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arXiv:math/0512265 · Russian Mathematical Surveys

Chaotic states and stochastic integration in quantum systems

V. P. Belavkin

Published Feb 28, 1992• 37 citations

Abstract

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.

DOI: 10.1070/rm1992v047n01abeh000862

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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.

In this thesis we develop the theory of quantum Wiener integrals on the bosonic Fock space. We study multiple quantum Wiener integrals as an algebra of unbounded operators, investigating its properties, including closedness, common domains and multiplication formulas. We show the applications of the new formalism by providing new proofs to the established theory of quantum stochastic calculus and new conditions for generating quantum stochastic cocycles and quantum stochastic evolutions. The corresponding quasifree case is also studied and the constructions extended to fit in that formalism. We construct the multiple quantum Wiener integral as one operator on a family of operators which we dub operator kernels. This in particular covers the case of quantum stochastic cocycles and evolutions. We show that the family of quantum Wiener integrals forms a WOT-dense algebra of unbounded operators on the bosonic Fock space. We provide more general conditions for an operator kernel to be multiple quantum Wiener integrable, which allows us to treat multiple quantum Wiener integrals as an algebra. We explore the influence of an initial space on the theory. Our setting gives natural conditions for a product of two cocycles (evolutions) to still be a cocycle (an evolution). We apply our theory by solving quantum stochastic differential equations (QSDEs) and by finding more elementary proofs of structure conditions on the generator of a quantum stochastic evolution and of the fundamental estimate in the proof of quantum stochastic Lie–Trotter formula. We also show how our theory unifies and generalises the theory of integral kernels and chaotic representation properties, proving in particular that every Hilbert–Schmidt operator is a quantum Wiener integral.

A simple axiomatic characterization of the general (infinite dimensional, noncommutative) Ito algebra is given and a pseudo-Euclidean fundamental representation for such algebra is described. The notion of Ito B*-algebra, generalizing the C*-algebra is defined to include the Banach infinite dimensional Ito algebras of quantum Brownian and quantum Levy motion, and the B*-algebras of vacuum and thermal quantum noise are characterized. It is proved that every Ito algebra is canonically decomposed into the orthogonal sum of quantum Brownian (Wiener) algebra and quantum Levy (Poisson) algebra. In particular, every quantum thermal noise is the orthogonal sum of a quantum Wiener noise and a quantum Poisson noise as it is stated by the Levy-Khinchin Theorem in the classical case .

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 in [10].The differential Q-adapted formula generalizing Itô product formula for adapted integrals is presented in both strong and weak sense as a particular case of the quantum stochastic nonadapted Itô formula.

A generalized definition of quantum stochastic (QS) integrals and differentials is given in the free of adaptiveness and dimensionality form in terms of Malliavin derivative on a projective Fock space, and their uniform continuity with respect to the inductive limite convergence is proved. A new form of QS calculus based on an inductive *-algebraic structure in an indefinite space is developed and a nonadaptive generalization of the QS Ito formula for its representation in Fock space is derived. The problem of solution of general QS evolution equations in a Hilbert space is solved in terms of the constructed operator representation of chronological products, defined in the indefinite space, and isometry and *-homomorphism property respectively for operators and maps of these solutions, corresponding to the peseudounitary and *-homomorphism property of the QS integrable generators is proved.

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A simple axiomatic characterization of the general (infinite dimensional, noncommutative) Ito algebra is given and a pseudo-Euclidean fundamental representation for such algebra is described. The notion of Ito B*-algebra, generalizing the C*-algebra is defined to include the Banach infinite dimensional Ito algebras of quantum Brownian and quantum Levy motion, and the B*-algebras of vacuum and thermal quantum noise are characterized. It is proved that every Ito algebra is canonically decomposed into the orthogonal sum of quantum Brownian (Wiener) algebra and quantum Levy (Poisson) algebra. In particular, every quantum thermal noise is the orthogonal sum of a quantum Wiener noise and a quantum Poisson noise as it is stated by the Levy-Khinchin theorem in the classical case.