arXiv:0804.3487

Comment on the Quantum Brachistochrone Problem

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

Published Apr 22, 2008• 6 citations

Abstract

In this brief comment we attempt to clarify the apparent discrepancy between the papers [1] and [2] on the quantum brachistochrone, namely whether it is possible to use a judicious mixture of Hermitian and non-Hermitian quantum mechanics to evade the standard lower limit on the time taken for evolution by a Hermitian Hamiltonian with given energy dispersion between two given states.

DOI: 10.48550/arxiv.0804.3487

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arXiv:0804.3487

Comment on the Quantum Brachistochrone Problem

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

Published Apr 22, 2008• 6 citations

arXiv HTML PDF

Abstract

DOI: 10.48550/arxiv.0804.3487

Related papers

Comments on "Observation of Fast Evolution in Parity-Time-Symmetric System"

Fabio Masillo

arXiv:1106.1550Jun 8, 20111 citation

The quantum brachistochrone problem for non-Hermitian Hamiltonians

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Some Remarks on Quantum Brachistochrone

Fabio Masillo

arXiv:1105.3332Open Systems & Information DynamicsMar 1, 20121 citation

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.