Optimal Time Evolution for Hermitian and Non-Hermitian Hamiltonians

Abstract

Consider the set of all Hamiltonians whose largest and smallest energy eigenvalues, E_max and E_min, differ by a fixed energy ω. Given two quantum states, an initial state |ψ_I> and a final state |ψ_F>, there exist many Hamiltonians H belonging to this set under which |ψ_I> evolves in time into |ψ_F>. Which Hamiltonian transforms the initial state to the final state in the least possible time τ? For Hermitian Hamiltonians, $\tau$ has a nonzero lower bound. However, among complex non-Hermitian PT-symmetric Hamiltonians satisfying the same energy constraint, τcan be made arbitrarily small without violating the time-energy uncertainty principle. The minimum value of τcan be made arbitrarily small because for PT-symmetric Hamiltonians the evolution path from the vector |ψ_I> to the vector |ψ_F>, as measured using the Hilbert-space metric appropriate for this theory, can be made arbitrarily short. The mechanism described here resembles the effect in general relativity in which two space-time points can be made arbitrarily close if they are connected by a wormhole. This result may have applications in quantum computing.

20 pages. To appear in: Time in Quantum Mechanics II, edited by J.G. Muga

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Cited by in corpus (9)

• Non-Hermitian Physics
• Shortcuts to adiabaticity for non-Hermitian systems
• Bypassing the bandwidth theorem with PT symmetry
• Time-dependent Hamiltonians with 100% evolution speed efficiency
• Chaotic diffusion of complex trajectory and its quantum signature
• Geometry of a two-spin quantum state in evolution
• Geometric properties of evolutionary graph states and their detection on a quantum computer
• Quantum speed limits for time evolution of a system subspace
• Optimal bounds on the speed of subspace evolution