Quantum Stochastic Calculus

Let , , be one-dimensional Brownian motion. Integration with respect to was defined by Itô (1951). A basic result of the theory is that stochastic integral equations of the form

can be interpreted as stochastic differential equations of the form

where differentials are handled with the use of Itô's formula

Hudson and Parthasarathy (1984) obtained a Fock space representation of Brownian motion and Poisson processes. The boson Fock space over is the Hilbert space completion of the linear span of the exponential vectors under the inner product

where and and is the complex conjugate of .

The annihilation, creation and conservation operators , and respectively, are defined on the exponential vectors of as follows,

The basic quantum stochastic differentials , , and are defined as follows,

Hudson and Parthasarathy (1984) defined stochastic integration with respect to the noise differentials of Definition 3 and obtained the Itô multiplication table

The two fundamental theorems of the Hudson-Parthasarathy quantum stochastic calculus give formulas for expressing the matrix elements of quantum stochastic integrals in terms of ordinary Lebesgue integrals. The first theorem states that is

where , , , are (in general) time-dependent adapted processes. Let also and be in the exponential domain of , then

The second theorem states that if

and

where , , , , , , , are (in general) time dependent adapted processes and also and be in the exponential domain of , then

The fundamental result that connects classical with quantum stochastics is that the processes and defined by

and

are identified, through their statistical properties, e.g., their vacuum characteristic functionals

and

with Brownian motion and a Poisson process of intensity , respectively.

Within the framework of Hudson-Parthasarathy quantum stochastic calculus, classical quantum mechanical evolution equations take the form

where, for each , is a unitary operator defined on the tensor product of a system Hilbert space and the noise (or reservoir) Fock space . Here, , , are in , the space of bounded linear operators on , with unitary and self-adjoint. Notice that for , equation (21) reduces to a classical stochastic differential equation of the form (2). Here and in what follows we identify time-independent, bounded, system space operators with their ampliation to .

The quantum stochastic differential equation (analogue of the Heisenberg equation for quantum mechanical observables) satisfied by the quantum flow

where is a bounded system space operator, is

for .

The commutation relations associated with the operator processes , are the canonical (or Heisenberg) commutation relations, namely