Faster than Hermitian quantum mechanics

• Published in: Physical review letters Vol. 98; no. 4; p. 040403
• Main Authors: Bender, Carl M, Brody, Dorje C, Jones, Hugh F, Meister, Bernhard K
• Format: Journal Article
• Language: English
• Published: United States 26.01.2007
• ISSN: 0031-9007
• DOI: 10.1103/PhysRevLett.98.040403

Given an initial quantum state |psi(I)> and a final quantum state |psi(F)>, there exist Hamiltonians H under which |psi(I)> evolves into |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 tau? For Hermitian Hamiltonians tau has a nonzero lower bound. However, among non-Hermitian PT-symmetric Hamiltonians satisfying the same energy constraint, tau can be made arbitrarily small without violating the time-energy uncertainty principle. This is because for such Hamiltonians the path from |psi(I)> to |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.