4.3 Experimental CKA
65
polarizing beam splitter (PBS)
5 and exits the PBS in path 2. Here, a half-wave
plate (HWP) oriented at 45
◦ rotates the photon’s polarization to |H before entering
the non-linear crystal. The photon undergoes SPDC in the PPKTP crystal and is
converted to two photons of lower energy in path 1 (red line), orthogonally polarized,
called signal and idler: |H s 1 |V i 1 . Similarly, an |H photon from the laser source gets
transmitted at the PBS and is down-converted to signal and idler, |H s |V i , circulating
the loop counter-clockwise. Their polarization gets rotated by the HWP in path 2 to:
|V s 2 |H i 2 . At this point, the two counter-propagating photon pairs interfere at the
PBS by entering through ports 1 and 2, respectively. The photon pair entering in 1
exits the PBS in the state |H s 3 |V i 4 , while the pair entering from 2 is mapped to
|V s 3 |H i 4 . The down-converted photons exiting in 3 are separated from the input
signal with a dichroic mirror (DM) and rotated by a HWP at 45
◦ . The final state of
the photon pair is a coherent superposition of the two considered cases: |V s 3 |V i 4
and |H s 3 |H i 4 , that is the Bell state |
+
:
|
+
=
|H s 3 |H i 4 + |V s 3 |V i 4
√
2
.
(4.11)
In order to produce a four-party GHZ state, one photon from each Bell pair (4.11) is
combined in a PBS before transmitting the photons to Alice and the Bobs (Fig. 4.2b).
Indeed, the state of the four photons after interfering at the PBS reads:
1
2
|H H H H AB 1 B 2 B 3 + |H A |0 B 1 |H V B 2 |V B 3 + |V A |H V B 1 |0 B 2 |H B 3
+|V V V V AB 1 B 2 B 3
,
(4.12)
where |0 indicates the vacuum state and |H V B 2 that there are two photons in path
B 2 with orthogonal polarizations. Thus, the parties can post-select the GHZ state
(|H H H H + |V V V V )/
√
2 by discarding all the events where one of them did not
have a detection, which occurs with probability 1 /2.
After interfering one photon per source in the PBS, the four photons are coupled to
single-mode fibres of total length up to 50 km and sent to the four parties. Each party
measures the incoming photon either in the Z basis ({|H , |V }) for key generation or
in the X basis ({|++, |−−}) for parameter estimation. Only the events with coincident
detections for all parties are retained, since those correspond to the post-selection of
a GHZ state.
The measurement setup of each party is depicted in Fig. 4.3. Each party measures
the incoming signal in either the X , Y or Z basis by using a quarter-wave plate
(QWP), a HWP and a PBS followed by two superconducting nanowire single-photon
detectors labelled D t and D r . For the Z and X measurements, the QWP is oriented
at 0
◦ and the HWP is oriented at 0
◦ and 22.5
◦ , respectively. A click in D t (D r )
indicates that the photon’s polarization has been projected on |H or |++ (|V or
5 In a polarizing beam splitter, light polarized horizontally is transmitted while light polarized
vertically is reflected.
65
polarizing beam splitter (PBS)
5 and exits the PBS in path 2. Here, a half-wave
plate (HWP) oriented at 45
◦ rotates the photon’s polarization to |H before entering
the non-linear crystal. The photon undergoes SPDC in the PPKTP crystal and is
converted to two photons of lower energy in path 1 (red line), orthogonally polarized,
called signal and idler: |H s 1 |V i 1 . Similarly, an |H photon from the laser source gets
transmitted at the PBS and is down-converted to signal and idler, |H s |V i , circulating
the loop counter-clockwise. Their polarization gets rotated by the HWP in path 2 to:
|V s 2 |H i 2 . At this point, the two counter-propagating photon pairs interfere at the
PBS by entering through ports 1 and 2, respectively. The photon pair entering in 1
exits the PBS in the state |H s 3 |V i 4 , while the pair entering from 2 is mapped to
|V s 3 |H i 4 . The down-converted photons exiting in 3 are separated from the input
signal with a dichroic mirror (DM) and rotated by a HWP at 45
◦ . The final state of
the photon pair is a coherent superposition of the two considered cases: |V s 3 |V i 4
and |H s 3 |H i 4 , that is the Bell state |
+
:
|
+
=
|H s 3 |H i 4 + |V s 3 |V i 4
√
2
.
(4.11)
In order to produce a four-party GHZ state, one photon from each Bell pair (4.11) is
combined in a PBS before transmitting the photons to Alice and the Bobs (Fig. 4.2b).
Indeed, the state of the four photons after interfering at the PBS reads:
1
2
|H H H H AB 1 B 2 B 3 + |H A |0 B 1 |H V B 2 |V B 3 + |V A |H V B 1 |0 B 2 |H B 3
+|V V V V AB 1 B 2 B 3
,
(4.12)
where |0 indicates the vacuum state and |H V B 2 that there are two photons in path
B 2 with orthogonal polarizations. Thus, the parties can post-select the GHZ state
(|H H H H + |V V V V )/
√
2 by discarding all the events where one of them did not
have a detection, which occurs with probability 1 /2.
After interfering one photon per source in the PBS, the four photons are coupled to
single-mode fibres of total length up to 50 km and sent to the four parties. Each party
measures the incoming photon either in the Z basis ({|H , |V }) for key generation or
in the X basis ({|++, |−−}) for parameter estimation. Only the events with coincident
detections for all parties are retained, since those correspond to the post-selection of
a GHZ state.
The measurement setup of each party is depicted in Fig. 4.3. Each party measures
the incoming signal in either the X , Y or Z basis by using a quarter-wave plate
(QWP), a HWP and a PBS followed by two superconducting nanowire single-photon
detectors labelled D t and D r . For the Z and X measurements, the QWP is oriented
at 0
◦ and the HWP is oriented at 0
◦ and 22.5
◦ , respectively. A click in D t (D r )
indicates that the photon’s polarization has been projected on |H or |++ (|V or
5 In a polarizing beam splitter, light polarized horizontally is transmitted while light polarized
vertically is reflected.