Explicit definition of $\mathcal{P}\mathcal{T}$ symmetry for non-unitary quantum walks with gain and loss

Abstract

$\mathcal{P}\mathcal{T}$ symmetry, that is, a combined parity and time-reversal symmetry is a key milestone for non-Hermite systems exhibiting entirely real eigenenergy. In the present work, motivated by a recent experiment, we study $\mathcal{P}\mathcal{T}$ symmetry of the time-evolution operator of non-unitary quantum walks. We present the explicit definition of $\mathcal{P}\mathcal{T}$ symmetry by employing a concept of symmetry time frames. We provide a necessary and sufficient condition so that the time-evolution operator of the non-unitary quantum walk retains $\mathcal{P}\mathcal{T}$ symmetry even when parameters of the model depend on position. It is also shown that there exist extra symmetries embedded in the time-evolution operator. Applying these results, we clarify that the non-unitary quantum walk in the experiment does have $\mathcal{P}\mathcal{T}$ symmetry.

arXiv comment14 pages, 8 figures

References in corpus (44)

• Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry
• Making Sense of Non-Hermitian Hamiltonians
• Nonreciprocal light transmission in parity-time-symmetric whispering-gallery microcavities
• Unidirectional Invisibility induced by PT-Symmetric Periodic Structures
• Complex Extension of Quantum Mechanics
• Pseudo-Hermiticity versus PT Symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian
• Quantum random walks - an introductory overview
• PT-Symmetric Quantum Mechanics
• Loss-induced suppression and revival of lasing
• Pseudo-Hermiticity versus PT-Symmetry II: A complete characterizatio n of non-Hermitian Hamiltonians with a real spectrum
• Pseudo-Hermiticity versus PT-Symmetry III: Equivalence of pseudo-Her miticity and the presence of antilinear symmetries
• Quantum Walk in Position Space with Single Optically Trapped Atoms
• Vortex Pinning and Non-Hermitian Quantum Mechanics
• Observation of non-Hermitian degeneracies in a chaotic exciton-polariton billiard
• Observation of topologically protected bound states in a one dimensional photonic system
• Photons Walking the Line: A quantum walk with adjustable coin operations
• Exploring Topological Phases With Quantum Walks
• Selective enhancement of topologically induced interface states in a dielectric resonator chain
• Realization of a quantum walk with one and two trapped ions
• Biorthogonal Quantum Mechanics
• Edge states and topological phases in non-Hermitian systems
• Anderson localization of entangled photons in an integrated quantum walk
• Quantum walk of a trapped ion in phase space
• A 2D Quantum Walk Simulation of Two-Particle Dynamics
• Discrete single-photon quantum walks with tunable decoherence
• Decoherence and disorder in quantum walks: From ballistic spread to localization
• Topologically protected midgap states in complex photonic lattices
• Non-Hermitian Delocalization and Eigenfunctions
• Exponentially Fragile PT-Symmetry in Lattices with Localized Eigenmodes
• Absence of topological insulator phases in non-Hermitian PT-symmetric Hamiltonians
• Bulk--Boundary Correspondence for Chiral Symmetric Quantum Walks
• Observation of Defect States in PT-Symmetric Optical Lattices
• Spectral singularities and Bragg scattering in complex crystals
• Symmetries, Topological Phases and Bound States in the One-Dimensional Quantum Walk
• Photon propagation in a discrete fiber network: An interplay of coherence and losses
• Invisibility and PT-symmetry
• Topological phases and delocalization of quantum walks in random environments
• Stimulation of the fluctuation superconductivity by the PT-symmetry
• Unveiling hidden topological phases of a one-dimensional Hadamard quantum walk
• Pseudo-Unitary Operators and Pseudo-Unitary Quantum Dynamics
• Bound states, scattering states and resonant states in PT-symmetric open quantum systems
• Systems of coupled PT-symmetric oscillators
• Trapping photons on the line: controllable dynamics of a quantum walk
• PT Meets Supersymmetry and Nonlinearity: An Analytically Tractable Case Example

Cited by in corpus (55)

