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Witnessing non-Markovianity and criticality in (anti-)parity-time-symmetric systems

Hossein Rangani Jahromi, Franco Nori, Rosario Lo Franco

Published Jan 12, 2021• 5 citations
OpenAlex

Abstract

Non-Hermitian systems with parity-time ($\mathcal{PT}$) symmetry and anti-$\mathcal{PT}$ symmetry give rise to exceptional points (EPs) with important properties due to the coalescence of eigenvectors. Because of the limitations on the Hermiticity of the Hamiltonian, these kinds of non-Hermitian systems have recently attracted extensive attention and emerged as a new way to engineer the response of open quantum systems. We propose an efficient and easily computable tool, based on the Hilbert-Schmidt speed (HSS), which does not require the diagonalization of the evolved density matrix, to characterize non-Markovian effects in (anti-)$\mathcal{PT}\!-$symmetric systems. Our proposed witness of non-Markovianity can also detect exactly the EPs and hence the critical behavior of the non-Hermitian systems, especially high-dimensional qudits. Our theoretical predictions are completely consistent with results extracted from recent experiments studying the non-Markovianity and criticality in (anti-)$\mathcal{PT}\!-$symmetric systems.

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Theoretical investigation of dynamics and concurrence of entangled $${{\mathcal {P}}}{{\mathcal {T}}}$$ and anti-$${{\mathcal {P}}}{{\mathcal {T}}}$$ symmetric polarized photons

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Non-Hermitian systems with parity-time [Formula: see text] symmetry and anti-parity-time [Formula: see text] symmetry have exceptional points (EPs) resulting from eigenvector co-coalescence with exceptional properties. In the quantum and classical domains, higher-order EPs for [Formula: see text] symmetry and [Formula: see text]-symmetry systems have been proposed and realized. Both two-qubits [Formula: see text]-[Formula: see text] and [Formula: see text]-[Formula: see text] symmetric systems have seen an increase in recent years, especially in the dynamics of quantum entanglement. However, to our knowledge, neither theoretical nor experimental investigations have been conducted for the dynamics of two-qubits entanglement in the [Formula: see text]-[Formula: see text] symmetric system. We investigate the [Formula: see text]-[Formula: see text] dynamics for the first time. Moreover, we examine the impact of different initial Bell-state conditions on entanglement dynamics in [Formula: see text]-[Formula: see text], [Formula: see text]-[Formula: see text] and [Formula: see text]-[Formula: see text] symmetric systems. Additionally, we conduct a comparative study of entanglement dynamics in the [Formula: see text]-[Formula: see text] symmetrical system, [Formula: see text]-[Formula: see text] symmetrical system, and [Formula: see text]-[Formula: see text] symmetrical systems in order to learn more about non-Hermitian quantum systems and their environments. Entangled qubits evolve in a [Formula: see text]-[Formula: see text] symmetric unbroken regime, the entanglement oscillates with two different oscillation frequencies, and the entanglement is well preserved for a long period of time for the case when non-Hermitian parts of both qubits are taken quite away from the exceptional points.

