Thus the state of system a is set back to j every

Thus the state of system A is set back to \<j>) every after r, while that of
B just evolves dynamically on the basis of the total Hamiltonian H. We
repeat the same measurement, represented by (3), N times and collect only
those events in which system A has been found in state \<j>) consecutively N
times; other events are discarded. The state of system B is then described
by the density matrix
P
{
B\N)
= ( W ) % ( 0 ) ( l ^ ( r ) ) V
( r )
W , (4)
where
V*{T) = (<f>\e-
iH
^) (5)
is an operator acting on B and
P
(r
> (N) = Tr \(Oe-
iHT
0)
N
p
0
(Oe
iHT
O)
N
= Tr
B
[(V
4>
(r))
N
p
B
(0)(Vl(r))
N
] (6)
is the success probability for these events to occur (yield). This normaliza-
tion factor in (4) reflects the fact that only right outcomes are collected in
this process.
In order to examine the asymptotic state of system B, consider the
spectral decomposition of the operator
V^(T),
which is not hermitian,
V^(T) ^ Vl(r). We therefore need to set up both the right- and left-
eigenvalue problems
V^(r)|u„) = \„\u
n
), {v
n
\Vd>{r) - A„(t>„|. (7)
The eigenvalue A„ is complex valued in general, but its absolute value is
bounded
12
0 < |A„| < 1, (8)
which is a reflection of the unitarity of the time evolution operator e~
lHr
.
These eigenvectors are assumed to form a complete orthonormal set in the
following sense
^2\u
n
)(Vn\ = I s , (V
n
\u
m
) - 8
nm
. (9)

262
Then the operator V^(r
N
itself is expanded in terms of these eigenvectors
^ ( r ) = ^ A > „ ) ( i ;
n
| . (
1 0
)
n
It is now easy to see that the TVth power of this operator is expressed as
n
and therefore it is dominated by a single term for large N
(V^f^^X^uoKvol (12)
when the largest (in magnitude) eigenvalue Ao is discrete, nondegenerate
and unique. If these conditions are satisfied, the density operator of system
B is driven to a pure state
p
W(
N
) 1^1^ |«o)(«
0
|/(uo|«o) (13)
with the probability
pW(iV) ^^U \\o\
2N
(u
0
\u
0
)(v
0
\p
B
(0)\v
0
). (14)
The pure state |u
0
), which is nothing but the right-eigenvector of the op-
erator V^(r) belonging to the largest (in magnitude) eigenvalue Ao, is thus
distilled in system B. This is the purification scheme proposed in
9
.
A few comments are in order. First, the final pure state |«o) toward
which system B is to be driven is dependent on the choice of the state \<p)
on which system A is projected every after measurement, the measurement
interval r and the Hamiltonian H, but does not depend on the initial state
of system B at all. In this sense, the purification is accomplished irrespec-
tively of the initial (mixed) state p
s
(0). Second, as is clear in the above
exposition, what is crucial in this purification scheme is the repetition of
one and the same measurement (more appropriately, spectral decomposi-
tion) and the measurement interval T need not be very small. It remains to
be an adjustable parameter. Third, if we can make other eigenvalues than
Ao much smaller in magnitude
|A„/A
0
|<1 for n ^ 0, (15)
by adjusting parameters, we will need fewer steps (i.e., smaller N) to purify
system B.