Faster than Hermitian Time Evolution

作者：Bender Carl M^*

来源：Symmetry Integrability and Geometry-Methods and Applications, 2007, 3: 126.

DOI：10.3842/SIGMA.2007.126

• brachistochrone
• PT quantum mechanics
• parity
• time reversal
• time evolution
• unitarity

For any pair of quantum states, an initial state |I > and a final quantum state |F >, in a Hilbert space, there are many Hamiltonians H under which |I > evolves into |F >. 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 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 PT-symmetric Hamiltonians the path from the vector |I > to the vector |F >, 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.

• 出版日期2007