ON STOCHASTIC GENERATORS OF POSITIVE DEFINITE EXPONENTS

This is the proceedings of the 2nd Japanese-German Symposium on Infinite Dimensional Harmonic Analysis held from September 20th to September 24th 1999 at the Department of Mathematics of Kyoto University.この論文集は, 1999年9月20日から9月24日の日程で京都大学理学研究科数学教室において開催された第2回日独セミナー「無限次元調和解祈」の成果をもとに編集されたものである.編集 : ハーバート・ハイヤー, 平井 武, 尾畑 信明Editors: Herbert Heyer, Takeshi Hirai, Nobuaki Obata #enA characterisation of quantum stochastic positive definite (PD) exponent is given in terms of the conditional positive definiteness (CPD) of their form-generator. The pseudo-Hilbert dilation of the stochastic form-generator and the pre-Hilbert dilation of the corresponding dissipator is found. The structure of quasi-Poisson stochastic generators giving rise to a quantum stochastic birth processes is studied

Multiple Q-Adapted Integrals and Ito Formula of Noncommutative Stochastic Calculus in Fock Space

We study the continuity property of multiple Q-adapted quantum stochastic integrals with respect to noncommuting integrands given by the non-adapted multiple integral kernels in Fock scale. The noncommutative algebra of relatively (exponentially) bounded nonadapted quantum stochastic processes is studied in the kernel form as introduced by Belavkin in 1991. The differential Q-adapted formula generalizing Ito product formula for adapted integrals is presented in both strong and weak sense as a particular case of the quantum stochastic nonadapted Ito formula.Comment: Due to appear in communications on stochastic analysis journal (KRP volume). 21 page

Dynamical programming of continuously observed quantum systems

We develop dynamical programming methods for the purpose of optimal control of quantum states with convex constraints and concave cost and bequest functions of the quantum state. We consider both open loop and feedback control schemes, which correspond respectively to deterministic and stochastic Master Equation dynamics. For the quantum feedback control scheme with continuous non-demolition observations we exploit the separation theorem of filtering and control aspects for quantum stochastic dynamics to derive a generalized Hamilton-Jacobi-Bellman equation. If the control is restricted to only Hamiltonian terms this is equivalent to a Hamilton-Jacobi equation with an extra linear dissipative term. In this work, we consider, in particular, the case when control is restricted to only observation. A controlled qubit is considered as an example throughout the development of the formalism. Finally, we discuss optimum observation strategies to obtain a pure state from a mixed state of a quantum two-level system.Comment: 11 pages, no figures, published versio

Operational distance and fidelity for quantum channels

We define and study a fidelity criterion for quantum channels, which we term the minimax fidelity, through a noncommutative generalization of maximal Hellinger distance between two positive kernels in classical probability theory. Like other known fidelities for quantum channels, the minimax fidelity is well-defined for channels between finite-dimensional algebras, but it also applies to a certain class of channels between infinite-dimensional algebras (explicitly, those channels that possess an operator-valued Radon--Nikodym density with respect to the trace in the sense of Belavkin--Staszewski) and induces a metric on the set of quantum channels which is topologically equivalent to the CB-norm distance between channels, precisely in the same way as the Bures metric on the density operators associated with statistical states of quantum-mechanical systems, derived from the well-known fidelity (`generalized transition probability') of Uhlmann, is topologically equivalent to the trace-norm distance.Comment: 26 pages, amsart.cls; improved intro, fixed typos, added a reference; accepted by J. Math. Phy