Quantum theory based on real numbers can be experimentally falsified

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.

Full Network Nonlocality

“…If a correlation pð ājxÞ does not admit a model of the form (1), it is called network nonlocal (NN). While it is generally believed that networks enable new forms of nonlocality as compared to standard Bell scenarios (see, e.g., [8][9][10][11]), it remains largely unclear how such phenomena arise and to what extent they are intrinsic to the network structure. This is partly due to properties inherent to the definition (1).…”

The operational foundations of PT-symmetric and quasi-Hermitian quantum theory

“…However, we eliminate the assumption that pure states in the new theory constitute a Hilbert space, as done, instead, in the existing literature. Finally, we consider the setting in which all observables are quasi-Hermitian as well as PT-symmetric, and show that the resulting theory has equivalent features to real quantum theory [48][49][50][51][52][53][54][55]. Our results show that neither PT-symmetry nor quasi-Hermiticity constraints are sufficient to extend standard quantum theory consistently.…”

“…The main finding of this section is that a GPT with η-Hermitian, K η -symmetric observables is isomorphic to real quantum theory. To prove this result, we first focus on the special case where the allowed observables are Hermitian (η = 1) as well as κ-symmetric, κ being complex conjugation in the canonical basis as in Section III, and show that we arrive at real quantum theory [48][49][50][51][52][53][54][55]. After that, we extend the analysis to observables being Hermitian as well as K-symmetric, where K is a valid PT-symmetry (cf.…”

Recycling nonlocality in a quantum network

“…Networks constitute natural generalisations of the traditional Bell scenario: several parties are connected via multiple independent sources that distribute entangled states in some network configuration [36,37]. They enable interest-ing new possibilities such as both stronger [36,38] and novel forms [39] of entanglement-swapping experiments, nonlocality without inputs [40], limitations on measurement dependence [41] and distinguishing the role of complex numbers in quantum theory [42]. This has motivated several implementations of network nonlocality experiments [43][44][45][46][47][48][49][50].…”