Reply to the Comment on ‘The operational foundations of PT-symmetric and quasi-Hermitian quantum theory’

We thank M. Znojil for reading our work with interest and for initiating a constructive exchange. Their Comment [1] on our work [2] consists of three addenda. In the first addendum, it is claimed that our work is ill-motivated as the motivating question, namely whether PT-symmetric quantum theory extends the standard quantum theory, has been answered over 12 years ago. The second addendum points to some missing references. Finally, the third addendum suggests what constraints could lead to an extension of standard quantum theory. In this reply, we comment on these three addenda. In particular, we emphasize that the author found our work ill-motivated because they have misinterpreted our motivating question to be alluding to rather restricted definitions (which we refer to as settings in the rest of this reply) of PT-symmetric quantum theory, which have been already investigated in the literature. In fact, the third addendum in their Comment in itself elaborates on why our motivating question is interesting and relevant even though their addendum is restricted in its scope as it pertains only to non-stationary version of quasi-Hermitian quantum theory (QHQT). As our response to the second addendum, we also explain our rationale behind citing certain references while leaving out others. Whereas some relevant references were omitted unintentionally, these missing references would have in fact served to strengthen the main message of our paper, as we discuss below.

Before diving into the three addenda mentioned above, we point out that the correctness of our results was not challenged in Comment [1]. Rather, the motivation behind the work and the interpretation of our results are criticized. We also clarify that we do not agree with the author’s claim that the main mathematical message of our paper is the “compatibility between the three alternative versions of quantum theory.” If “three alternative versions” refer to the three settings we examine in our paper, then quite clearly they lead to very different physical systems, as stated therein. These three settings pertain to three different physical theories: one with PT-symmetric observables, one with quasi-Hermitian observables, and one with quasi-Hermitian observables with PT symmetry, given in §3, §4 and §5 of [2], respectively. We show that these settings are respectively equivalent to theories with only one trivial state, all states allowed by standard quantum theory and all real quantum states. The meaning of equivalence in this context is clearly explained in our paper, the idea being that there is a linear bijection between the sets of states and effects of corresponding systems. Thus, the main message of our paper is conveyed accurately in the abstract of our paper and is quite different from the one claimed in the Comment under consideration.

In the first addendum provided in §2 of the Comment, the author claims that the question ‘whether PT-symmetric or quasi Hermitian quantum theories extend standard quantum theory’ has been answered more than 12 years ago, and provides some references to back this claim. In making this claim, the author is assuming that there exists a certain well-defined theory called PT-symmetric quantum theory in the literature. We reiterate, as we did in §I and §II of our paper, that postulates or axioms of PT-symmetric quantum theory have never been written down and/or agreed upon. Any work attempting to answer the question given above must choose a setting that defines PT-symmetric quantum theory. For instance, Ref. 3 cited in the Comment provides a discussion of various settings that were motivated by PT symmetry and investigated to form consistent extensions of standard quantum theory [3]. Some of these settings pertain to the use of indefinite, $CPT$ and other positive-definite inner product spaces, respectively. The last setting in fact gives rise to QHQT, which is also considered in our paper. However, Ref. [3] does not preclude the possibility of other settings leading to consistent extensions of standard quantum theory. Other examples from the literature include Ref. [4] (Ref. 4 in the Comment) and Ref. [5], which consider a setting in which all observables are quasi-Hermitian with respect to a possibly time-dependent metric operator.

In fact, we raised the above question (not for the first time) without keeping any particular setting in mind. The only guide to our exploration was to base our assumptions on the concepts developed in the field of PT-symmetry. It is obvious, then, that the question we raised cannot be answered in full generality, as any such answer will always depend on the setting under consideration. Even our work does not answer this motivating question in its entirety, as newer settings can lead to different conclusions. In the third addendum, the author of the Comment seems to have misinterpreted our motivation and work to be only pertaining to the stationary version of QHQT. We never made any statement to this effect. This appears to be the root cause of confusion, which might have motivated a comment from the author in the first place. Contrary to the claims in the Comment, even the second setting in our paper (see §IV of [2]), which is the closest to QHQT, need not be restricted to the stationary case. We now discuss how the characterization of states in this setting is equally applicable to the non-stationary case. Consider a finite-dimensional system with time-dependent metric operator $\eta(t)$ and evolving under some time-dependent Hamiltonian $H(t)$. $H(t)$ is quasi-Hermitian with respect to $\eta(t)$ (see Eq. (4) in [2]). In other words, $H(t)$ is such that the unitarity of the evolution is preserved with respect to $\eta(t)$. At any time $t$, the states of the system are linear functionals on the observables at time $t$, which are also quasi-Hermitian with respect to the instantaneous $\eta(t)$. Then, by our analysis in §IV, these states are represented by $\eta(t)$-density operators. This is in agreement with the literature on QHQT with non-stationary or time-dependent metric operator [5]. Furthermore, by employing the tools developed in [6, 7, 8], one can see that for finite-dimensional systems a time-dependent metric operator is equivalent to representing the system in a time-dependent Hilbert space. Thus, our analysis in §IV dictates that the non-stationary version of QHQT for finite-dimensional systems does not extend the standard quantum theory; rather, similar to the stationary QHQT, it is equivalent to representing standard quantum theory in a non-orthogonal basis, albeit a time-dependent one.