• Exceptional Topology of Non-Hermitian Systems
• Non-Bloch Band Theory of Non-Hermitian Systems
• Topological Phase Transition driven by Infinitesimal Instability: Majorana Fermions in Non-Hermitian Spintronics
• Non-Hermitian Floquet topological phases with arbitrarily many real-quasienergy edge states
• Generalized bulk-edge correspondence for non-hermitian topological systems
• Review: Quantum Metrology and Sensing with Many-Body Systems
• Non-Bloch $\mathcal{P}\mathcal{T}$ symmetry breaking in non-Hermitian Photonic Quantum Walks
• Non-Hermitian Floquet topological phases in the double-kicked rotor
• Higher winding number in a non-unitary photonic quantum walk
• 2D Solitons in PT-symmetric photonic lattices
• Observation of slowly decaying eigenmodes without exceptional points in Floquet, dissipative, synthetic circuits
• Topological phases protected by shifted sublattice symmetry in dissipative quantum systems
• Quantum dynamics on a lossy non-Hermitian lattice
• Persistence of Topological Phases in Non-Hermitian Quantum Walks
• Bulk-edge correspondence in nonunitary Floquet systems with chiral symmetry
• Anomalous-order exceptional point and non-Markovian Purcell effect at threshold in one-dimensional continuum systems
• Generalized Bloch band theory for non-Hermitian bulk-boundary correspondence
• Reservoir-assisted symmetry breaking and coalesced zero-energy modes in an open PT-symmetric Su-Schrieffer-Heeger model
• Topological quantum walks: theory and experiments
• Critical non-Hermitian topology induced quantum sensing
• Quantum centrality testing on directed graphs via PT-symmetric quantum walks
• Statistical properties of eigenvalues of the non-Hermitian Su-Schrieffer-Heeger model with random hopping terms
• Emergent conservation in Floquet dynamics of integrable non-Hermitian models
• Bloch-like super-oscillations and unidirectional motion of phase driven quantum walkers
• The Witten Index for 1D Supersymmetric Quantum Walks with Anisotropic Coins
• Bulk-edge correspondence and stability of multiple edge states of a $\mathcal{P}\mathcal{T}$ symmetric non-Hermitian system by using non-unitary quantum walks
• Delocalization of non-Hermitian Quantum Walk on Random Media in One Dimension
• Stability of topologically protected edge states in nonlinear quantum walks: Additional bifurcations unique to Floquet systems
• Entanglement transitions in a periodically driven non-Hermitian Ising chain
• Proposal of a quantum version of active particles via a nonunitary quantum walk
• Quantum walk on the Bloch sphere
• Dynamic conditioning of two particle discrete-time quantum walks
• Hilbert space fragmentation imposed real spectrum of non-Hermitian systems
• Finite-dimensional pseudo-bosons: a non-Hermitian version of the truncated harmonic oscillator
• Bulk-Edge Correspondence for Point-Gap Topological Phases in Junction Systems
• The Quantum Skin Hall Effect
• Impurity-induced counter skin-effect and linear modes in non-Hermitian systems
• Effects of disorder on non-unitary $\mathcal{P}\mathcal{T}$ symmetric quantum walks
• Non-Hermitian quantum walks and non-Markovianity: the coin-position interaction
• Dual Symmetry Classification of Non-Hermitian Systems and ${\mathbb{Z}}_2$ Point-Gap Topology of a Non-Unitary Quantum Walk
• Finite-element assembly approach of optical quantum walk networks
• Distinguishability Transitions in Non-Unitary Boson Sampling Dynamics
• Entanglement entropy distinguishes PT-symmetry and topological phases in a class of non-unitary quantum walks
• Band structures under non-Hermitian periodic potentials: Connecting nearly-free and bi-orthogonal tight-binding models
• Bulk--Boundary Correspondence and Boundary Zero Modes in a Non-Hermitian Kitaev Chain Model
• Absorption to Fluctuating Bunching States in Non-Unitary Boson Dynamics
• Exceptional mode topological surface laser
• Quantum walk for SU(1,1)
• Investigating the quench dynamics of the bound states in a spin-orbital coupling system using a trapped ion
• Observation of novel robust edge states without bulk-boundary correspondence in non-Hermitian quantum walks
• Topological Phase Transitions and Edge-State Transfer in Time-Multiplexed Quantum Walks
• Index Theorems for One-dimensional Chirally Symmetric Quantum Walks with Asymptotically Periodic Parameters
• Reduced dynamics in quasi-Hermitian systems
• Shannon Entropy and Diffusion Coeffcient in Parity-Time Symmetric Quantum Walks
• The Witten Index for One-dimensional Non-unitary Quantum Walks with Gapless Time-evolution