Cryptography Reference
In-Depth Information
3.2.2 Purifying Quantum Entanglement
Owing to unavoidable decoherence in the quantum communication channel,
the quality of entangled states generally decreases with the channel length.
Entanglement purification schemes [29] allow two spatially separated parties
to convert an ensemble of partially entangled states (which result from trans-
mission through noisy channels) to a set of almost perfectly entangled states
by performing local unitary operations and measurements on the shared
pairs, and coordinating their actions with a classical channel. One thus sim-
ulates a noiseless quantum channel by a noisy one, supplemented by local
actions and classical communication. In a recent experiment, entanglement
purification could be demonstrated for the first time experimentally for mixed
polarization-entangled two-particle states [30].
The crucial operation for a successful purification step is a bilateral condi-
tional NOT (CNOT) gate, which effectively detects single bit-flip errors in the
channel by performing local CNOT operations (see Section 3.2.3) at Alice's
and Bob's side between particles of shared entangled states. The outcome of
these measurements can be used to correct for such errors and eventually
leads to a less noisy quantum channel [29]. For the case of polarization en-
tanglement, such a parity check on the correlations can be performed in a
straightforward way by using polarizing beamsplitters (PBS) [31] that trans-
mit horizontally polarized photons and reflect vertically polarized ones.
Consider the situation in which Alice and Bob have established a noisy
quantum channel, i.e., they share a set of equally mixed, entangled states
AB
.
At both sides the two particles of two shared pairs are directed into the input
ports
a
1
,a
2
and
b
1
,b
2
of a PBS (see Figure 3.2). Only if the entangled input
states have the same correlations, i.e., they have the same parity with respect
to their polarization correlations, will the four photons exit in four different
outputs (four-mode case), and a projection of one of the photons at each side
will result in a shared two-photon state with a higher degree of entanglement.
All single bit-flip errors are effectively suppressed.
For example, they might start with the mixed state
ρ
ρ
=
·|
+
+
|
F
AB
AB
+
(
1
−
F
)
·|
−
−
|
AB
, where
|
+
=
(
|
HH
+|
VV
)
is one of the four maxi-
|
+
a
1
,a
2
mally
entangled
Bell
states.
Then
only
the
combinations
⊗
|
+
b
1
,b
2
and
|
−
a
1
,a
2
⊗|
−
b
1
,b
2
will lead to a four-mode case, while
|
+
a
1
,a
2
⊗|
−
b
1
,b
2
and
|
−
a
1
,a
2
⊗|
+
b
1
,b
2
will be rejected. Finally, a projection of the
output modes
a
4
,b
4
into the basis
1
|± =
√
2
(
|
H
±|
V
)
will create the pure
|
+
a
3
,b
3
with probability
F
=
F
2
[
F
2
2
] and
|
+
a
3
,b
3
states
/
+
(
1
−
F
)
with
F
, respectively. The fraction
F
|
+
probability 1
−
of the desired state
be-
1
comes larger for each purification step if
F
2
. In other words, the new state
ρ
AB
shared by Alice and Bob after the bilateral parity operation demonstrates
an increased fidelity with respect to a pure, maximally entangled state. This
is the purification of entanglement.
Typically, in the experiment, one photon pair of fidelity 92% could be
obtained from two pairs, each of fidelity 75%. Also, although only bit-flip
errors in the channel have been discussed, the scheme works for any general
>
