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The internal nonlocality in general dilated Hermiticity

Minyi Huang, Ray‐Kuang Lee

Published Nov 9, 2021• 0 citations
OpenAlex

Abstract

$\cal PT$-symmetric systems can be viewed as effective models in the sense of open systems. This allows people to simulate a $\cal PT$-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.

Related papers

Internal nonlocality in generally dilated Hermiticity

Minyi Huang, Ray‐Kuang Lee

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 $\mathcal{P}T$-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.

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Extracting the internal nonlocality from the dilated Hermiticity

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To effectively realize a $\mathcal{P}\mathcal{T}$-symmetric system, one can dilate a $\mathcal{P}\mathcal{T}$-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 $\mathcal{P}\mathcal{T}$-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 $\mathcal{P}\mathcal{T}$-symmetric (sub)system.

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Simulating Broken PT -Symmetric Hamiltonian Systems by Weak Measurement

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By embedding a PT-symmetric (pseudo-Hermitian) system into a large Hermitian one, we disclose the relations between PT-symmetric quantum theory and weak measurement theory. We show that the weak measurement can give rise to the inner product structure of PT-symmetric systems, with the preselected state and its postselected state resident in the dilated conventional system. Typically in quantum information theory, by projecting out the irrelevant degrees and projecting onto the subspace, even local broken PT-symmetric Hamiltonian systems can be effectively simulated by this weak measurement paradigm.

arXiv:1809.09918•Physical Review Letters•Aug 23, 2019•36 citations
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.
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.
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.
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.
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.
By embedding a PT-symmetric (pseudo-Hermitian) system into a large Hermitian one, we disclose the relations between PT-symmetric quantum theory and weak measurement theory. We show that the weak measurement can give rise to the inner product structure of PT-symmetric systems, with the preselected state and its postselected state resident in the dilated conventional system. Typically in quantum information theory, by projecting out the irrelevant degrees and projecting onto the subspace, even local broken PT-symmetric Hamiltonian systems can be effectively simulated by this weak measurement paradigm.