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arXiv · 0808.1823

Optimal Time Evolution for Hermitian and Non-Hermitian Hamiltonians

Carl M. Bender·Dorje C. Brody
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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, $τ$ 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.

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Carl M. Bender, Dorje C. Brody. 2008-08-13. Optimal Time Evolution for Hermitian and Non-Hermitian Hamiltonians. https://doi.org/10.1007/978-3-642-03174-8_12

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