Cryptography Reference
In-Depth Information
but the operators
S
1
,
S
2
, and
S
3
satisfy the commutation relations of the SU(2)
Lie algebra:
[
S
k
, S
l
]
S
m
.
=
2
i
ε
klm
(5.12)
Apart from the factor of 2 and the absence of Planck's constant, this is iden-
tical to the commutation relation for components of the angular momentum
operator. Simultaneous exact measurements of the quantities represented by
these Stokes operators are thus impossible in general, and their means and
variances are restricted by the uncertainty relations
S
1
S
2
S
3
2
,
2
,
2
,
V
2
V
3
≥|
|
3
V
1
≥|
|
1
V
2
≥|
|
(5.13)
S
j
−
S
j
2
of the
quantum Stokes parameter
S
j
. The angle brackets denote expectation values
with respect to the state of interest.
Figure 5.2 witnesses the convenience of using polarization encoding in
coherent state cryptography in place of the quadrature encoding traditional
for continuous variable schemes. In a measurement of a Stokes parameter
S
j
,
the mode with high photon number
a
x
is used as a phase reference to de-
termine the photon number in the dark mode
a
y
of orthogonal polarization.
Note that in conventional homodyne detection, the measurement of conju-
gate quadratures (e.g., amplitude and phase) of an optical mode normally
requires a separate phase reference (local oscillator). The signal and the local
oscillator are in two spatially separated modes. Thus the spatial overlap and
the phase stability limit the efficiency of such a setup. The use of the quantum
polarization of a two-mode coherent state in our cryptographic system pro-
vides a clear practical advantage of having its own built-in strong reference
where
V
j
is a convenient shorthand notation for the variance
Detector 1
Detector 1
S
detector
S
detector
0
1
-
+
Detector 2
Detector 2
PBS
PBS
Detector 1
Detector 1
S
detector
S
detector
2
3
-
-
Detector 2
Detector 2
λ
/2
22.5°
λ
/2
22.5°
λ
/4
45°
PBS
PBS
Figure 5.2
Schematical setups to detect all four Stokes parameters. (PBS: polarizing
beam splitter.)
