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Quantum Stochastic Calculus
المؤلف:
Itô, K.
المصدر:
"On Stochastic Differential Equations." Mem. Amer. Math. Soc. No. 4, 1951.
الجزء والصفحة:
...
14-7-2020
950
Let {B_t(omega)/omega in Omega}" src="https://mathworld.wolfram.com/images/equations/QuantumStochasticCalculus/Inline1.gif" style="height:15px; width:117px" />,
, 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
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(1) |
can be interpreted as stochastic differential equations of the form
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(2) |
where differentials are handled with the use of Itô's formula
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(3) |
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(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
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(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,
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(6) |
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(7) |
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(8) |
The basic quantum stochastic differentials ,
, and
are defined as follows,
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(9) |
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(10) |
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(11) |
Hudson and Parthasarathy (1984) defined stochastic integration with respect to the noise differentials of Definition 3 and obtained the Itô multiplication table
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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
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(12) |
where ,
,
,
are (in general) time-dependent adapted processes. Let also
and
be in the exponential domain of
, then
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(13) |
The second theorem states that if
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(14) |
and
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(15) |
where ,
,
,
,
,
,
,
are (in general) time dependent adapted processes and also
and
be in the exponential domain of
, then
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(16) |
The fundamental result that connects classical with quantum stochastics is that the processes and
defined by
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(17) |
and
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(18) |
are identified, through their statistical properties, e.g., their vacuum characteristic functionals
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(19) |
and
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(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
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(21) |
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(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
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(23) |
where is a bounded system space operator, is
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(24) |
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(25) |
for .
The commutation relations associated with the operator processes ,
are the canonical (or Heisenberg) commutation relations, namely
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(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.