×

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.

MSC:

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

References:

[1] DOI: 10.1103/PhysRevLett.96.060503
[2] DOI: 10.1088/0305-4470/39/11/L02 · Zbl 1085.81022
[3] DOI: 10.1088/0305-4470/36/20/314 · Zbl 1042.81011
[4] DOI: 10.1103/PhysRevLett.65.1697 · Zbl 0960.81524
[5] DOI: 10.1103/PhysRevLett.80.5243 · Zbl 0947.81018
[6] DOI: 10.1063/1.532860 · Zbl 1057.81512
[7] DOI: 10.1088/0305-4470/34/28/102 · Zbl 0982.81022
[8] DOI: 10.1088/0305-4470/34/28/305 · Zbl 0982.81021
[9] DOI: 10.1103/PhysRevLett.89.270401 · Zbl 1267.81234
[10] DOI: 10.1119/1.1574043 · Zbl 1219.81125
[11] DOI: 10.1063/1.1489072 · Zbl 1061.81075
[12] DOI: 10.1103/PhysRevLett.93.251601
[13] DOI: 10.1063/1.1418246 · Zbl 1059.81070
[14] DOI: 10.1088/0305-4470/36/25/312 · Zbl 1048.81041
[15] DOI: 10.1063/1.1703673 · Zbl 0096.43402
[16] DOI: 10.1103/PhysRevLett.61.1446
[17] DOI: 10.1119/1.15620 · Zbl 0957.83529
[18] DOI: 10.1126/science.1104149
[19] DOI: 10.1126/science.1121541 · Zbl 1226.81049
[20] DOI: 10.1063/1.2203236 · Zbl 1112.81098
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.