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Q-adapted quantum stochastic integrals and differentials in Fock scale

Viacheslav Belavkin, Matthew Brown (2011)

Banach Center Publications

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

Franco Fagnola, Veronica Umanità (2011)

Banach Center Publications

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 t r ( ρ 1 / 2 x ρ t 1 / 2 ( y ) ) = t r ( ρ 1 / 2 θ y * θ ρ t 1 / 2 ( θ x * θ ) ) 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 Itô algebra and quantum martingale

Viacheslav Belavkin, Un Cig Ji (2007)

Banach Center Publications

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 random walk revisited

Kalyan B. Sinha (2006)

Banach Center Publications

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.

Quantum stochastic calculus on full Fock space

Michael Skeide (1998)

Banach Center Publications

We present a new version of integration of time-adapted processes with respect to creation, annihilation and conservation processes on the full Fock space. Among the new features, in the first place, there is a new formulation of adaptedness which is both simpler and more general than the known ones. The new adaptedness allows for processes which are not restricted to be elements of some norm closure of the ∗-algebra which is generated by the basic creation processes.

Quantum stochastic convolution cocycles -algebraic and C*-algebraic

J. Martin Lindsay, Adam G. Skalski (2006)

Banach Center Publications

We summarise recent results concerning quantum stochastic convolution cocycles in two contexts-purely algebraic and C*-algebraic. In each case the class of cocycles arising as the solution of a quantum stochastic differential equation is characterised and the form taken by the stochastic generator of a *-homomorphic cocycle is described. Throughout the paper a common viewpoint on the algebraic and C*-algebraic situations is emphasised; the final section treats the unifying example of convolution...

Quantum stochastic processes arising from the strong resolvent limits of the Schrödinger evolution in Fock space

Alexander Chebotarev, Dmitry Victorov (1998)

Banach Center Publications

By using F. A. Berezin's canonical transformation method [5], we derive a nonadapted quantum stochastic differential equation (QSDE) as an equation for the strong limit of the family of unitary groups satisfying the Schrödinger equation with singularly degenerating Hamiltonians in Fock space. Stochastic differentials of QSDE generate a nonadapted associative Ito multiplication table, and the coefficients of these differentials satisfy the formal unitarity conditions of the Hudson-Parthasarathy type...

q-White noise and non-adapted stochastic integral

Un Cig Ji, Byeong Su Min (2006)

Banach Center Publications

The q-white noise is studied as the time derivative of the q-Brownian motion. As an application of the q-white noise, a non-adapted (non-commutative) stochastic integral with respect to the q-Brownian motion is constructed.

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