Faster than Hermitian Quantum Mechanics

Abstract

Given an initial quantum state |psi_I> and a final quantum state |psi_F> in a Hilbert space, 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.

4 pages

Cited by in corpus (12)

• Making Sense of Non-Hermitian Hamiltonians
• The Naimark dilated PT-symmetric brachistochrone
• Interface between Hermitian and non-Hermitian Hamiltonians in a model calculation
• Interactions of Hermitian and non-Hermitian Hamiltonians
• The PT-symmetric brachistochrone problem, Lorentz boosts and non-unitary operator equivalence classes
• The quantum brachistochrone problem for non-Hermitian Hamiltonians
• Non-Hermitian Quantum Systems and Time-Optimal Quantum Evolution
• Lower bound of minimal time evolution in quantum mechanics
• Is PT-symmetric quantum mechanics just quantum mechanics in a non-orthogonal basis?
• Physical Meaning of Hermiticity and Shortcomings of the Composite (Hermitian + non-Hermitian) Quantum Theory of Gunther and Samsonov
• Faster than Hermitian Time Evolution
• Energy Optimal Interpolation in Quantum Evolution