Quantum Control and Manipulation of Systems and Processes at Molecular Scale - Advanced Models of Neural Networks - page 251

associated with the probability of locating the particle at a certain eigenstate.
The Lindblad and Belavkin equations are actually the quantum analogous of
the Kushner-Stratonovich stochastic differential equation which denotes that the
change of the probability of the state vector x to take a particular value depends
on the difference (innovation) between the measurement y.x/ and the mean value
of the estimation of the measurement EŒy.x/. It is also known that the Kushner-
Stratonovich SDE can be written in the form of a Langevin SDE [ 155 ]

dx D ˛.x/ dt C b.x/ dv (13.10)

which finally means that the Lindblad and Belavkin description of a quantum system
are a generalization of Langevin's SDE for quantum systems [ 210 ]. For a quantum
system with state vector x and eigenvalues .x/2R, the Lindblad equation is written
as [ 27 , 210 ]

„P DiŒ H;C DŒc (13.11)

where is the associated probability density matrix for state x, i.e. it defines the
probability to locate the particle at a certain eigenstate of the quantum system and
the probabilities of transition to other eigenstates. The variable H is the system's
Hamiltonian, operator ŒA;B is a Lie bracket defined as ŒA;B D

AB BA ,the

c D .c 1 ; ; c L / T is also a vector of operators, variable D is defined as vector

DŒc D P lD1 DŒc l , and finally „ is Planck's constant.

13.3.2 The Belavkin Description of Quantum Systems

The Lindblad equation (also known as stochastic master equation), given in
Eq. ( 13.11 ), is actually a differential equation which can be also written in the form
of a stochastic differential equation that is known as Belavkin equation .Themost
general form of the Belavkin equation is:

„d c D dt DŒc c C HŒi H dt C dz C .t/c c

(13.12)

Variable H is an operator which is defined as follows:

HŒr DrC r C Tr ŒrC r C (13.13)

Variable H stands for the Hamiltonian of the quantum system. Variable c is
an arbitrary operator obeying c C c D R, where R is an hermitian operator.
The infinite dimensional complex variables vector dz is defined as dz D
. dz 1 ; ; dz L / T , and in analogy to the innovation dv of the Langevin equation
(see Eq. ( 13.10 )), variable dz expresses innovation for the quantum case. Variable
dz C denotes the conjugate-transpose . dz / T . The statistical characteristics of dz are