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QP-PQ, quantum probability and white noise analysis

Kernel Representations of ☆-semigroups Associated With Infinitely Divisible States

V. P. Belavkin

Published Jul 1, 1992• 12 citations
Publisher
DOI
10.1142/9789814354783_0003

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Quantum chaotic states over a noncommutative monoid, a unitalization of a noncommutative Ito algebra parametrizing a quantum stochastic Levy process, are described in terms of their infinitely divisible generating functionals over the monoid-valued processes on an atomless `space-time' set. A canonical decomposition of the logarithmic conditionally posive-definite generating functional is constructed in a pseudo-Euclidean space, given by a quadruple defining the monoid triangular operator representation and a cyclic zero pseudo-norm state in this space. It is shown that the exponential representation in the corresponding pseudo-Fock space yields the infinitely-divisible generating functional with respect to the exponential state vector, and its compression to the Fock space defines the cyclic infinitly-divisible representation associated with the Fock vacuum state. The structure of states on an arbitrary Ito algebra is studied with two canonical examples of quantum Wiener and Poisson states. A generalized quantum stochastic nonadapted multiple integral is explicitly defined in Fock scale, its continuity and quantum stochastic differentiability is proved. A unified non-adapted and functional quantum Ito formula is discovered and established both in weak and strong sense, and the multiplication formula on the exponential Ito algebra is found for the relatively bounded kernel-operators in Fock scale. The unitarity and projectivity properties of nonadapted quantum stochastic linear differential equations are studied, and their solution is constructed for the locally bounded nonadapted generators in terms of the chronological products in the underlying kernel algebra canonically represented by triangular operators in the pseudo-Fock space.
We present two projects concerning the main part of my PhD work. In the first one we study quantum channels, which are the most general operations mapping quantum states into quantum states, from the point of view of their divisibility properties. We introduced tools to test if a given quantum channel can be implemented by a process described by a Lindblad master equation. This in turn defines channels that can be divided in such a way that they form a one-parameter semigroup, thus introducing the most restricted studied divisibility type of this work. Using our results, together with the study of other types of divisibility that can be found in the literature, we characterized the space of qubit quantum channels. We found interesting results connecting the concept of entanglement-breaking channel and infinitesimal divisibility. Additionally we proved that infinitely divisible channels are equivalent to the ones that are implementable by one-parameter semigroups, opening this question for more general channel spaces. In the second project we study the functional forms of one-mode Gaussian quantum channels in the position state representation, beyond Gaussian functional forms. We perform a black-box characterization using complete positivity and trace preserving conditions, and report the existence of two subsets that do not have a functional Gaussian form. The study covers as particular limit the case of singular channels, thus connecting our results with the known classification scheme based on canonical forms. Our full characterization of Gaussian channels without Gaussian functional form is completed by showing how Gaussian states are transformed under these operations, and by deriving the conditions for the existence of master equations for the non-singular cases.
The Fock space of bosons and fermions and its underlying superalgebra are represented by algebras of functions on a superspace. We define Gaussian integration on infinite-dimensional superspaces, and construct super-analogs of the classical function spaces with a reproducing kernel — including the Bargmann–Fock representation — and of the Wiener–Segal representation. The latter representation requires the investigation of Wick ordering on Z2-graded algebras. As application we derive a Mehler formula for the Ornstein–Uhlenbeck semigroup on the Fock space.