Page 189 - Principles and Applications of NanoMEMS Physics

4. NANOMEMS APPLICATIONS: CIRCUITS AND SYSTEMS 177

quantum gate to an initial state 000 ... 000 [193]. The result is the same,
namely, p = (1− ( aa ′ )) 2 , except that now the deviation in question takes
1
the form =δ ε 000 ... 000 000 ... 000 .
In general, if a state has a deviation proportional to a pure state ψ ψ ,
in particular, =δ ε ψ ψ , it is called a pseudo-pure state. Physically, Cory

et al. [200] stated that the justification for constructing a pseudo-pure state
derives from the fact that the spins in the different molecules of a liquid are
virtually independent of one another and that, as a result, they may be
construed as a large number of copies of a single type of molecule, thus
permitting the liquid to be approximated by a Gibbs ensemble. Because of
N
this, instead of dealing with a density matrix of size 2 , which is the total
n
number of molecules, one can deal with a reduced density matrix of size 2 ,
where n is the number of spins in a single molecule. Analytically, instead of
the density matrix [200],
Ψ = ³ p () ψψ ψ d ψ , (3 )
5
{}
ψ
ψ
p
where () is the probability density of the pure state described by the
ψ
spinor ψ and {} denotes the set of all unit norm spinors, one uses the
approximation [200],
(1 − α ) +I 2α ψ ψ
Ψ = ( 1 ≤− α ≤ ) 1 , (3 )
6
(1 − α )2 + 2α
n
where ψ is a unit spinor. Thus, since the ensemble average of an
observable O is obtained by taking the trace of its product by the density
matrix, ( ΨOtr ), a simplification is obtained from using the pseudo-state,
since the ensemble average is now given by,
O
tr
tr ( ΨO ) ( −∝ 1 α ) ( )+ 2Otr α ψ O ψ , where () is known. While the
pseudo-pure state continues to be made up of a statistical mixture of
molecules, since by Eq. (3 ), each spinor determines a unique psudo-pure
6
density matrix, and each pseudo-pure density matrix determines a spinor that
is unique to within an overall phase factor (assuming the polarization is α
known), each addition of the magnetizations of all the molecules reveals the
predominance of one particular state present, in effect capturing each
molecule’s state for the final spectrum without the necessity of wavefunction
collapse [200]. The price paid as a result of using pseudo-pure states is the
loss of a factor of the order of one million in the effective number of