Cryptography Reference
In-Depth Information
the appropriate value of the postselection threshold
x
0
to eliminate the most
inconclusive results
x
(see Figure 5.1). As already mentioned, an impor-
tant point is that for coherent states, Eve's and Bob's probability distributions
of measurement outcomes
x
are independent. Owing to this fact, the shared
knowledge between Alice and Bob about the selected events
x
<
|
x
0
|
is larger
than that shared between Alice and Eve, allowing a secret key to be distilled
(compare Equations (5.4) and (5.5)). The process of sorting out those events
with
I
AB
>
>
|
x
0
|
I
AE
, i.e., the events that are favorable for Alice and Bob after the
data have been recorded is called postselection. The detailed description of
this procedure can be found in Ref. 20, which includes the derivation of the
relevant formulae.
Postselection procedure for a secure key distillation is not limited to co-
herent state cryptography. It can be extended to enhance the security of contin-
uous variable quantum cryptography [10-17,26] in general. The postselection
analysis [20,29] was already applied to one particular QKD scheme [15] based
on the entanglement of intense beams. The error probability in (Equation (5.4))
is then determined by the ratio between the signal level and the noise level
(signal-to-noise ratio) [10], which depends on the transmitivity of the channel
and on the quality of entanglement. It was shown [29] that postselection of
the data for such an entanglement-based scheme enables one to sort out the
events with
I
AB
>
I
AE
and thus to distill the secret key, even in the presence
of high losses.
5.3 Polarization Encoding
Continuous variable cryptography with coherent states and postselection can
be implemented with traditional polarization variables of the BB84 proto-
col. The role of two incompatible nonorthogonal bases is then taken up by
the noncommuting quantum polarization variables, Stokes parameters. The
hermitian Stokes operators [30,31] are defined as quantum versions of their
classical counterparts [32]:
S
0
=
a
x
a
x
+
a
y
a
y
=
n
x
+
n
y
=
n
(5.6)
S
1
=
a
x
a
x
−
a
y
a
y
=
n
x
−
n
y
(5.7)
S
2
a
x
a
y
a
y
a
x
=
+
(5.8)
a
y
a
x
(5.9)
where the
x
and
y
subscripts label the creation, destruction, and number
operators of quantum harmonic oscillators associated with the
x
and
y
photon
polarization modes, and
n
is the total photon number operator. The creation
and destruction operators have the usual commutation relations,
a
j
, a
k
=
δ
i
a
x
a
y
S
3
=
−
j, k
=
x, y
.
(5.10)
jk
The Stokes operator
S
0
commutes with all the others:
[
S
0
, S
i
]
=
0
i
=
1
,
2
,
3
(5.11)
