Faster than Hermitian Quantum Mechanics

Carl M. Bender, Dorje C. Brody, Hugh F. Jones, and Bernhard K. Meister
Phys. Rev. Lett. 98, 040403 – Published 24 January 2007

Abstract

Given an initial quantum state |ψI and a final quantum state |ψF, there exist Hamiltonians H under which |ψI evolves into |ψF. 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 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. This is because for such Hamiltonians the path from |ψI to |ψF 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.

  • Received 5 September 2006

DOI:https://doi.org/10.1103/PhysRevLett.98.040403

©2007 American Physical Society

Authors & Affiliations

Carl M. Bender1,2, Dorje C. Brody2, Hugh F. Jones3, and Bernhard K. Meister4

  • 1Physics Department, Washington University, St. Louis, Missouri 63130, USA
  • 2Department of Mathematics, Imperial College, London SW7 2BZ, United Kingdom
  • 3Blackett Laboratory, Imperial College, London SW7 2BZ, United Kingdom
  • 4Department of Physics, Renmin University of China, Beijing 100872, China

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Issue

Vol. 98, Iss. 4 — 26 January 2007

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