1
Ph 12b
Homework Assignment No. 4
Due: 5pm, Thursday, 4 February 2010
1. Weaker decoherence
. In class we discussed the
phase damping
of a
qubit that results when the qubit scatters a photon with probability
p
.
The scattered photon is knocked into one of two mutually orthogonal
states
{
0
a
E
,

1
a
E
}
, correlated with the qubit’s state, both of which are
orthogonal to the state

un
a
E
of the unscattered photon. If the initial
state of the qubit is

ψ
a
S
=
a

0
a
S
+
b

1
a
S
, then the joint state of the
qubit and photon evolves as

ψ
a
S
⊗
un
a
E
→
r
1

p

ψ
a
S
⊗
un
a
E
+
√
p
(
a

0
a
S
⊗ 
0
a
E
+
b

1
a
S
⊗ 
1
a
E
)
.
(1)
Thus the qubit density operator ˆ
ρ
evolves as
ˆ
ρ
=
p
ρ
00
ρ
01
ρ
10
ρ
11
P
→
ˆ
ρ
p
=
p
ρ
00
(1

p
)
ρ
01
(1

p
)
ρ
10
ρ
11
P
.
Now consider a diFerent model of decoherence, in which photon scat
tering does not perfectly resolve the state of the qubit. The scattered
photon is knocked to the normalized state

γ
a
E
if the qubit’s state is

0
a
S
and it is knocked to the normalized state

η
a
E
if the photon’s state
is

1
a
S
; thus eq.(1) is replaced by

ψ
a
S
⊗
un
a
E
→
r
1

p

ψ
a
S
⊗
un
a
E
+
√
p
(
a

0
a
S
⊗ 
γ
a
E
+
b

1
a
S
⊗ 
η
a
E
)
.
(2)
Both

γ
a
E
and

η
a
E
are orthogonal to the state

un
a
E
of the unscattered
photon, but they are not necessarily mutually orthogonal; rather
E
A
η

γ
a
E
= 1

e,
where
e
is a real number. Thus for
e
= 1, the states

γ
a
E
and

η
a
E
are
orthogonal, and we recover the model considered previously, while for
e
= 0, the scattered photon remains uncorrelated with the qubit, and
there is no decoherence at all.
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 Spring '09
 DUDKO
 Linear Algebra, mechanics, Work, Photon, Hilbert space, density operator, state vector ψ

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