On the relationship between a quantum Markov semigroup and its representation via linear stochastic Schrödinger equations (original) (raw)
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We study the structure of generic quantum Markov semigroups, arising from the stochastic limit of a discrete system with generic Hamiltonian interacting with a Gaussian gauge invariant reservoir. We show that they can be essentially written as the sum of their irreducible components determined by closed classes of states of the associated classical Markov jump process. Each irreducible component turns out to be recurrent, transient or have an invariant state if and only if its classical (diagonal) restriction is recurrent, transient or has an invariant state, respectively. We classify invariant states and study convergence towards invariant states as time goes to infinity.
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We study a class of generic quantum Markov semigroups on the algebra of all bounded operators on a Hilbert space h arising from the stochastic limit of a discrete system with generic Hamiltonian H S , acting on h, interacting with a Gaussian, gauge invariant, reservoir. The self-adjoint operator H S determines a privileged orthonormal basis of h. These semigroups leave invariant diagonal and off-diagonal bounded operators with respect to this basis. The action on diagonal operators describes a classical Markov jump process. We construct generic semigroups from their formal generators by the minimal semigroup method and discuss their conservativity (uniqueness). When the semigroup is irreducible we prove uniqueness of the equilibrium state and show that, starting from an arbitrary initial state, the semigroup converges towards this state. We also prove that the exponential speed of convergence of the quantum Markov semigroup coincides with the exponential speed of convergence of the classical (diagonal) semigroup towards its unique invariant measure. The exponential speed is computed or estimated in some examples.
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Quantum Markov Semigroups and Quantum Flows
It is a great pleasure to thank also R. Soto and the Editorial Board of "Proyecciones" for publication of this unusually lengthy article. 3 k=0 i −k v + i k u, (T x α)(v + i k u) it follows then that (1.2) holds for each v, u ∈ h. 4 implies 3. In fact 4 implies that (1.2) holds for each v, u in the dense subset of h linearly spanned by the total set. This linear span is obviously dense. 3 implies 2. Let (v n) n≥0 , (u n) n≥0 be two sequences of vectors in h such that the sequences (v n) n≥0 , (u n) n≥0 are square-summable. We must show that lim α n≥0 v n , (T x α)u n = n≥0 v n , (T x)u n. To this end, for every ε > 0, take an integer ν such that n>ν u n 2 < ε, n>ν v n 2 < ε. Remark. Notice that, under the assumptions of Theorem A.1, P (t)u = T (t)u for every u ∈ L 2 (IR d ; I C) ∩ C 0 0 (IR d ; IR) and t ≥ 0. In fact P (t) and T (t) are bounded operators coinciding on the dense (in L 2 (IR d ; I C) and C 0 0 (IR d ; IR)) subset C ∞ c (IR d ; IR). We refer to [43] for the proofs of the above results.
Quantum Stochastic Dynamical Semigroup
Dynamics of Dissipation, 2002
We review the quantization of dynamics of stochastic models whose equilibrium states are the classical Gibbs ones. To proceed with the study of correctness of the quantization we indicate how the obtained dynamicals maps are related to quantum correlations and quantum entanglement.
The Feller Property of a Class of Quantum Markov Semigroups II
Quantum Probability and Infinite-Dimensional Analysis - Proceedings of the Conference, 2003
Let h be a Hilbert space and let B(h) be the von Neumann algebra of all bounded operators on h. We characterise w *-continuous Quantum Markov Semigroups (T t) t≥0 enjoying the Feller property with respect to the C *-algebra K(h) of compact operators i.e. such that K(h) is T t-invariant and (T t|K(h)) t≥0 is a strongly continuous semigroup on K(h). When (T t) t≥0 is the minimal Quantum Markov Semigroup associated with quadratic forms L −(x) (x ∈ B(h)) given by L −(x)[v, u] = ⟨Gv, xu⟩ + ∑ ℓ ⟨L ℓ v, xL ℓ u⟩ + ⟨v, xGu⟩ with possibly unbounded operators G, L ℓ we show that the Feller property with respect to K(h) holds under a summability condition on the L * ℓ. We also show that the quantum Ornstein-Uhlenbeck semigroup enjoys the Feller property with respect to a bigger C *-algebra including K(h) and functions of position and momentum operators.