Regarding the third addendum given in §4 of the Comment, we welcome the author’s optimism about a possible extension of the standard quantum theory via non-stationary version of QHQT in infinite dimensions. However, we caution that such an extension must also have a consistent interpretation in the general probabilistic theories (GPT) framework, unless the notions behind GPTs are being challenged. Exploring such a consistent interpretation in the GPT framework is a worthwhile exercise in our opinion. It is then apparent that the third and the heaviest addendum in the Comment in fact serves to elaborate on the validity and the importance of the motivating question we raise. We thank M. Znojil for this contribution.

We now address the second addendum in §3 of the Comment. The author has expressed disappointment over the apparent lack of relevant references on the GPT framework. We cited those references from the GPT literature that were the most significant for our analysis. We claim neither to provide a historical overview of the field nor that the list of references in our paper is exhaustive. For the same reason, Gudder’s seminal work that contributed to the development of GPT is not cited. This special issue is dedicated to Gudder’s work, and we believe that the importance of the contributions by Gudder is beyond dispute, and it does not need further evidence in the form of citations of works that do not impact our analysis directly.

We thank the author, however, for bringing to our attention two papers [9, 10], namely Refs. [18,19] in the Comment, that discuss the PT symmetric quantum theory in terms of generalized effect algebras, which are a particular approach to GPTs. We clarify that these references pertain to setting 2 of our paper (see §4), and they were unintentionally omitted from our list of references. We are apologetic for these omissions. Nevertheless, the results of our paper are in complete agreement with the results in Refs. [9, 10]. Therefore, they strengthen the main message of our paper. On the other hand, it is important to note that settings given in §3 and §4 of our paper and the approach used to study them are novel and previously unexplored in the literature.

In the introduction of the Comment, the author of the Comment claims that some of the statements in our paper could mislead the reader into believing that PT-symmetric quantum theory is not useful. Throughout the paper, we did not make any comment on the utility of PT-symmetric quantum theory; we restricted our statements to the possibility of an extension of the standard quantum theory, and backed our claims with rigorous mathematics. We completely agree with the author of the Comment that PT-symmetric quantum theory, in its various forms, has engendered new ideas and their realizations both in mathematics and physics.

Having finished commenting on all three addenda given in the Comment, we now turn to addressing a particular point made therein. That is, “the concept of the ‘extension’ of the existing quantum theory is vague”. The author implies that the ambiguity in the meaning of extension stems from a myriad formulations that are prevalent under the name of standard quantum theory. We completely disagree with this statement in the Comment. Quantum theory as a physical theory has precisely stated postulates, and while it has several formulations, they are all equivalent. Indeed, the author correctly notes that various formulations of quantum theory “differ dramatically in mathematical and conceptual overview, yet each one makes identical predictions for all experimental results”. Thus, what quantum theory is is not vague. On the contrary, it provides a concrete prescription for predicting observable quantities irrespective of the formalism being used. Perhaps the author means to argue that, while standard quantum theory is not vague, the meaning of ‘extending’ this theory is unclear. A point of confusion here is the meaning of the word ‘extension’ itself. The author of the Comment, on multiple occasions therein, has used this term to mean both an ‘interpretation of a physical theory’ (e.g. “benchmark models still wait for a ‘meaningful extension’ of their fully consistent GPT interpretation”) and, more vaguely, to mean a new theory that subsumes the old theory. As for the former context, the word ‘interpretation’ is more appropriate and accepted in the community. What we mean by an extension of the standard quantum theory is another theory that consists of new states that are not in bijection with the states of the former (cf. Def. 2.10 in [2]). In other words, if any state in the new theory does not have a counterpart in the standard quantum theory, then that theory would be an extension of the latter. We therefore disagree with the author’s statement that “the concept of the ‘extension’ of the existing quantum theory is vague”. We however agree that one formulation of quantum theory may turn out to be more suited than others in the quest for such extensions.

In conclusion, much of the content of the Comment under consideration appears to have stemmed from a misinterpretation of the motivating question we pose in our original paper. Ironically, the final addendum in the Comment reiterates the validity and importance of the motivating question pursued in our work. While we disagree with the author of the Comment on a couple of points, none of the criticisms provided therein has any implications on the correctness of our results.