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

(1)

can be interpreted as stochastic differential equations of the form

(2)

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

			(3)
			(4)

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

(5)

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,

			(6)
			(7)
			(8)

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

			(9)
			(10)
			(11)

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

(12)

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

(13)

The second theorem states that if

(14)

and

(15)

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

(16)

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

(17)

and

(18)

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

(19)

and

(20)

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

			(21)
			(22)

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

(23)

where  is a bounded system space operator, is

			(24)
			(25)

for .

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

(26)

REFERENCES:

Hudson, R. L. and Parthasarathy, K. R. "Quantum Ito's Formula and Stochastic Evolutions." Comm. Math. Phys. 93, 301-323, 1984.

Itô, K. "On Stochastic Differential Equations." Mem. Amer. Math. Soc. No.  4, 1951.

Parthasarathy, K. R. An Introduction to Quantum Stochastic Calculus. Boston, MA: Birkhäuser, 1992.