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arXiv:2111.05270 · Physical Review A

Internal nonlocality in generally dilated Hermiticity

Minyi Huang, Ray‐Kuang Lee

Published May 17, 2022• 7 citations
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Abstract

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.

DOI
10.1103/physreva.105.052210

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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.
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.
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.
Non-unitary dynamics in quantum mechanics arise in open quantum systems, where the interaction between the system and its environment is not negligible. One way this coupling can be realised is through measurement, where the backaction from the measurement apparatus induces non-unitary dynamics in the system. An effective non-Hermitian description arises within the no-jump approximation of a measured system, in which the time-evolved states are explicitly normalised at all times. Non-Hermiticity in quantum Hamiltonians gives rise to non-unitary time evolution and possibly complex energy eigenvalues, which can lead to a rich phenomenology with no Hermitian counterpart. In this thesis, we explore these two interconnected concepts related to non-unitary dynamics: non-Hermiticity and measurements. However, instead of effective non-Hermiticity, we consider the cases where non-Hermiticity is realised exactly. This setup necessitates a treatment under the self-consistent frameworks of non-Hermitian quantum mechanics in order to restore its probabilistic interpretation. Concretely, the self-consistent metric formalism restores unitarity of the evolution through a modified Hilbert space endowed with a dynamical metric. Through comparative studies using exactly solvable models, we find that the physics of the two approaches of non-Hermitian quantum mechanics is fundamentally different. This has a profound physical impact in terms of defect production in quenches across an exceptional point (EP). We demonstrate this through a non-Hermitian system hosting both PT-symmetric and PT-broken modes subject to a linear quench. Our results highlight the role played by the EP and the spontaneous breaking of the PT-symmetry in contributing to a non-zero defect density, even in the limit of an infinitely slow drive. Further, we uncover generalised transformations underpinning the results, which are incorporated in the dynamical metric. Going beyond purely theoretical considerations, we implement two novel dilation schemes to emulate the metric dynamics in a digital quantum simulator. Our protocols enable controlled access to all PT-symmetry regimes and to a direct tomography of the dynamical metric, opening the possibility of studying a new kind of non-Hermitian physics in engineered matter. To explore the second theme of the thesis, we investigate the dynamics of entanglement entropy in the continuously monitored Bose-Hubbard model. Our numerical data reveals a counterintuitive enhancement of superfluid correlations in a parameter regime, in which the entanglement entropy increases with the onsite repulsion strength in the presence of measurement. This phenomenon arising from the interplay between hopping, interaction, and measurement, is also present in the effective central charge extracted from a fit of the logarithmic scaling form of the entropy. In summary, this thesis reveals the richness of non-unitary quantum dynamics. Through an in-depth study of the different aspects of non-Hermitian systems, we illustrate the non-trivial physics uncovered only with the consideration of the dynamical metric. By bridging theoretical insights and physical realisation through novel dilation protocols realisable in engineered matter, our work lays the groundwork for further exploration of dynamical aspects in non-Hermitian and monitored quantum systems.
Non-Hermitian quantum systems have recently attracted considerable attention due to their exotic properties. Though many experimental realizations of non-Hermitian systems have been reported, the non-Hermiticity usually resorts to the hard-to-control environments and cannot last for too long times. An alternative approach is to use quantum simulation with the closed system, whereas how to simulate non-Hermitian Hamiltonian dynamics remains a great challenge. To tackle this problem, we propose a protocol which combines a dilation method with the variational quantum algorithm. The dilation method is used to transform a non-Hermitian Hamiltonian into a Hermitian one through an exquisite quantum circuit, while the variational quantum algorithm is for efficiently approximating the complex entangled gates in this circuit. As a demonstration, we apply our protocol to simulate the dynamics of an Ising chain with nonlocal non-Hermitian perturbations, which is an important model to study quantum phase transition at nonzero temperatures. The numerical simulation results are highly consistent with the theoretical predictions, revealing the effectiveness of our protocol. The presented protocol paves the way for practically simulating small-scale non-Hermitian dynamics.
Local non-Hermitian (NH) quantum systems generically exhibit breakdown of Lieb-Robinson (LR) bounds, motivating study of whether new locality measures might shed light not seen by existing measures. In this paper we discuss extensions of the connected correlation function (CC) as measures of locality and information spreading in both Hermitian and NH systems. We find that in Hermitian systems, $\ensuremath{\delta}\ensuremath{\rho}=\ensuremath{\rho}\ensuremath{-}{\ensuremath{\rho}}_{A}\ensuremath{\bigotimes}{\ensuremath{\rho}}_{B}$ can be written as a linear combination of CCs, allowing placement of an LR bound on ${\ensuremath{\parallel}\ensuremath{\delta}\ensuremath{\rho}\ensuremath{\parallel}}_{2}$, which we show generically extends to an LR bound on mutual information. Additionally, we extend the CC to NH systems in a form that recovers locality, and use the metric formalism to derive a modified CC which recovers not just locality but even LR bounds in local PTPTPT-Symmetric systems. We find that even with these CCs, the bound on ${\ensuremath{\parallel}\ensuremath{\delta}\ensuremath{\rho}\ensuremath{\parallel}}_{2}$ breaks down in certain NH cases, which can be used to place a necessary condition on which local NH Hamiltonians are capable of nonlocal entanglement generation. Numerical simulations are provided by means of exact diagonalization for the NH Transverse-Field Ising Model, demonstrating both breakdown and recovery of LR bounds.