Scientific Reports•May 26, 2023•6 citations
Non-Hermitian systems with parity-time (PT\mathcal{PT}PT) symmetry and anti-PT\mathcal{PT}PT symmetry give rise to exceptional points (EPs) with important properties due to the coalescence of eigenvectors. Because of the limitations on the Hermiticity of the Hamiltonian, these kinds of non-Hermitian systems have recently attracted extensive attention and emerged as a new way to engineer the response of open quantum systems. We propose an efficient and easily computable tool, based on the Hilbert-Schmidt speed (HSS), which does not require the diagonalization of the evolved density matrix, to characterize non-Markovian effects in (anti-)PT ⁣−\mathcal{PT}\!-PT−symmetric systems. Our proposed witness of non-Markovianity can also detect exactly the EPs and hence the critical behavior of the non-Hermitian systems, especially high-dimensional qudits. Our theoretical predictions are completely consistent with results extracted from recent experiments studying the non-Markovianity and criticality in (anti-)PT ⁣−\mathcal{PT}\!-PT−symmetric systems.
Non-Hermitian systems with parity-time (PT\mathcal{PT}PT) symmetry give rise to exceptional points (EPs) with exceptional properties that arise due to the coalescence of eigenvectors. Such systems have been extensively explored in the classical domain, where second- or higher-order EPs have been proposed or realized. In contrast, quantum information studies of PT\mathcal{PT}PT-symmetric systems have been confined to systems with a two-dimensional Hilbert space. Here, by using a single-photon interferometry setup, we simulate the quantum dynamics of a four-dimensional PT\mathcal{PT}PT-symmetric system across a fourth-order exceptional point. By tracking the coherent, nonunitary evolution of the density matrix of the system in PT\mathcal{PT}PT-symmetry unbroken and broken regions, we observe the entropy dynamics for both the entire system, and the gain and loss subsystems. Our setup is scalable to the higher-dimensional PT\mathcal{PT}PT-symmetric systems, and our results point towards the rich dynamics and critical properties.
In this paper, we demonstrate that the trace speed (TS), a statistical measure based on the trace norm of the derivative of the density matrix, serves as a powerful and precise tool for locating exceptional points (EPs) in quantum systems with parity-time (PT) and anti-PT symmetry. This detection of EPs is performed by benchmarking TS against the Hilbert-Schmidt speed (HSS), another efficient statistical measure. Analytical derivations and numerical simulations reveal that the dynamics of TS undergo characteristic changes when traversing EPs, enabling their exact positions to be determined unambiguously. Furthermore, TS can be more readily accessible experimentally than HSS, since it relates directly to trace-norm changes that can be estimated from feasible measurement schemes. Owing to its high sensitivity to small parameter variations and straightforward laboratory implementation, TS is a promising figure of merit for ultra-sensitive quantum sensing and for the optimization of EP-based photonic architectures. These results provide a foundation for exploring more complex non-Hermitian systems and their technological applications.
Non-Markovian effects can speed up the dynamics of quantum systems while the limits of the evolution time can be derived by quantifiers of quantum statistical speed. We introduce a witness for characterizing the non-Markovianity of quantum evolutions through the Hilbert-Schmidt speed (HSS), which is a special type of quantum statistical speed. This witness has the advantage of not requiring diagonalization of the evolved density matrix. Its sensitivity is investigated by considering several paradigmatic instances of open quantum systems, such as one qubit subject to phase-covariant noise and a Pauli channel, two independent qubits locally interacting with leaky cavities, and V-type and $\mathrm{\ensuremath{\Lambda}}$-type three-level atoms (qutrits) in a dissipative cavity. We show that the proposed HSS-based non-Markovianity witness detects memory effects, in agreement with the well-established trace-distance-based witness, being sensitive to system-environment information backflows.
Hilbert-Schmidt speed (HSS)-based witness, a quantifier of quantum statistical speed having the advantage of avoiding the diagonalization of the evolved density matrix, has been introduced as a trustful witness of non-Markovianity in low-dimensional systems. We investigate the sensitivity of this HSS-based witness to detect non-Markovianity in high-dimensional and multipartite open quantum systems. We find that the time behaviors of the HSS-based witness are always in agreement with those of quantum negativity or quantum correlation measures. These results show that the HSS-based witness is a faithful identifier of the memory effects appearing in the quantum evolution of a high-dimensional system.
A witness of non-Markovianity based on the Hilbert-Schmidt speed (HSS), a special type of quantum statistical speed, has been recently introduced for low-dimensional quantum systems. Such a non-Markovianity witness is particularly useful, being easily computable since no diagonalization of the system density matrix is required. We investigate the sensitivity of this HSS-based witness to detect non-Markovianity in various high-dimensional and multipartite open quantum systems with finite Hilbert spaces. We find that the time behaviors of the HSS-based witness are always in agreement with those of quantum negativity or quantum correlation measure. These results show that the HSS-based witness is a faithful identifier of the memory effects appearing in the quantum evolution of a high-dimensional system with a finite Hilbert space.
Theoretical investigation of dynamics and concurrence of entangled PT{{\mathcal {P}}}{{\mathcal {T}}}PT and anti-PT{{\mathcal {P}}}{{\mathcal {T}}}PT symmetric polarized photons
Non-Hermitian systems with parity-time [Formula: see text] symmetry and anti-parity-time [Formula: see text] symmetry have exceptional points (EPs) resulting from eigenvector co-coalescence with exceptional properties. In the quantum and classical domains, higher-order EPs for [Formula: see text] symmetry and [Formula: see text]-symmetry systems have been proposed and realized. Both two-qubits [Formula: see text]-[Formula: see text] and [Formula: see text]-[Formula: see text] symmetric systems have seen an increase in recent years, especially in the dynamics of quantum entanglement. However, to our knowledge, neither theoretical nor experimental investigations have been conducted for the dynamics of two-qubits entanglement in the [Formula: see text]-[Formula: see text] symmetric system. We investigate the [Formula: see text]-[Formula: see text] dynamics for the first time. Moreover, we examine the impact of different initial Bell-state conditions on entanglement dynamics in [Formula: see text]-[Formula: see text], [Formula: see text]-[Formula: see text] and [Formula: see text]-[Formula: see text] symmetric systems. Additionally, we conduct a comparative study of entanglement dynamics in the [Formula: see text]-[Formula: see text] symmetrical system, [Formula: see text]-[Formula: see text] symmetrical system, and [Formula: see text]-[Formula: see text] symmetrical systems in order to learn more about non-Hermitian quantum systems and their environments. Entangled qubits evolve in a [Formula: see text]-[Formula: see text] symmetric unbroken regime, the entanglement oscillates with two different oscillation frequencies, and the entanglement is well preserved for a long period of time for the case when non-Hermitian parts of both qubits are taken quite away from the exceptional points.