Quasi-Hermitian quantum systems, including $\mathcal{PT}$ -symmetric ones, can be mapped to equivalent Hermitian systems via a similarity transformation that redefines the inner product with a positive-definite metric operator. Although an instantaneous algebraic Hermitianization can be obtained locally from a positive metric operator, a stronger requirement is needed for dynamical equivalence: the similarity transformation must be proper, globally single-valued, and compatible with the modified quasi-Hermitian Schrodinger equation. We identify two distinct obstructions: geometric obstructions arising from the curvature of a metric-induced connection, and topological obstructions originating from non-trivial holonomies around non-contractible loops in parameter space. We derive explicit criteria for these obstructions and illustrate them with concrete examples. Our results establish a geometric and topological foundation for the Hermitianization of quasi-Hermitian systems, clarifying when they can be globally reduced to Hermitian ones and when intrinsic non-Hermitian features persist.Quasi-Hermitian quantum systems, including $\mathcal{PT}$-symmetric ones, can be mapped to equivalent Hermitian systems via a similarity transformation that redefines the inner product with a positive-definite metric operator. Although an instantaneous algebraic Hermitianization can be obtained locally from a positive metric operator, a stronger requirement is needed for dynamical equivalence: the similarity transformation must be proper, globally single-valued, and compatible with the modified quasi-Hermitian Schrodinger equation. We identify two distinct obstructions: geometric obstructions arising from the curvature of a metric-induced connection, and topological obstructions originating from non-trivial holonomies around non-contractible loops in parameter space. We derive explicit criteria for these obstructions and illustrate them with concrete examples. Our results establish a geometric and topological foundation for the Hermitianization of quasi-Hermitian systems, clarifying when they can be globally reduced to Hermitian ones and when intrinsic non-Hermitian features persist.

quant-pharXiv:2605.05335May 8, 20260 citations

Steady states and non‐Hermitian dynamics

Peter Schmitteckert

Annalen der PhysikMay 3, 20122 citations

Quasi-Hermitian proxy for the conditional no-jump dynamics in a three-site model

Aritra Ghosh, Miloslav Znojil

Abstract In this work, we show that in the regime of real and non-degenerate spectrum, the quasi-Hermitian framework provides an effective description of the conditional no-jump evolution of certain Markovian open systems within invariant subspaces while the full Lindblad dynamics governs the unconditional dynamics which includes quantum jumps. We present a three-site fermionic model with asymmetric hopping amplitudes whose single-particle generator is intrinsically non-Hermitian yet supports an all-real spectrum in a finite region of the parameter space. We construct an explicit Lindbladian-based embedding in which the same Hamiltonian arises as the no-jump generator of a completely positive and trace-preserving Lindblad equation, up to a particle-decay term. Due to particle-number conservation, this shift reduces to a scalar within each fixed-particle-number sector. We further analyze the single-particle spectrum and identify a third-order exceptional point that lies on the boundary marking the breakdown of quasi-hermiticity in a given fixed-particle-number sector.Abstract In this work, we show that in the regime of real and non-degenerate spectrum, the quasi-Hermitian framework provides an effective description of the conditional no-jump evolution of certain Markovian open systems within invariant subspaces while the full Lindblad dynamics governs the unconditional dynamics which includes quantum jumps. We present a three-site fermionic model with asymmetric hopping amplitudes whose single-particle generator is intrinsically non-Hermitian yet supports an all-real spectrum in a finite region of the parameter space. We construct an explicit Lindbladian-based embedding in which the same Hamiltonian arises as the no-jump generator of a completely positive and trace-preserving Lindblad equation, up to a particle-decay term. Due to particle-number conservation, this shift reduces to a scalar within each fixed-particle-number sector. We further analyze the single-particle spectrum and identify a third-order exceptional point that lies on the boundary marking the breakdown of quasi-hermiticity in a given fixed-particle-number sector.

Europhysics Letters (EPL)Mar 1, 20260 citations

Non-Hermitian Dynamics of Open Quantum Systems

Matt Lane

Jan 1, 20240 citations

Emulating Non-Hermitian Dynamics in a Finite Non-Dissipative Quantum System

Eloi Flament, François Impens, David Guéry-Odelin

We discuss the emulation of non-Hermitian dynamics during a given time window using a low-dimensional quantum system coupled to a finite set of equidistant discrete states acting as an effective continuum. We first emulate the decay of an unstable state and map the quasi-continuum parameters, enabling the precise approximation of non-Hermitian dynamics. The limitations of this model, including in particular short- and long-time deviations, are extensively discussed. We then consider a driven two-level system and establish criteria for non-Hermitian dynamics emulation with a finite quasi-continuum. We quantitatively analyze the signatures of the finiteness of the effective continuum, addressing the possible emergence of non-Markovian behavior during the time interval considered. Finally, we investigate the emulation of dissipative dynamics using a finite quasi-continuum with a tailored density of states. We show through the example of a two-level system that such a continuum can reproduce non-Hermitian dynamics more efficiently than the usual equidistant quasi-continuum model.We discuss the emulation of non-Hermitian dynamics during a given time window using a low-dimensional quantum system coupled to a finite set of equidistant discrete states acting as an effective continuum. We first emulate the decay of an unstable state and map the quasi-continuum parameters, enabling the precise approximation of non-Hermitian dynamics. The limitations of this model, including in particular short- and long-time deviations, are extensively discussed. We then consider a driven two-level system and establish criteria for non-Hermitian dynamics emulation with a finite quasi-continuum. We quantitatively analyze the signatures of the finiteness of the effective continuum, addressing the possible emergence of non-Markovian behavior during the time interval considered. Finally, we investigate the emulation of dissipative dynamics using a finite quasi-continuum with a tailored density of states. We show through the example of a two-level system that such a continuum can reproduce non-Hermitian dynamics more efficiently than the usual equidistant quasi-continuum model.

arXiv:2502.02507EntropyAug 24, 20230 citations

Quasi-Hermitian quantum systems, including

\mathcal{PT}

-symmetric ones, can be mapped to equivalent Hermitian systems via a similarity transformation that redefines the inner product with a positive-definite metric operator. Although an instantaneous algebraic Hermitianization can be obtained locally from a positive metric operator, a stronger requirement is needed for dynamical equivalence: the similarity transformation must be proper, globally single-valued, and compatible with the modified quasi-Hermitian Schrodinger equation. We identify two distinct obstructions: geometric obstructions arising from the curvature of a metric-induced connection, and topological obstructions originating from non-trivial holonomies around non-contractible loops in parameter space. We derive explicit criteria for these obstructions and illustrate them with concrete examples. Our results establish a geometric and topological foundation for the Hermitianization of quasi-Hermitian systems, clarifying when they can be globally reduced to Hermitian ones and when intrinsic non-Hermitian features persist.