Faster than Hermitian quantum mechanics. (English) Zbl 1228.81027

Summary: Given an initial quantum state \(|\psi_I\rangle\) and a final quantum state \(|\psi_F\rangle\), there exist Hamiltonians \(H\) under which \(|\psi_I\rangle\) evolves into \(|\psi_F\rangle\). 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 \(\mathcal P\mathcal T\)-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 \(|\psi_I\rangle\) to \(|\psi_F\rangle\) 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 the points are connected by a wormhole. This result may have applications in quantum computing.


81P05 General and philosophical questions in quantum theory
81P68 Quantum computation
Full Text: DOI arXiv


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