Random Quantum Maps and a quantum analogue of Frobenius-Perron theorem

Speaker(s)

Karol Życzkowski

Affiliation

Uniwersytet Jagielloński / Polska Akademia Nauk

Date

April 23, 2010, 10:15 a.m.

Room

room 5840

Seminar

Seminar of Dynamical Systems Group

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Statistical properties of periodically driven quantum chaotic systems described in a finite dimensional Hilbert space can be mimicked by random unitary matrices distributed according to the Haar measure on the unitary group.

To describe the effect of a possible interaction of the system in question with an environment one needs to work with density operators, which are Hermitian, positive and normalized. Discrete time evolution of a density matrix can be represented by quantum operation (completely positive, trace preserving map).

We introduce ensemble of random operations and investigate spectral properties of the corresponding superoperators with spectrum consisting of N² eigenvalues inside the unit disk. A quantum analogue of the Frobenius-Perron theorem concerning the spectrum of stochastic matrices is formulated. Obtained predictions for random operations are compared with spectral properties of quantized chaotic systems, interacting with an environment.

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