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

TL;DR

This paper investigates whether PT-symmetry and quasi-Hermiticity can extend standard quantum theory and finds that these constraints do not lead to a non-trivial extension, effectively reducing to standard or real quantum systems.

Contribution

The paper demonstrates that PT-symmetry and quasi-Hermiticity constraints do not produce a consistent, non-trivial extension of standard quantum theory within the general probabilistic framework.

Findings

PT-symmetric states are trivial under the probabilistic framework

Quasi-Hermiticity with PT-symmetry is equivalent to standard quantum theory

All observables being quasi-Hermitian and PT-symmetric reduces to real quantum systems

Abstract

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.