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The internal nonlocality in $\cal PT$-symmetric systems

Minyi Huang, Ray‐Kuang Lee, Junde Wu

Published Sep 14, 2020• 0 citations
OpenAlex

Abstract

In this paper, we investigate the properties of $\cal PT$-symmetric quantum systems, proposing a new way to illustrate its novelty and measure its departure from classical and Hermitian quantum systems. We argue that there exists some internal nonlocal structure when simulating $\cal PT$-symmetric quantum systems. This is done by unveiling its merit in a concrete scenario. In addition, by using the dilation method, it is clearly shown that there exists some bounds of the correlation expectations. Moreover, it is shown that the exceptional point is some extreme point in this case. When the parameters are adjusted and the Hamiltonian tends to the exceptional point, it also tends to a largest departure from the usual Hermitian systems. This also provides us with a new perspective to discuss the properties of the exceptional point.

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The internal nonlocality in PT\cal PTPT-symmetric systems
In this paper, we investigate the properties of PT\cal PTPT-symmetric quantum systems, proposing a new way to illustrate its novelty and measure its departure from classical and Hermitian quantum systems. We argue that there exists some internal nonlocal structure when simulating PT\cal PTPT-symmetric quantum systems. This is done by unveiling its merit in a concrete scenario. In addition, by using the dilation method, it is clearly shown that there exists some bounds of the correlation expectations. Moreover, it is shown that the exceptional point is some extreme point in this case. When the parameters are adjusted and the Hamiltonian tends to the exceptional point, it also tends to a largest departure from the usual Hermitian systems. This also provides us with a new perspective to discuss the properties of the exceptional point.
To effectively realize a PT\mathcal{P}\mathcal{T}PT-symmetric system, one can dilate a PT\mathcal{P}\mathcal{T}PT-symmetric Hamiltonian to some global Hermitian one and simulate its evolution in the dilated Hermitian system. However, with only a global Hermitian Hamiltonian, how do we know whether it is a dilation and is useful for simulation? To answer this question, we consider the problem of how to extract the internal nonlocality in the Hermitian dilation. We unveil that the internal nonlocality brings nontrivial correlations between the subsystems. By evaluating the correlations with local measurements in three different pictures, the resulting different expectations of the Bell operator reveal the distinction of the internal nonlocality. When the simulated PT\mathcal{P}\mathcal{T}PT-symmetric Hamiltonian approaches its exceptional point, such a distinction tends to be most significant. Our results clearly make a distinction between the Hermitian dilation and other global Hamiltonians without internal nonlocality. They also provide the figure of merit to test the reliability of the simulation, as well as to verify a PT\mathcal{P}\mathcal{T}PT-symmetric (sub)system.
PT\cal PTPT-symmetric systems can be viewed as effective models in the sense of open systems. This allows people to simulate a PT\cal PTPT-symmetric system by dilating it to a global Hermitian one. Recently, the internal nonlocality in such dilated Hermitian systems was discussed for a special case with some two fold structure. In this paper, we extend such a discussion to the general case, by utilizing a general property of Hermitian dilation Hamiltonians. Different correlation pictures are proposed and the corresponding correlation bounds are obtained.
According to von Neumann, the global Hamiltonian of the whole universe must be Hermitian in order to keep the eigenvalues real and to construct a self-consistent quantum theory. In addition to the open system approach by introducing environmental degrees of freedom to a small system, a global Hermitian Hamiltonian can also be generated through the dilation from a small Hilbert space. For example, a local non-Hermitian PT\mathcal{P}TPT-symmetric system can be simulated with a global Hermitian one by the Naimark dilation. Recently, by introducing local measurements and investigating the correlation functions of outcomes, the internal nonlocality in such dilated Hermitian systems has been revealed, but only for a special case with a twofold structure. In this paper, we extend such a discussion to the generalized case when the twofold structure breaks. The internal nonlocality is discussed with different correlation pictures and the corresponding correlation bounds. Our results provide a device-independent test of the reliability of the simulation in the global Hermiticity.
The dilation method is a practical way to experimentally simulate non-Hermitian, especially PT\mathcal{PT}PT-symmetric, quantum systems. However, the time-dependent dilation problem cannot be explicitly solved in general. In this paper, we present a simple yet nontrivial exactly solvable dilation problem for two-dimensional time-dependent PT\mathcal{PT}PT-symmetric Hamiltonians. Our system is initially set in the unbroken PT\mathcal{PT}PT-symmetric phase, then goes across the so-called exceptional point, and ends in the broken PT\mathcal{PT}PT-symmetric phase. For this system, the dilated Hamiltonian and the evolution of PT\mathcal{PT}PT-symmetric system are analytically worked out. By investigating the large-time behaviors, we give an effective method to choose and adjust the dilation parameters. Our result also shows that the exceptional points do not have much physical relevance in a time-dependent system.
The dilation method is an important and useful way in experimentally simulating non-Hermitian, especially PT\cal PTPT-symmetric systems. However, the time dependent dilation problem cannot be explicitly solved in general. In this paper, we consider a special two dimensional time dependent PT\cal PTPT-symmetric system, which is initially set in the unbroken PT\cal PTPT-symmetric phase and later goes across the exceptional point and enters the broken PT\cal PTPT-symmetric phase. For this system, the dilation Hamiltonian and the evolution of PT\cal PTPT-symmetric system are analytically worked out.
We investigate the theory of exceptional points (EPs); degeneracies of non-hermitian systems famous for the swaps of eigenstates that arise when encircling the EP according to the adiabatic approximation. We review various characterizations of EPs that are used in the literature. We observe that they are not equivalent and consequently a unanimous definition is lacking. The theory of PT symmetric systems is studied as an aid in the study of EPs. In particular, we deal explicitly with a three-channel waveguide system originating from PT theory and supporting various EPs. We show how one can properly compose the eigenvalue permutations arising from EPs and how this generates a non-abelian group which we call the Λ-group. We finish with the needed geometry as induced by the adiabatic approximation. The adiabatic change as measured in experiment can be modelled naturally on a principal C× o Sn-bundle consisting of the eigenframes of the operator family. To the best of our knowledge this bundle is not found in the literature to date. The adiabatic approximation induces a canonical connection on this bundle. The special shape of the holonomy group induces three groups, one of which is the Λ-group, that fit together in a short exact sequence (SES). This SES can be used to describe and distinguish degeneracy structures. In particular, the Λ-group allows one to phrase a new definition of exceptional point that specializes existing definitions. Explicitly, one can define an EP by demanding that swaps of eigenvalues occur arbitrarily close to the EP. This definition is widely applicable, yet guarantees that the eigenvalue and eigenstates sheets are connected in a non-trivial way.