Faster than Hermitian time evolution
type: Item
instance of: scholarly article
title: Faster than Hermitian time evolution
zbMATH Open document ID: 1136.81022
DOI: 10.3842/SIGMA.2007.126
author: Carl M. Bender
published in: SIGMA. Symmetry, Integrability and Geometry: Methods and Applications
publication date: 2008-02-27T00:00:00Z
full work available at URL: https://arxiv.org/abs/0712.3910
full work available at URL: https://eudml.org/doc/54077
full work available at URL: http://www.emis.de/journals/SIGMA/2007/126/
review text: For any pair of quantum states, an initial state \(|I\rangle\) and a final quantum state \(|F\rangle\), in a Hilbert space, there are many Hamiltonians \(H\) under which \(|I\rangle\) evolves into \(|F\rangle\). 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 \(\tau\). For Hermitian Hamiltonians, \(\tau\) has a nonzero lower bound. However, among non-Hermitian \({\mathcal {PT}}\)-symmetric Hamiltonians satisfying the same energy constraint, \(\tau\) can be made arbitrarily small without violating the time-energy uncertainty principle. The minimum value of \(\tau\) can be made arbitrarily small because for \({\mathcal {PT}}\)-symmetric Hamiltonians the path from the vector \(|I\rangle\) to the vector |\(|F\rangle\), 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.
Mathematics Subject Classification ID: 81Q10
Mathematics Subject Classification ID: 81S99
zbMATH DE Number: 5241639
zbMATH Keywords: brachistochrone
zbMATH Keywords: PT quantum mechanics
zbMATH Keywords: parity
zbMATH Keywords: time reversal
zbMATH Keywords: time evolution
zbMATH Keywords: unitarity
MaRDI profile type: Publication
arXiv ID: 0712.3910
description: scientific article