Abstract
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 spaces1,2. 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 natural3. 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 states4. 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.
Highlights
In its Hilbert space formulation, quantum theory is defined in terms of the following postulates5,6. (1) For every physical system S, there corresponds a Hilbert space HS and its state is represented by a normalized vector φ in HS, that is, ⟨φ|φ⟩ = 1. (2) A measurement Π in S corresponds to an ensemble {Πr}r of projection operators, indexed by the measurement result r and acting on HS, with ∑r Πr = IS. (3) Born rule: if we measure Π when system S is in state φ, the probability of obtaining result r is given byPr(r) = ⟨φ|Πr|φ⟩. (4) The Hilbert space HST corresponding to the composition of two systems S and T is HS ⊗ HT
Owing to the controversy surrounding their irruption in mathematics and their almost total absence in classical physics, the occurrence of complex numbers in quantum theory worried some of its founders, for whom a formulation in terms of real operators seemed much more natural
Ψ is surely fundamentally a real function.” (Letter from Schrödinger to Lorentz, 6 June 1926; ref. 3)). This is precisely the question we address in this work: whether complex numbers can be replaced by real numbers in the Hilbert space formulation of quantum theory without limiting its predictions
Summary
Marc-Olivier Renou[1], David Trillo[2], Mirjam Weilenmann[2], Thinh P. 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[1,2]. We investigate whether complex numbers are 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. It is noted that, assuming a fixed Hilbert space dimension, Conan could come up with single-site experiments where real and complex quantum theory differ, for instance, because the former does not satisfy local tomography, or even leads to different experimental predictions Conan may consider experiments involving several distant labs, where phenomena such as entanglement[21] and Bell non-locality[22] can Complex
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