Biomedical Engineering Reference
In-Depth Information
1
0.8
0.6
0.4
0.2
0
0
1
2
3
4
0
1
2
3
4
t [ps]
t [ps]
Fig. 9.9
Evolution of entanglement of the two-qubit system for various distances between the dots
at
T
0(0nm—
red curve
,2nm—
green curve
,6nm—
blue curve
).
Left panel
corresponds to the initial state (
9.28
) and the right panel corresponds to the initial state (
9.29
)
=
100 K and
V
B
=
0. Such an
energy shift leads to an entanglement-generating evolution. This mechanism is used
for performing nontrivial two-qubit gates (controlled-shift) in many proposals for
semiconductor-based quantum information processing [
115
,
116
]. As can be seen
in Fig.
9.8
(oscillating green line in left panel), in the presence of phonon-induced
pure dephasing the cyclic evolution of entanglement is damped and the maximum
achievable level of entanglement is reduced. Moreover, extended periods of time
appear when the entanglement remains zero.
The appearance of complete disentanglement for some initial states under suffi-
ciently strong partial pure dephasing may be understood with the help of Eq. (
9.18
).
If the completely dephased state (a state where the density matrix is diagonal) has
λ
An important case is that of a nonzero biexcitonic shift,
V
B
=
0 then, by continuity, it will be surrounded by states with van-
ishing concurrence, so that entanglement vanishes for sufficiently strongly dephased
states, before the complete dephasing is reached. From the Woot
ters fo
rm
ula for
a
diagonal density matrix one finds
−
λ
−
λ
−
λ
<
0
1
2
3
(
√
ρ
,
√
ρ
,
meaning that the above condition may only be satisfied, if all four diagonal elements
are nonzero, which is the case for the initial state (
9.28
) but not the case of
state (
9.29
).
The time at which the entanglement of the state (
9.28
) vanishes completely
depends on temperature and on the distance between the systems. This time
becomes finite only at a certain temperature (that depends on the coupling strength).
Slightly above this critical temperature, complete disentanglement takes place only
for strongly separated systems. For higher temperatures the disentanglement time
for non-overlapping systems depends very weakly on the distance. It should be
stressed that the appearance of complete disentanglement at increased temperatures
is only related to stronger dephasing in the system at higher temperatures and, in
principle, the state might become separable already at
T
λ
−
λ
−
λ
−
λ
=
−
2min
ρ
ρ
)
0
1
2
3
00
33
11
22
=
0 if the coupling were
sufficiently strong.
On the contrary, the dependence on the separation between the two subsystems
reflects a more fundamental crossover, from the regime of a common reservoir to
that of independent reservoirs [
21
]. In our model, the off-diagonal element
ρ
12
is
unaffected for completely overlapping systems (
D
=
0) which is related to the fact
