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arXiv:quant-ph/0609032 · Physical Review Letters

Faster than Hermitian Quantum Mechanics

Carl M. Bender, Dorje C. Brody, Hugh F Jones, Bernhard K. Meister

Published Jan 24, 2007• 414 citations
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Abstract

Given an initial quantum state $|{\ensuremath{\psi}}_{I}⟩$ and a final quantum state $|{\ensuremath{\psi}}_{F}⟩$, there exist Hamiltonians $H$ under which $|{\ensuremath{\psi}}_{I}⟩$ evolves into $|{\ensuremath{\psi}}_{F}⟩$. Consider the following quantum brachistochrone problem: subject to the constraint that the difference between the largest and smallest eigenvalues of $H$ is held fixed, which $H$ achieves this transformation in the least time $\ensuremath{\tau}$? For Hermitian Hamiltonians $\ensuremath{\tau}$ has a nonzero lower bound. However, among non-Hermitian $\mathcal{P}\mathcal{T}$-symmetric Hamiltonians satisfying the same energy constraint, $\ensuremath{\tau}$ can be made arbitrarily small without violating the time-energy uncertainty principle. This is because for such Hamiltonians the path from $|{\ensuremath{\psi}}_{I}⟩$ to $|{\ensuremath{\psi}}_{F}⟩$ can be made short. The mechanism described here is similar to that in general relativity in which the distance between two space-time points can be made small if they are connected by a wormhole. This result may have applications in quantum computing.

DOI
10.1103/physrevlett.98.040403

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Given an initial quantum state ∣\ensuremathψI⟩|{\ensuremath{\psi}}_{I}⟩∣\ensuremathψI​⟩ and a final quantum state ∣\ensuremathψF⟩|{\ensuremath{\psi}}_{F}⟩∣\ensuremathψF​⟩, there exist Hamiltonians HHH under which ∣\ensuremathψI⟩|{\ensuremath{\psi}}_{I}⟩∣\ensuremathψI​⟩ evolves into ∣\ensuremathψF⟩|{\ensuremath{\psi}}_{F}⟩∣\ensuremathψF​⟩. Consider the following quantum brachistochrone problem: subject to the constraint that the difference between the largest and smallest eigenvalues of HHH is held fixed, which HHH achieves this transformation in the least time \ensuremathτ\ensuremath{\tau}\ensuremathτ? For Hermitian Hamiltonians \ensuremathτ\ensuremath{\tau}\ensuremathτ has a nonzero lower bound. However, among non-Hermitian PT\mathcal{P}\mathcal{T}PT-symmetric Hamiltonians satisfying the same energy constraint, \ensuremathτ\ensuremath{\tau}\ensuremathτ can be made arbitrarily small without violating the time-energy uncertainty principle. This is because for such Hamiltonians the path from ∣\ensuremathψI⟩|{\ensuremath{\psi}}_{I}⟩∣\ensuremathψI​⟩ to ∣\ensuremathψF⟩|{\ensuremath{\psi}}_{F}⟩∣\ensuremathψF​⟩ can be made short. The mechanism described here is similar to that in general relativity in which the distance between two space-time points can be made small if they are connected by a wormhole. This result may have applications in quantum computing.
For any pair of quantum states, an initial state |I and a final quantum state |F , in a Hilbert space, there are many Hamiltonians H under which |I evolves into |F .Let us impose the constraint that the difference between the largest and smallest eigenvalues of H, E max and E min , is held fixed.We can then determine the Hamiltonian H that satisfies this constraint and achieves the transformation from the initial state to the final state in the least possible time τ .For Hermitian Hamiltonians, τ has a nonzero lower bound.However, among non-Hermitian PT -symmetric Hamiltonians satisfying the same energy constraint, τ can be made arbitrarily small without violating the time-energy uncertainty principle.The minimum value of τ can be made arbitrarily small because for PT -symmetric Hamiltonians the path from the vector |I to the vector |F , as measured using the Hilbert-space metric appropriate for this theory, can be made arbitrarily short.The mechanism described here is similar to that in general relativity in which the distance between two space-time points can be made small if they are connected by a wormhole.This result may have applications in quantum computing.
Recently the quantum brachistochrone problem has been considered for non-Hermitian PT\cal PTPT-symmetric quantum system, and it has been shown that the optimal evolution time required to transform a given initial state ∣ψi>|\psi_i>∣ψi​> into specific final state ∣ψf>|\psi_f>∣ψf​> can be made arbitrary small. Here we study the similar problem for generic non-Hermitian Hamiltonian and find that the same answer can be obtained. We show that at the exceptional point the distance between ∣ψi>|\psi_i>∣ψi​> and ∣ψf>|\psi_f>∣ψf​> becomes virtually vanishing and quantum tunneling occurring at the exceptional point yields the shortest timed required to evolve from the initial state ∣ψi>|\psi_i>∣ψi​> into the final state ∣ψf>|\psi_f>∣ψf​>. These results may have applications in holonomic quantum computation and quantum information processing.
Recently, Bender et al. have considered the quantum brachistochrone problem for the non-Hermitian PT -symmetric quantum system and have shown that the optimal time evolution required to transform a given initial state |ψ i into a specific final state |ψ f can be made arbitrarily small.Additionally, it has been shown that finding the shortest possible time requires only the solution of the two-dimensional problem for the quantum system governed by the effective Hamiltonian acting in the subspace spanned by |ψ i and |ψ f .In this paper, we study a similar problem for the generic non-Hermitian Hamiltonian, focusing our attention on the geometric aspects of the problem.
Recently Bender, Brody, Jones and Meister found that in the quantum brachistochrone problem the passage time needed for the evolution of certain initial states into specified final states can be made arbitrarily small, when the time-evolution operator is taken to be non-Hermitian but PT-symmetric. Here we demonstrate that such phenomena can also be obtained for dissipative systems. We conclude that the effect of a tunable passage time can be attributed to the non-Hermitian nature of the time-evolution operator rather than to its PT-symmetry.