The PT-symmetric brachistochrone problem, Lorentz boosts and non-unitary operator equivalence classes

Abstract

The PT-symmetric (PTS) quantum brachistochrone problem is reanalyzed as quantum system consisting of a non-Hermitian PTS component and a purely Hermitian component simultaneously. Interpreting this specific setup as subsystem of a larger Hermitian system, we find non-unitary operator equivalence classes (conjugacy classes) as natural ingredient which contain at least one Dirac-Hermitian representative. With the help of a geometric analysis the compatibility of the vanishing passage time solution of a PTS brachistochrone with the Anandan-Aharonov lower bound for passage times of Hermitian brachistochrones is demonstrated.

12 pages, 2 figures, strongly extended version

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• Simulating broken $\mathcal{P}\mathcal{T}$-symmetric Hamiltonian systems by weak measurement
• Interpretation of quantum mechanics with indefinite norm
• A third-order exceptional point effect on the dynamics of a single particle in a time-dependent harmonic trap
• Fundamental length in quantum theories with PT-symmetric Hamiltonians II: The case of quantum graphs
• Embedding, simulation and consistency of $\mathcal{P}\mathcal{T}$ -symmetric quantum Theory
• Scattering cross section resonance originating from a spectral singularity
• PT symmetry, Cartan decompositions, Lie triple systems and Krein space related Clifford algebras
• Quantum state discrimination in a PT-symmetric system
• On the construction of pseudo-hermitian quantum system with a pre-determined metric in the Hilbert space
• Observing a Changing Hilbert-Space Inner Product
• Quantum deleting and cloning in a pseudo-unitary system
• Non-Hermitian Quantum Systems and Time-Optimal Quantum Evolution
• Equivalent Hermitian operator from supersymmetric quantum mechanics
• Resolutions of Identity for Some Non-Hermitian Hamiltonians. I. Exceptional Point in Continuous Spectrum
• Entanglement Efficiencies in PT-Symmetric Quantum Mechanics
• Recognizing critical lines via entanglement in non-Hermitian systems
• The stabilizer group of honeycomb lattices and its application to deformed monolayers
• Faster than Hermitian Time Evolution
• Physical Meaning of Hermiticity and Shortcomings of the Composite (Hermitian + non-Hermitian) Quantum Theory of Gunther and Samsonov
• Quantum speed limit time of a qubit system with non-Hermitian detuning