Biomedical Engineering Reference
In-Depth Information
Neumann entropy minimized over all ensemble preparations of the state [
110
].
Such a measure may be interpreted as the asymptotic number of pure singlets
necessary to prepare the state by local operations and classical communication.
A practical characterization for mixed state entanglement is available for small
systems [
111
,
112
], but an explicit formula for calculating the EOF for a given
density matrix is known only for a pair of two-level systems [
113
,
114
]. This is
given by
EOF
[
ρ
(
t
)] =
−
x
+
log
2
x
+
−
x
−
log
2
x
−
,
1
C
2
where
x
±
=(
is the concurrence. The concur-
rence is an entanglement measure itself and is equal to
1
±
−
[
ρ
(
t
)])
/
2, and
C
[
ρ
(
t
)]
C
[
ρ
(
t
)] =
max
(
0
,
λ
0
−
λ
1
−
λ
2
−
λ
3
)
,
(9.18)
where
λ
i
are the square roots of the eigenvalues of the non-Hermitian matrix
)(
σ
y
⊗
σ
y
)
ρ
∗
(
ρ
(
t
t
)(
σ
y
⊗
σ
y
)
in decreasing order. Here,
ρ
(
t
)
is the density matrix
ρ
∗
(
of the potentially entangled 2
×
2system,
t
)
is its complex conjugate, and
0
−
i
σ
y
=
(9.19)
i
0
is one of the Pauli matrices.
Let us study the decay of entanglement under phonon-induced pure dephasing.
Since local unitary transformations do not change the amount of entanglement
in the system, we can use the density matrix ˜
e
−
iH
L
t
e
iH
L
t
, with
H
L
=
ρ
(
t
)=
ρ
(
t
)
E
1
(
|
in the formulas for entanglement (in
this section, we use the explicit tensor product notation for the states of the two-
dot system). Using the method introduced in Sect.
9.2.2.2
and assuming a separable
initial system-reservoir state, we find the elements of the density matrix ˜
1
1
|⊗
I)+
E
2
(I
⊗|
1
1
|
)
, instead of
ρ
(
t
)
ρ
(
t
)
.These
are equal to
ρ
0
]
ij
e
−
iA
ij
(
t
)+
B
ij
(
t
)
,
[
ρ
(
˜
t
)]
ii
=[
ρ
0
]
ii
;
˜
[
ρ
(
˜
t
)]
ij
=[
˜
(9.20)
with
A
02
=
∑
|
g
q
|
2
sin
w
q
t
A
01
=
,
(9.21)
∑
|
g
q
|
2
cos
2
A
03
=
4
(
q
z
D
/
2
)
sin
w
q
t
−
Δ
Et
,
(9.22)
A
12
=
0
,
(9.23)
=
=
−
,
A
13
A
23
A
03
A
01
(9.24)
=
∑
|
g
q
|
2
=
=
=
(
−
)(
2
n
q
+
)
,
B
01
B
02
B
13
B
23
cos
w
q
t
1
1
(9.25)
∑
|
g
q
|
2
cos
2
B
03
=
4
(
q
z
D
/
2
)(
cos
w
q
t
−
1
)(
2
n
q
+
1
)
,
(9.26)
B
12
=
4
B
01
−
B
03
,
(9.27)
