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

TL;DR

This paper critiques recent claims about quasi-Hermitian quantum theory, clarifying that its main findings are not new and discussing alternative frameworks and constraints relevant to the theory's extensions.

Contribution

It clarifies misconceptions about the novelty of QHQT's limitations and discusses alternative mathematically consistent theories and potential constraints for extending the framework.

Findings

The main discovery about QHQT not extending standard quantum theory is not new.

Existing GPT-like theories, such as effect algebras, are already available.

A tentative answer is provided for the constraints needed for meaningful extensions.

Abstract

An elucidation of the current state of art in quasi-Hermitian quantum theory (QHQT) as inspired by the recent paper by Alase et al (J. Phys. A: Math. Theor. 55 (2022) 244003, paper [1]) is offered. We point out that the author's main discovery (viz., that the QHQT does not extend the standard quantum theory) is not new. In a related comment on the author's method of proof performed in ``the framework of general probabilistic theories'' (GPT) we add that also in this context a few other, mathematically consistent GPT-like theories are already available in the literature (pars pro toto we mention the results using the so called effect algebras). Thirdly, the ``intriguing open question'' about ``what possible constraints, if any, could lead to such a meaningful extension'' is given a tentative answer: The constraint could be just the generalized, non-stationary version of the quasi-Hermiticity.