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arXiv:1106.1550

Comments on "Observation of Fast Evolution in Parity-Time-Symmetric System"

Fabio Masillo

Published Jun 8, 2011• 1 citation
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Abstract

In the paper "Observation of Fast Evolution in Parity-Time-Symmetric System" the authors propose a physical apparatus for the realization of a faster than Hermitian evolution. This last appears in contrast with the conclusions obtained in our paper "Some Remarks on Quantum Brachistochrone". We will clarify this apparent contradiction and some problematic aspects of the treatment in [1].

DOI
10.48550/arxiv.1106.1550

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In his fresh "Comment" (arXiv:0711.0137v1), A. Mostafazadeh reacts on my very recent letter (arXiv:0710.5653v1) where I tried to clarify certain misunderstandings which occurred in A. M., Phys. Lett. B \textbf{650}, 208 (2007) [arXiv:0706.1872v2, "Paper"]. As long as the "Comment" offers a new support of the original assertions made in the "Paper", I feel obliged to re-clarify the matter by extending my argumentation. I insist that it is possible to escape the main conclusion of the "Paper", indeed. In particular, I point out a gap in the new calculations in "Comment", add a few remarks on the notation and reconfirm that the unitarity of the time-evolution DOES NOT require the time-independence of the metric operator.

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Comment on "Reply to Comment on Time-dependent Quasi-Hermitian Hamiltonians and the Unitary Quantum Evolution"

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I point out that if one defines the operator $U_R(t)$ as done by M. Znojil in his reply [arXiv:0711.0514v1] to my comment [arXiv:0711.0137v1] and also accepts the validity of the defining relation of $U_R(t)$ as given in his paper [arXiv:0710.5653v1], one finds that the time-evolution of the associated quantum system is not governed by the Schrödinger equation for the Hamiltonian operator $H$ but an operator $H'$ which differs from $H$ if the metric operator is time-dependent. In the latter case this effective Hamiltonian $H'$ is not observable. This is consistent with the conclusions of my paper [Phys. Lett. B 650, 208 (2007), arXiv:0706.1872v2] which allow for unitary time-evolution generated by unobservable Hamiltonians.

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Advanced Science•Aug 24, 2025•3 citations
In the paper "Observation of Fast Evolution in Parity-Time-Symmetric System" the authors propose a physical apparatus for the realization of a faster than Hermitian evolution. This last appears in contrast with the conclusions obtained in our paper "Some Remarks on Quantum Brachistochrone". We will clarify this apparent contradiction and some problematic aspects of the treatment in [1].
We explain why the main conclusion of Bender et al, hep-th/0511229 [J. Phys. A 39 (2006) 1657] regarding the practical superiority of the non-Hermitian description of PT-symmetric quantum systems over their Hermitian description is not valid. Recalling the essential role played by the Hermitian description in the characterization and interpretation of the physical observables, we maintain that as far as the physical aspects of the theory are concerned the Hermitian description is not only unavoidable but also indispensable.
In his fresh "Comment" (arXiv:0711.0137v1), A. Mostafazadeh reacts on my very recent letter (arXiv:0710.5653v1) where I tried to clarify certain misunderstandings which occurred in A. M., Phys. Lett. B \textbf{650}, 208 (2007) [arXiv:0706.1872v2, "Paper"]. As long as the "Comment" offers a new support of the original assertions made in the "Paper", I feel obliged to re-clarify the matter by extending my argumentation. I insist that it is possible to escape the main conclusion of the "Paper", indeed. In particular, I point out a gap in the new calculations in "Comment", add a few remarks on the notation and reconfirm that the unitarity of the time-evolution DOES NOT require the time-independence of the metric operator.
I point out that if one defines the operator UR(t)U_R(t)UR​(t) as done by M. Znojil in his reply [arXiv:0711.0514v1] to my comment [arXiv:0711.0137v1] and also accepts the validity of the defining relation of UR(t)U_R(t)UR​(t) as given in his paper [arXiv:0710.5653v1], one finds that the time-evolution of the associated quantum system is not governed by the Schrödinger equation for the Hamiltonian operator HHH but an operator H′H'H′ which differs from HHH if the metric operator is time-dependent. In the latter case this effective Hamiltonian H′H'H′ is not observable. This is consistent with the conclusions of my paper [Phys. Lett. B 650, 208 (2007), arXiv:0706.1872v2] which allow for unitary time-evolution generated by unobservable Hamiltonians.
Unambiguous Quantum State Discrimination in a PTPT\mathcal {PT}PT‐Symmetric System of a Single Trapped Ion
Abstract An unambiguous quantum state discrimination of two qubit states is experimentally demonstrated under a non‐Hermitian Hamiltonian with parity‐time‐reversal () symmetry in a single trapped 40 Ca + ion. It is shown that any two non‐orthogonal states can become orthogonal subjected to time evolution of a ‐symmetric Hamiltonian in both the ‐symmetry preserving and broken regimes, thus can be discriminated unambiguously. For a given pair of candidate states, it is shown that the parameters of the Hamiltonian must be confined in a proper range, within which there exists an optimal choice to realize quantum brachistochrone for the fastest orthogonalization. Besides, a clear geometric picture and some analytic results are provided to understand the main conclusions. This work shows a promising application of non‐Hermitian physics in quantum information processing.