Itô calculus and quantisation of Lie bialgebras
Itô versus Woronowicz calculus in Itô Hopf algebras
Markov product of positive definite kernels and applications to Q-matrices of graph products
We show positivity of the Q-matrix of four kinds of graph products: direct product (Cartesian product), star product, comb product, and free product. During the discussion we give an alternative simple proof of the Markov product theorem on positive definite kernels.
Martin boundary theory of some quantum random walks
Méthodes de noyaux et de symboles en calcul stochastique non commutatif
Mild solutions of quantum stochastic differential equations.
On Markovian cocycle perturbations in classical and quantum probability.
On the second quantisation of Hilbert-Schmidt processes
On two quantum versions of the detailed balance condition
Quantum detailed balance conditions are often formulated as relationships between the generator of a quantum Markov semigroup and the generator of a dual semigroup with respect to a certain scalar product defined by an invariant state. In this paper we survey some results describing the structure of norm continuous quantum Markov semigroups on ℬ(h) satisfying a quantum detailed balance condition when the duality is defined by means of pre-scalar products on ℬ(h) of the form (s ∈ [0,1]) in order...
Opérateurs d'échange et quelques applications des méthodes de noyaux
Photoemissive sources and quantum stochastic calculus
Poisson stochastic integration in Hilbert spaces
Probability and quantum symmetries. I. The theorem of Noether in Schrödinger's euclidean quantum mechanics
Progrès récents en calcul stochastique quantique
Q-adapted quantum stochastic integrals and differentials in Fock scale
In this paper we first introduce the Fock-Guichardet formalism for the quantum stochastic (QS) integration, then the four fundamental processes of the dynamics are introduced in the canonical basis as the operator-valued measures, on a space-time σ-field , of the QS integration. Then rigorous analysis of the QS integrals is carried out, and continuity of the QS derivative D is proved. Finally, Q-adapted dynamics is discussed, including Bosonic (Q = I), Fermionic (Q = -I), and monotone (Q = O) quantum...
Quantum detailed balance conditions with time reversal: the finite-dimensional case
We classify generators of quantum Markov semigroups on (h), with h finite-dimensional and with a faithful normal invariant state ρ satisfying the standard quantum detailed balance condition with an anti-unitary time reversal θ commuting with ρ, namely for all x,y ∈ and t ≥ 0. Our results also show that it is possible to find a standard form for the operators in the Lindblad representation of the generators extending the standard form of generators of quantum Markov semigroups satisfying the usual...
Quantum fluctuations of elementary excitations in discrete media.
Quantum Itô algebra and quantum martingale
In this paper, we study a representation of the quantum Itô algebra in Fock space and then by using a noncommutative Radon-Nikodym type theorem we study the density operators of output states as quantum martingales, where the output states are absolutely continuous with respect to an input (vacuum) state. Then by applying quantum martingale representation we prove that the density operators of regular, absolutely continuous output states belong to the commutant of the ⋆-algebra parameterizing the...
Quantum probability, renormalization and infinite-dimensional *-Lie algebras.
Quantum random walk revisited
In the framework of the symmetric Fock space over L²(ℝ₊), the details of the approximation of the four fundamental quantum stochastic increments by the four appropriate spin-matrices are studied. Then this result is used to prove the strong convergence of a quantum random walk as a map from an initial algebra 𝓐 into 𝓐 ⊗ ℬ (Fock(L²(ℝ₊))) to a *-homomorphic quantum stochastic flow.