this is that quantum physics requires some possible ways for events to happen to have negative probabilities of happening, although the ultimate observable result, which is calculated as the sum of all possible ways that something can happen, is still always between zero and one hundred percent.

Science News explains the limits of these findings as well in its article published today:

[T]he results don’t rule out all theories that eschew imaginary numbers, notes theoretical physicist Jerry Finkelstein of Lawrence Berkeley National Laboratory in California, who was not involved with the new studies. The study eliminated certain theories based on real numbers, namely those that still follow the conventions of quantum mechanics. It’s still possible to explain the results without imaginary numbers by using a theory that breaks standard quantum rules. But those theories run into other conceptual issues, making them “ugly,” he says. But “if you’re willing to put up with the ugliness, then you can have a real quantum theory.”

The article recaps conclusions from three recent physics papers, specifically:

Although complex numbers are essential in mathematics, they are not needed to describe physical experiments, as those are expressed in terms of probabilities, hence real numbers. Physics, however, aims to explain, rather than describe, experiments through theories. Although 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 studies have shown that such a ‘real quantum theory’ can reproduce the outcomes of any multipartite experiment, as long as the parts share arbitrary real quantum states. Here we investigate whether complex numbers are actually needed in the quantum formalism. We show this to be case by proving that real and complex Hilbert-space formulations of quantum theory make different predictions in network scenarios comprising independent states and measurements. This allows us to devise a Bell-like experiment, the successful realization of which would disprove real quantum theory, in the same way as standard Bell experiments disproved local physics.

M.-O. Renou et al. Quantum theory based on real numbers can be experimentally falsified. Nature. Published online December 15, 2021. doi: 10.1038/s41586-021-04160-4.

Standard quantum theory was formulated with complex-valued Schrdinger equations, wave functions, operators, and Hilbert spaces. Previous work attempted to simulate quantum systems using only real numbers by exploiting an enlarged Hilbert space. A fundamental question arises: are the complex numbers really necessary in the standard formalism of quantum theory? 

To answer this question, a quantum game has been developed to distinguish standard quantum theory from its real-number analogue, by revealing a contradiction between a high-fidelity multi-qubit quantum experiment and players using only real-number quantum theory. Here, using superconducting qubits, we faithfully realize the quantum game based on deterministic entanglement swapping with a state-of-the-art fidelity of 0.952. Our experimental results violate the real-number bound of 7.66 by 43 standard deviations. Our results disprove the real-number formulation and establish the indispensable role of complex numbers in the standard quantum theory.

M.-C. Chen et al. Ruling out real-valued standard formalism of quantum theory. Physical Review Letters. In press, 2021.

Quantum theory is commonly formulated in complex Hilbert spaces. However, the question of whether complex numbers need to be given a fundamental role in the theory has been debated since its pioneering days. Recently it has been shown that tests in the spirit of a Bell inequality can reveal quantum predictions in entanglement swapping scenarios that cannot be modelled by the natural real-number analog of standard quantum theory. Here, we tailor such tests for implementation in state-of-the-art photonic systems. We experimentally demonstrate quantum correlations in a network of three parties and two independent EPR sources that violate the constraints of real quantum theory by over 4.5 standard deviations, hence disproving real quantum theory as a universal physical theory.

Z.-D. Li et al. Testing real quantum theory in an optical quantum network. Physical Review Letters. In press, 2021.