A real-valued description of multipartite quantum experiments

Quantum mechanical predictions and outcomes of measurements are real-valued, while the abstract quantum mechanical formalism typically relies on using complex numbers. However, there have been several suggestions how to reformulate quantum mechanics in a (higher-dimensional) real Hilbert space: A quantum state in a d-dimensional complex Hilbert space can alternatively be represented as a vector in a 2d-dimensional real Hilbert space. Particular care has to be taken when describing state vectors of composite quantum systems: in a complex Hilbert space, an undetectable global phase factor of a product state can be ascribed to one or another subsystem, or even be ”split up” between the subsystems (into suitable local phase factors), without changing any observable quantities. Thus, these different quantum states are indistinguishable, or in other words equivalent. We build a different variant of real quantum mechanics in a way that it transfers this property of the state of a composite quantum system to a real Hilbert space.

We revisit recent claims about the necessity of complex numbers in the context of multipartite Bell-type experiments, and an experimental demonstration thereof. A subtle but essential point in these works is the following postulate which the authors require for both the complex-valued and the real-valued description of quantum mechanics: ”The Hilbert space corresponding to the composition of two systems is given by the tensor product . ... Similarly, the state representing two independent preparations of the two systems is the tensor product of the two preparations.” However, this postulate is not justified for composite quantum systems described in a real Hilbert space. Therefore, any conclusion drawn - using this postulate - will also not be justified. Our conclusion instead is that real and complex Hilbert space formulations of quantum theory do not make different predictions. Thus, complex numbers are not necessary, even in the context of multipartite quantum experiments.