Cryptography Reference
In-Depth Information
rotated by 45
◦
(the rotation is represented by the circle drawn inside the sym-
bol of the PBS), which works as a destructive CNOT gate on the polarization
qubits, as was experimentally demonstrated in [42]. The upper part, compris-
ing the entangled state and the PBS1, is meant to encode the control bit in the
two channels
a
4
and
b
1
.
Owing to the PBS operation, which transmits horizontally polarized pho-
tons and reflects vertically polarized ones, the successful detection of the state
|+
at the port
b
3
postselects the following transformation of the arbitrary in-
put state in
a
1
:
b
1
. The control bit
is thus encoded in
a
4
and in
b
1
. The photon in
a
4
serves as the control input to
the destructive CNOT gate and will be destroyed, while the second photon
in
b
1
serves as the output control qubit.
For the gate to work properly, one has to demonstrate that the most gen-
eral input state,
α
|
H
a
1
+
β
|
V
a
1
→
α
|
H
a
4
|
H
b
1
+
β
|
V
a
4
|
V
i
a
1
,a
2
|
=|
H
(α
|
H
+
α
|
V
)
+|
V
(α
|
H
+
α
|
V
)
,
a
1
1
a
2
2
a
2
a
1
3
a
2
4
a
2
can be converted to the output state,
out
b
2
).
Let us consider first the case where the control photon is in the logical
zero or horizontally polarized. The control photon will then travel undis-
turbed through the PBS, arriving in the spatial mode
b
1
. As required, the
output photon is
|
b
1
,b
2
=|
H
b
1
(α
1
|
H
b
2
+
α
2
|
V
b
2
)
+|
V
b
1
(α
3
|
V
b
2
+
α
4
|
H
. In order for the scheme to work, a photon needs to
arrive also in mode
b
3
: given that the input photon is already in mode
b
1
,
the additional photon will necessarily be provided by the EPR pair and is
|
|
H
after transmission through PBS1. We know that the photons in
a
3
and
a
4
are entangled, so the photon in
a
4
is also in the horizontal polarization
state. For a
H
) target photon, taking into account the 45
◦
rotation of the
polarization on the paths
a
2
,a
4
due to the half-wave plates, the input at PBS2
will then be in the state
|
|
H
(
V
. This state will give rise, with
a probability of 50%, to the state where two photons go through the PBS2,
namely
|+
|+
(
|+
|−
a
4
)
a
2
a
4
a
2
|
φ
±
b
2
,b
4
=
1
. After the additional rotation
of the polarization and after the subsequent change to the H/V basis (where
the measurement will be performed), this state acquires the form
√
2
(
|
H
b
2
|
H
b
4
±|
V
b
2
|
V
b
4
)
|
φ
+
b
2
,b
4
=
1
|
ψ
+
1
√
2
(
|
H
|
H
+|
V
|
V
)
(
=
√
2
(
|
H
|
V
+|
V
|
H
)
).
b
2
b
4
b
2
b
4
b
2
,b
4
b
2
b
4
b
2
b
4
), in the mode
b
2
is found for the case where
the photon in
b
4
is horizontally polarized. We can see in a similar way that the
gate works also for the cases where the control photon is vertically polarized
or is polarized at 45
◦
.
The experimental setup for the CNOT gate is shown in Figure 3.5. An
ultraviolet pulsed laser, centered at a wavelength of 398 nm, with pulse du-
ration 200 fs and a repetition rate of 76 MHz, impinges on a nonlinear BBO
crystal [43], in which it produces probabilistically the first photon pair in
the spatial modes
a
1
and
a
2
. They serve as input qubits to the gate. The UV
laser is reflected back by the mirror M1 and, on passing through the crystal a
second time, produces the entangled ancilla pair in spatial modes
a
3
and
a
4
.
The expected result,
|
H
(
|
V
