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Decoherence of quantum Markov semigroups. (English) Zbl 1075.81040
This paper presents a probabilistic interpretation of decoherence based on the reduction of a quantum Markov semigroup $𝒯$ on an operator algebra $𝒜$ to a classical Markov semigroup on an abelian subalgebra of $𝒜$ which is $𝒯$-invariant. The author determines several conditions on the generator $ℒ$ of $𝒯$ for the existence of invariant abelian subalgebra. When $𝒜$ is the algebra of all bounded operator on a Hilbert space $h$, he investigates decoherence as a limit behaviour, namely vanishing of the off-diagonal terms $〈{e}_{m},{𝒯}_{*t}\left(\rho \right){e}_{n}〉$ (${𝒯}_{*}$ is the predual semigroup), in a given basis, of the evolution ${𝒯}_{*t}\left(\rho \right)$ at time $t$ of a density matrix $\rho$ applying results on the asymptotic behavior of $𝒯$ [J. Math. Phys. 42, No. 3, 1296–1308 (2001; Zbl 1013.81031), J. Math. Phys. 43, No. 2, 1074–1082 (2002; Zbl 1059.47049), Stochastic analysis and mathematical physics II. 4th international ANESTOC workshop in Santiago, Chile, January 5–11, 2000. Basel: Birkhäuser. Trends in Mathematics, 77–128 (2003; Zbl 1143.81309)]. Two natural possibilities for decoherence appear: 1) the semigroup has an invariant density matrix which is diagonal in the given basis, 2) ${𝒯}_{*t}\left(\rho \right)$ converges weakly to 0 as $t$ goes to infinity for any density matrix $\rho$. The exposition is clear, quite detailed and accompanied by several examples including the damped harmonic oscillator, quantum Brownian motion, the quantum exclusion semigroup.
MSC:
 81S25 Quantum stochastic calculus 82C10 Quantum dynamics and nonequilibrium statistical mechanics (general) 46N50 Applications of functional analysis in quantum physics 81P99 Axiomatics, foundations, philosophy of quantum theory 81R15 Operator algebra methods (quantum theory) 81P68 Quantum computation 60J65 Brownian motion 47D07 Markov semigroups of linear operators and applications to diffusion processes