Generators of KMS Symmetric Markov Semigroups on {\mathcal{B}({\rm h})} Symmetry and Quantum Detailed Balance

We find the structure of generators of norm-continuous quantum Markov semigroups on B(h){\mathcal{B}({\rm h})} that are symmetric with respect to the scalar product tr (ρ ^1/2 x*ρ ^1/2 y) induced by a faithful normal invariant state ρ and satisfy two quantum generalisations of the classical detailed balance condition related with this non-commutative notion of symmetry: the so-called standard detailed balance condition and the standard detailed balance condition with an antiunitary time reversal.     

Detailed balance, time reversal, and generators of quantum Markov semigroups

We characterize generators ? of norm continuous quantum Markov semigroups satisfying the quantum detailed balance condition with respect to an antiunitary time reversal in terms of the operators H and L _k in the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) representation ?(x) = i[H, x] ? (1/2)gZ _k (L* _k L _k x ? 2L* _k x L _k + x L* _k L _k ).     

Transience and recurrence of quantum Markov semigroups

 This article introduces a concept of transience and recurrence for a Quantum Markov Semigroup and explores its main properties via the associated potential. We show that an irreducible semigroup is either recurrent or transient and characterize transient semigroups by means of the existence of non trivial superharmonic operators. Received: 27 January 2003 / Revised version: 19 February 2003 / Published online: 12 May 2003 This research has been partially supported by the ``Cátedra Presidencial en Análisis Cualitativo de Sistemas Dinámicos Cuánticos', DIPUC, FONDECYT project 1030552, MIUR program ``Probabilità Quantistica e Applicazioni', 2003-2004 and MECESUP PUC 0103. Mathematics Subject Classification (2000): 60J45, 81S25, 60J99, 37A50, 47A40 Key words or phrases: Quantum Markov semigroups – Potential theory – Markov processes     

A Lévy theorem for free noises

Summary We prove a Lévy type characterization theorem for the free Brownian motion and the free Poisson process using martingale and convariance conditions and some assumption on fourth order conditional moments.     

On quantum stochastic differential equations with unbounded coefficients

Summary We prove an existence, uniqueness and unitarity theorem for quantum stochastic differential equations with unbounded coefficients which satisfy an analyticity condition on a common dense invariant domain. This result, applied to the quantum harmonic oscillator, gives a rigorous meaning to a large class of stochastic differential equations that have been considered formally in quantum probability.     

Quadratic open quantum harmonic oscillator

Letters in Mathematical Physics - We study the quantum open system evolution described by a Gorini–Kossakowski–Sudarshan–Lindblad generator with creation and annihilation...     

Spectral Analysis and Feller Property for Quantum Ornstein–Uhlenbeck Semigroups

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 C²-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.     

Are sufficient conditions for conservativity of minimal quantum semigroups necessary?

We show by a counter-example that the two basic operator inequalities in the typical sufficient conditions for conservativity of minimal quantum dynamical semigroups are, in fact, both necessary.     

Solving quantum stochastic differential equations with unbounded coefficients

We demonstrate a method for obtaining strong solutions to the right Hudson-Parthasarathy quantum stochastic differential equation