Dilations of quantum dynamical semigroups with classical Brownian motion (original) (raw)
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 | #).
Markov quantum semigroups admit covariant MarkovC*-dilations
Communications in Mathematical Physics, 1986
Through a Danietl-Kolmogorov type construction, to any Markov quantum semigroup on a C*-algebra there is associated a quantum stochastic process which is a dilation of the semigroup, and satisfies a covariance rule which implies the weak Markov property.
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.
Invitation to Quantum Dynamical Semigroups
Lecture Notes in Physics, 2002
The theory of quantum dynamical semigroups within the mathematically rigorous framework of completely positive dynamical maps is reviewed. First, the axiomatic approach which deals with phenomenological constructions and general mathematical structures is discussed. Then basic derivation schemes of the constructive approach including singular coupling, weak coupling and low density limits are presented in their higly simplified versions. Two-level system coupled to a heat bath, damped harmonic oscillator, models of decoherence, quantum Brownian particle and Bloch-Boltzmann equations are used as illustrations of the general theory. Physical and mathematical limitations of the quantum open system theory, the validity of Markovian approximation and alternative approaches are discussed also.
Indian Journal of Pure and Applied Mathematics
A quantum Markov semigroup can be represented via classical diffusion processes solving a stochastic Schroedinger equation. In this paper we first prove that a quantum Markov semigroup is irreducible if and only if classical diffusion processes are total in the Hilbert space of the system. Then we study the relationship between irreducibility of a quantum Markov semigroup and properties of these diffusions such as accessibility, the Lie algebra rank condition, and irreducibility. We prove that all these properties are, in general, stronger than irreducibility of the quantum Markov
A resolution of quantum dynamical semigroups
Eprint Arxiv Math 0505384, 2005
We consider a class of quantum dissipative systems governed by a one parameter completely positive maps on a von-Neumann algebra. We introduce a notion of recurrent and metastable projections for the dynamics and prove that the unit operator can be decomposed into orthogonal projections where each projections are recurrent or metastable for the dynamics.
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 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.
Stability of Quantum Dynamical Semigroups
Operator Semigroups Meet Complex Analysis, Harmonic Analysis and Mathematical Physics, 2015
A one parameter semigroup of maps is said to be stable if it eventually decays to zero. Generally different topologies for convergence to zero give rise to different notions of stability. Stability is also connected with absence of fixed points. We examine these concepts in the context of quantum dynamical semigroups and dilation theory.
Dilation of a class of quantum dynamical semigroups with unbounded generators on UHF algebras
Annales de l'Institut Henri Poincare (B) Probability and Statistics, 2005
Evans-Hudson flows are constructed for a class of quantum dynamical semigroups with unbounded generator on UHF algebras, which appeared in [Rev. Math. Phys. 5 (3) (1993) 587-600]. It is shown that these flows are unital and covariant. Ergodicity of the flows for the semigroups associated with partial states is also discussed.
A comparison of quantum dynamical semigroups obtainable by mixing or partial tracing
Some simple examples of quantum systems are collected to illustrate requirements suffi-cient for the evolution of a subsystem according to a quantum dynamical semigroup. For this, a class of quantum dynamics of a system S coupled to a reservoir R is analyzed in the Hilbert space H SR = H S ⊗ H R , where H R = L 2 (R) and H S = l 2 I , with I standing for a complete at most countable set of pure orthogonal states of S. The Hamiltonian of SR is built of tensor products of multipliers acting on H S and H R . The chosen linear coupling implies the exponential decoherence of the reduced evolution of S if and only if the occupation density in R is of the Cauchy type. Then the system indicates the expo-nential decoherence. On the other hand, since the occupation density in S is discrete, the reduced evolution of R is never governed by a semigroup (unless there is no coupling). In the considered case, the reduced evolution of the subsystem S as well as of the reservoir R can be equivalently...
On the Long-Time Asymptotics of Quantum Dynamical Semigroups
Quantum Probability and Related Topics - Proceedings of the 30th Conference, 2011
We consider semigroups {αt : t ≥ 0} of normal, unital, completely positive maps αt on a von Neumann algebra M. The (predual) semigroup νt(ρ) := ρ • αt on normal states ρ of M leaves invariant the face Fp := {ρ : ρ(p) = 1} supported by the projection p ∈ M, if and only if αt(p) ≥ p (i.e., p is sub-harmonic). We complete the arguments showing that the sub-harmonic projections form a complete lattice. We then consider ro, the smallest projection which is larger than each support of a minimal invariant face; then ro is subharmonic. In finite dimensional cases sup αt(ro) = 1 and ro is also the smallest projection p for which αt(p) → 1. If {νt : t ≥ 0} admits a faithful family of normal stationary states then ro = 1 is useless; if not, it helps to reduce the problem of the asymptotic behaviour of the semigroup for large times.
Covariant Uniformly Continuous Quantum Markov Semigroups
Reports on Mathematical Physics, 2019
In this paper we analyze the structure of decoherence-free subalgebra N (T) of a uniformly continuous covariant semigroup with respect to a representation π of a compact group G on h. In particular, we obtain that, when π is irreducible, N (T) is isomorphic to (B(k) ⊗ 1 m) d for suitable Hilbert spaces k and m, and an integer d related to the connected components of G. We extend this result when π is reducible and N (T) is atomic by the decomposition of h due to the Peter-Weyl theorem.
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 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.
Spectral analysis and Feller property for quantum Ornstein–Uhlenbeck semigroups
2000
A class of dynamical semigroups arising in quantum optics models of masers and lasers is investigated. The semigroups are constructed, by means of noncommutative Dirichlet forms, on the full algebra of bounded operators on a separable Hilbert space. The explicit action of their generators on a core in the domain is used to demonstrate the Feller property of the semigroups, with respect to the C *-subalgebra of compact operators. The Dirichlet forms are analysed and the L 2-spectrum together with eigenspaces are found. When reduced to certain maximal abelian subalgebras, the semigroups give rise to the Markov semigroups of classical Ornstein-Uhlenbeck processes on the one hand, and of classical birth-and-death processes on the other.