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Proceedings of the Royal Society of Edinburgh Section A Mathematics

Exponential formulae in quantum stochastic calculus

Alexander Semenovich Holevo

Published Jan 1, 1996• 16 citations
Publisher

Abstract

The rigorous definition of time-ordered exponentials, solving quantum linear stochastic differential equations, is extended to Boson and Fermion stochastic calculi with infinitely many degrees of freedom. The relation to the classicalmultiplicative stochastic integral, solving the Doleans exponential equation, is discussed.
DOI
10.1017/s0308210500022794

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Dynamics of quantum systems which are stochastically perturbed by linear coupling to the reservoir can be studied in terms of quantum stochastic differential equations (for example, quantum stochastic Liouville equation and quantum Langevin equation).In order to work it out one needs to define the quantum Brownian motion.As far as only its boson version has been known until recently, in the present paper we present the definition which makes it possible to consider the fermion Brownian motion as well.
Quantum stochastic calculus is extended in a new formulation in which its stochastic integrals achieve their natural and maximal domains. Operator adaptedness, conditional expectations and stochastic integrals are all defined simply in terms of the orthogonal projections of the time filtration of Fock space, together with sections of the adapted gradient operator. Free from exponential vector domains, our stochastic integrals may be satisfactorily composed yielding quantum Itô formulas for operator products as sums of stochastic integrals. The calculus has seen two reformulations since its discovery---one closely related to classical Itô calculus; the other to noncausal stochastic analysis and Malliavin calculus. Our theory extends both of these approaches and may be viewed as a synthesis of the two. The main application given here is existence and uniqueness for the Attal--Meyer equations for implicit definition of quantum stochastic integrals.
We discuss stochastic derivations, stochastic Hamiltonians and the flows that they generate, algebraic fluctuaion-dissipation theorems, etc., in a language common to both classical and quantum algebras. It is convenient to define distinct notions of time-ordered exponentials to take account of the breakdown of the Leibniz rule in the Ito calculus. We introduce a notion of quantum Stratonovich calculus and show how it relates to Stratonovich-Dyson time ordered exponentials. We then use it to demonstrate a natural way to add stochastic derivations.
The basic integrator processes of quantum stochastic calculus, namely, creation, conservation, and annihilation, are introduced in the Hilbert space of square integrable Brownian functionals. Stochastic integrals with respect to these processes and a quantum Itô’s formula are described. As an application two examples of quantum stochastic differential equations are discussed. A continuous time version of Stinespring’s theorem on completely positive maps in C *-algebras is exploited to formulate the notion of a quantum Markov process and indicate how classical Markov processes are woven into the fabric of the Schodinger-Heisenberg dynamics of quantum theory.
A quantum stochastic calculus for fermions is developed where the basic integrators are based on Dirac fields and the charge operator. The associated Ito formula has seven non-trivial correction terms. Conditions are found for the solutions of stochastic differential equations to be unitary and it is shown that the corresponding quantum stochastic flow manifests a broken symmetry whereby the particle and antiparticle noises no longer balance each other. An abstract theory of such flows is then developed. By employing the unification between boson and fermion stochastic calculi, we are able to develop the entire theory using boson Fock spaces.