Pseudo-Hermiticity versus PT-Symmetry III: Equivalence of pseudo-Her miticity and the presence of antilinear symmetries

Abstract

We show that a diagonalizable (non-Hermitian) Hamiltonian H is pseudo-Hermitian if and only if it has an antilinear symmetry, i.e., a symmetry generated by an invertible antilinear operator. This implies that the eigenvalues of H are real or come in complex conjugate pairs if and only if H possesses such a symmetry. In particular, the reality of the spectrum of H implies the presence of an antilinear symmetry. We further show that the spectrum of H is real if and only if there is a positive-definite inner-product on the Hilbert space with respect to which H is Hermitian or alternatively there is a pseudo-canonical transformation of the Hilbert space that maps H into a Hermitian operator.

Slightly expanded version, accepted for publication in J. Math. Phys

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• Electron Self-energy in Pseudo-Hermitian Quantum Electrodynamics with a Maximal Mass M
• Spherical-separablility of non-Hermitian Hamiltonians and pseudo-PT-symmetry
• Metric Operator For The Non-Hermitian Hamiltonian Model and Pseudo-Supersymmetry
• Dissolution of the non-Hermitian skin effect in one-dimensional lattices with linearly varying nonreciprocal hopping
• Canonical Expansion of PT-Symmetric Operators and Perturbation Theory
• Pseudo-Hermitian systems with PT-symmetry: Degeneracy and Krein space
• Non-Hermitian tearing by dissipation
• Composition of Transfer Matrices for Potentials with Overlapping Support
• Three-Dimensional Shape Invariant Non-Separable Model With Equidistant Spectrum
• Local preservation of no-signaling in multiparty PT-symmetric evolutions
• Parallel dynamics between non-Hermitian and Hermitian systems
• Linked $\mathcal{P}\mathcal{T}$-symmetry to Supersymmetry in a class of non-Hermitian Hamiltonians
• $\mathcal{P}\mathcal{T}$-symmetric mapping of three states and its implementation on a cloud quantum processor
• Probability Interpretation for Klein-Gordon Fields and the Hilbert Space Problem in Quantum Cosmology
• Conditional observability versus self-duality in a schematic model
• Direct measurement of the quantum geometric tensor in pseudo-Hermitian systems
• Geometry of PT-symmetric quantum mechanics
• Band topology of pseudo-Hermitian phases through tensor Berry connections and quantum metric
• Two dimensional non-Hermitian harmonic oscillator: coherent states
• Reality from maximizing overlap in the periodic complex action theory
• Pseudo-Hermiticity and Electromagnetic Wave Propagation in Dispersive Media
• Extracting the internal nonlocality from the dilated Hermiticity
• $\mathcal{P}\mathcal{T}$-Symmetric Quantum Discrimination of Three States
• Eigenvalues collision for PT-symmetric waveguide
• Lorentz Quantum Computer
• Reality from maximizing overlap in the future-included real action theory
• Gaussian eigenstate pinning in non-Hermitian quantum mechanics
• Path-integral Bosonization of $d=2$ PT symmetric models
• Fixed lines in a non-Hermitian Kitaev chain with spatially balanced pairing processes
• Real spectra and phase transition of skin effect in nonreciprocal systems
• Some Remarks on Quantum Brachistochrone
• A Hermitian bypass to the non-Hermitian quantum theory
• Entanglement of pseudo-Hermitian random states
• Planarizable Supersymmetric Quantum Toboggans
• Automatic hermiticity for mixed states
• Momentum relation and classical limit in the future-not-included complex action theory
• $\mathcal{P}\mathcal{T}$-Symmetric Generalized Extended Momentum Operator
• Quantum backreaction effect in optical solitons
• Consistent treatment of quantum systems with a time-dependent Hilbert space
• Noise-Canceling Quantum Feedback: non-Hermitian Dynamics with Applications to State Preparation and Magic State Distillation
• Reciprocity Theorem and Fundamental Transfer Matrix
• Third-order exceptional surface in a pseudo-Hermitian superconducting circuit
• Complementarity versus coordinate transformations: mapping between pseudo-Hermiticity and weak pseudo-Hermiticity
• The Supersymmetric Origin of Chaos and its Hidden Topological Order
• Nonlocality of Observables in Quasi-Hermitian Quantum Theory
• Non-real Eigenvalues for PT-Symmetric Double Wells
• Neutrino Pair Cerenkov Radiation for Tachyonic Neutrinos
• Topological Classification of Gapped/Gap-preserving Rational Space-Time Crystal Systems
• On the construction of pseudo-Hermitian Hamiltonians by means of similarity transformations
• Instantons and transseries of the Mathieu potential deformed by a $\mathcal{P}\mathcal{T}$-symmetry parameter
• Vacuum Stability of the wrong sign $(-{\varphi }^6)$ Scalar Field Theory
• Stability band structure for periodic states in periodic potentials
• Quantization of pseudo-hermitian systems
• On the $\mathcal{P}\mathcal{T}$-symmetric parametric amplifier
• Oscillation probabilities for a PT-symmetric non-Hermitian two-state system
• $\mathcal{P}\mathcal{T}$-Symmetric Quantum State Discrimination for Attack on BB84 Quantum Key Distribution
• Balancing gain and loss in symmetrised multi-well potentials
• Topological Field Theory and Stochastic Dynamics
• A new Berry phase term in parity-time symmetric non-Hermitian spin-1/2 quantum systems
• $\mathcal{P}\mathcal{T}-$Symmetric Two-Level Open Quantum Systems: Information Theoretic Facets
• Reduced dynamics in quasi-Hermitian systems
• Anisotropic oscillator with pseudo-hermiticity and its noncommutative extension
• PT-symmetry and Integrability