Spectral analysis and Feller property for quantum Ornstein–Uhlenbeck semigroups (original) (raw)

Quantum Ornstein–Uhlenbeck semigroups

Quantum Studies: Mathematics and Foundations, 2014

Based on nuclear infinite-dimensional algebra of entire functions with a certain exponential growth condition with two variables, we define a class of operators which gives in particular three semigroups acting on continuous linear operators, called the quantum Ornstein-Uhlenbeck (O-U) semigroup, the left quantum O-U semigroup and the right quantum O-U semigroup. Then, we prove that the solution of the Cauchy problem associated with the quantum number operator, the left quantum number operator and the right quantum number operator, respectively, can be expressed in terms of such semigroups. Moreover, probabilistic representations of these solutions are given. Eventually, using a new notion of positive white noise operators, we show that the aforementioned semigroups are Markovian.

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.

On the recurrence of Quantum Dynamical Semigroups

The mathematical description of the evolution of Quantum Open Systems seems to reach a suitable formalism within the Theory of Quantum Dynamical Semigroups extensively developed during the last two decades. Moreover, from a probabilistic point of view, this theory provides a natural non commutative extension of Markov Processes. Following that line, we discuss the notion of recurrence and summarize a number of results on the large time behavior of Quantum Dynamical Semigroups.

Quantum Poisson processes and dilations of dynamical semigroups

Probability Theory and Related Fields, 1989

The notion of a quantum Poisson process over a quantum measure space is introduced. This process is used to construct new quantum Markov processes on the matrix algebra M, with stationary faithful state qS. If (Jd, #) is the quantum measure space in question (Jt a yon Neumann algebra and # a faithful normal weight), then the semigroup e tL of transition operators on (M,, ~b) has generator L: M,~M,: a~i[h,a]+(id|174174 where u is an arbitrary unitary element of the centraliser of (M, | Jg, ~b | #).

The Decoherence-Free Subalgebra of a Quantum Markov Semigroup with Unbounded Generator

Infinite Dimensional Analysis, Quantum Probability and Related Topics, 2010

Let [Formula: see text] be a quantum Markov semigroup on [Formula: see text] with a faithful normal invariant state ρ. The decoherence-free subalgebra [Formula: see text] of [Formula: see text] is the biggest subalgebra of [Formula: see text] where the completely positive maps [Formula: see text] act as homomorphisms. When [Formula: see text] is the minimal semigroup whose generator is represented in a generalised GKSL form [Formula: see text], with possibly unbounded H, Lℓ, we show that [Formula: see text] coincides with the generalised commutator of [Formula: see text] under some natural regularity conditions. As a corollary we derive simple sufficient algebraic conditions for convergence towards a steady state based on multiple commutators of H and Lℓ. We give examples of quantum Markov semigroups [Formula: see text], with h infinite-dimensional, having a non-trivial decoherence-free subalgebra.

The Structures of State Space Concerning Quantum Dynamical Semigroups

Reviews in Mathematical Physics, 2012

Each semigroup describing time evolution of an open quantum system on a finite dimensional Hilbert space is related to a special structure of this space. It is shown how the space can be decomposed into orthogonal subspaces: One part is related to decay, some subspaces of the other subspace are ranges of the stationary states. Specialities are highlighted where the complete positivity of evolutions is actually needed for analysis, mainly for evolution of coherence. Decompositions are done the same way for discrete as for continuous time evolutions, but they may show differences: Only for discrete semigroups there may appear cases of sudden decay and of perpetual oscillation. Concluding the analysis, we identify the relation of the state space structure to the processes of decay, decoherence, dissipation and dephasing.

The role of the atomic decoherence-free subalgebra in the study of quantum Markov semigroups

Journal of Mathematical Physics, 2019

We show that for a Quantum Markov Semigroup (QMS) with a faithful normal invariant state, atomicity of the decoherence-free subalgebra and environmental decoherence are equivalent. Moreover, we prove that the predual of the decoherence-free subalgebra is isometrically isomorphic to the subspace of reversible states. We also describe, in an explicit and constructive way, the relationship between the decoherence-free subalgebra and the fixed point subalgebra.