Quantum theory was originally formulated using complex numbers. Nonetheless, when replying to a letter by Hendrik Lorenz, Erwin Schrödinger (one of its founding fathers) wrote: “Using complex numbers in quantum theory is unpleasant and should be objected to. The wave function is surely fundamentally a real function.”
In recent years, scientists successfully ruled out any local hidden variable explanation of quantum theory using Bell tests. Later, such tests were generalized to a network with multiple independent hidden variables. In such a quantum network, quantum theory with only real numbers, or “real quantum theory,” and standard quantum theory make quantitatively different predictions in some scenarios, enabling experimental tests of the validity of real quantum theory.
Researchers at Southern University of Science and Technology in China, the Austrian Academy of Sciences, and other institutes worldwide have recently adapted one of these tests so that they could be implemented in state-of-the-art photonic systems, Phys.org reports. Their paper, published in Physical Review Letters, experimentally demonstrates the existence of quantum correlations in an optical network that cannot be explained by real quantum theory.
“From the early days of quantum theory, complex numbers were treated more as a mathematical connivence than a fundamental building block,” Zizhu Wang, one of the researchers who carried out the study, told Phys.org. “The general debate on the role of complex numbers in quantum theory has continued into the present.”
In the 1960s, the Swiss physicist Ernst Stueckelberg and his colleagues successfully formulated quantum theory in real Hilbert spaces. While this was an important milestone in the field, their formulation did not use the renowned, so-called “tensor product” to compose different systems. This essentially means that their formulation is not consistent with what is known as “real quantum theory.”
“Interest in this question was revived when we started looking at quantum theory from an information-theoretic perspective,” Wang explained. “Some generalized probabilistic theories (GPTs), formulated using only real numbers, turn out to be as powerful as quantum theory in some information-processing tasks, and even outperform quantum theory in some others. Even though we know GPTs contain correlations beyond quantum theory, we did not have the tools to definitively rule out real quantum theory as a viable alternative to complex quantum theory, until now.”
The recent paper by Fan and his colleagues draws inspiration from a long-standing debate in the physics field, namely that pertaining to the existence of local hidden variables in quantum theory. Physicists Albert Einstein, Boris Podolsky, and Nathan Rosen posed this important question in one of their seminal papers, published in 1935. While many physicists explored this question in later years, for decades no one was able to devise a concrete method to test whether these local hidden variables exist.
“In 1964, John Bell came up with the revolutionary idea of using correlations functions of probabilities, which can be tested and analyzed in a laboratory, to infer underlying properties of physical systems,” Jingyun Fan, another researcher involved in the study, said. “It took another 50 years to finally settle this debate and systematically rule out local hidden variable explanations of quantum theory.”
While it has been successfully applied in many studies, Bell’s theorem alone is not powerful enough to accurately predict the differences between real and complex quantum theories. In their recent study, Fan and his colleagues were able to assess these differences by considering a quantum network with multiple, independent sources.