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

PT-symmetric quantum theory was originally proposed with the aim of extending standard quantum theory by relaxing the Hermiticity constraint on Hamiltonians. However, no such extension has been formulated that consistently describes states, transformations, measurements and composition, which is a requirement for any physical theory. We aim to answer the question of whether a consistent physical theory with PT-symmetric observables extends standard quantum theory. We answer this question within the framework of general probabilistic theories, which is the most general framework for physical theories. We construct the set of states of a system that result from imposing PT-symmetry on the set of observables, and show that the resulting theory allows only one trivial state. We next consider the constraint of quasi-Hermiticity on observables, which guarantees the unitarity of evolution under a Hamiltonian with unbroken PT-symmetry. We show that such a system is equivalent to a standard quantum system. Finally, we show that if all observables are quasi-Hermitian as well as PT-symmetric, then the system is equivalent to a real quantum system. Thus our results show that neither PT-symmetry nor quasi-Hermiticity constraints are sufficient to extend standard quantum theory consistently.

Automated summary

This study aims to answer the question of whether there exists a consistent physical theory that extends standard quantum theory with PT-symmetric observables. The study finds that, according to the framework of general probabilistic theories, the resulting theory from imposing PT-symmetry on observables only allows for one trivial state. Moreover, considering the constraint of quasi-Hermiticity on observables, the study shows that such a system is equivalent to a standard quantum system. If all observables are both quasi-Hermitian and PT-symmetric, then the system is equivalent to a real quantum system.