The density-matrix and Heisenberg formulations of quantum mechanics follow—for unitary evolution—directly from the Schrödinger equation. Nevertheless, the symmetries of the corresponding evolution operator, the Liouvillian L = i[⋅, H], need not be limited to those of the Hamiltonian H. This is due to L only involving eigenenergy differences, which can be degenerate even if the energies themselves are not. Remarkably, this possibility has rarely been mentioned in the literature, and never pursued more generally. We consider an example involving mesoscopic Josephson devices, but the analysis only assumes familiarity with basic quantum mechanics. Subsequently, such L-symmetries are shown to occur more widely, in particular also in classical mechanics. The symmetry's relevance to dissipative systems and quantum-information processing is briefly discussed.
Quantum Information Processing – Springer Journals
Published: Oct 13, 2004
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