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Mathematical Proceedings of the Cambridge Philosophical Society

Quantum stochastic integration and quantum stochastic differential equations

Chris Barnett, Stanisław Goldstein, Ivan F. Wilde

Published Nov 1, 1994• 6 citations
Publisher

Abstract

Abstract Quantum stochastic integrals are constructed using the non-commutative L p -space theory of Haagerup. The existence and uniqueness of the solution to quantum stochastic differential equations driven by quasi-Wiener noises, or noises satisfying generalized standing hypotheses, is established as is the Markov behaviour of the solution. Various examples of the theory are discussed, and quantum Ornstein-Uhlenbeck processes are obtained as explicit solutions.

DOI
10.1017/s0305004100072807

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Abstract Quantum stochastic integrals are constructed using the non-commutative L p -space theory of Haagerup. The existence and uniqueness of the solution to quantum stochastic differential equations driven by quasi-Wiener noises, or noises satisfying generalized standing hypotheses, is established as is the Markov behaviour of the solution. Various examples of the theory are discussed, and quantum Ornstein-Uhlenbeck processes are obtained as explicit solutions.
Many examples of quantum independent stationary increment processes are solutions of quantum stochastic differential equations. We give a common characterization of these examples by a quantum stochastic differential equation on an abstract *-bialgebra. Specializing this abstract *-bialgebra and the coefficients of the equation, we obtain the equations for the Unitary Noncommutative Stochastic processes of [12], the Quantum Wiener Process [2], the Azéma martingales [11] and for other examples. The existence and uniqueness of a solution of the general equation is shown. Assuming the boundedness of this solution, we prove that it is a continuous and stationary independent increment process.
Quite a few researchers have contributed to the development of quantum stochastic differential equations driven by quantum noises, initiated by Hudson and Parthasarathy [HP84a]. A nonexhaustive list of contributors includes Fagnola and Wills [FW00], [FW03], Obata [Oba97], and Linsay and Skalski [LS10]. Based on the results presented in Chapter 3, this chapter derives and considers a general form of linear (left) and (right) Hudson-Parthasarathy quantum stochastic differential equations driven by quantum noises in symmetric Fock space with operator-valued matrices as coefficients. Specifically, it studies the existence and uniqueness of the strong and mild solution processes for both the left and right quantum stochastic differential equations and conditions under which the solution processes are unitary, isometry, co-isometry, and regular. These results make extensive use of the properties of stochastic integral driven by quantum noise and its quantum stochastic calculus in the context of a symmetric Fock space, which was explored in Chapter 3. Various discrete approximation schemes, including Bouten et al. [BvH07], of the left Hudson-Parthasarathy QSDE are also explored for numerical computation. Specifically, it is shown that the solution of the Hudson- Parthasarathy QSDE can be approximated by a sequence of discrete interaction models with decreasing time step. In order to study this problem, discrete interaction models are embedded in a limiting space. This allows one to present strong convergence of the embedded discrete cocycles to the solution of the Hudson-Parthasarathy QSDE. It will be pointed out that the way in which the embedding is done does not affect the proof of the main result for discrete approximations. Examples of a QSDE that is analytically solvable and a QSDE that possesses a mild solution but not a strong solution are also given. QSDE of Hudson-Parthasarathy Type It has been mentioned earlier in Chapter 3 that an open quantum systems treated in this monograph are to be formulated as a quantum system in which the underlined (or the initial) quantum system is perturbed by quantum noises that represent the environment.
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White noise theory allows to formulate quantum white noises explicitly as elemental quantum stochastic processes. A traditional quantum stochastic differential equation of Itô type is brought into a normal-ordered white noise differential equation driven by lower powers of quantum white noises. The class of normal-ordered white noise differential equations covers quantum stochastic differential equations with highly singular noises such as higher powers or higher order derivatives of quantum white noises, which are far beyond the traditional Itô theory. For a general normal-ordered white noise differential equation unique existence of a solution is proved in the sense of white noise distribution. Its regularity properties are investigated by means of weighted Fock spaces interpolating spaces of white noise distributions and associated characterization theorems for S-transform and for operator symbols.
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.