Cryptography Reference
In-Depth Information
p
e
is modified correspondingly. Figure 5.1 illustrates the dependence of the er-
ror probability
p
e
on
x,
. The probability distributions for quantum states
prepared as 0 and 1 are overlapping, and the degree of the overlap is defined
by
α
,
η
exhibits a certain pattern, which can be ex-
ploited in the postselection process taking into account the independence of
Bob's and Eve's measurement outcomes
x
. For Bob's measurement results
x
falling in the region around the zero point, the error probability of his decision
on 0 or 1 on the basis of his measurement is high. On the left and right wings
the error probability decreases as the results might be assigned to 0 or 1 with
more certainity.
The mutual information between Alice and Bob,
I
AB
, can be determined
and depends on Alice's amplitude
α
,
η
. The dependence
p
e
(
x
)
|
α
|
, Bob's amplitude measurement result
x
, and the transmission
η
of the channel between Alice and Bob [20,24,28]:
=
+
(
α
η)
(
α
η)
I
AB
1
p
e
x,
,
log
2
p
e
x,
,
+
(
1
−
p
e
(
x,
α
,
η))
log
2
(
1
−
p
e
(
x,
α
,
η)).
(5.4)
It is not possible yet to give a general statement about the security of the
protocol. However, it is possible to show the security of the scheme against
beam-splitting attacks, which are the basic kinds of attacks for a potential
eavesdropper (Eve) who wants to utilize nonzero quantum channel losses.
Eve could split off the part of the signal that is lost in the channel with trans-
mitivity
to obtain that
part of Alice's signal that would be lost normally. Eve transmits the rest of the
signal over a perfect (lossless) channel, so that her presence is undetectable.
She can then make measurements on her part of the signal, e.g., an amplitude
measurement like Bob's, and try to infer Bob's measurement results from her
own results. For coherent states, Eve's and Bob's probability distributions of
measurement outcomes are independent. This means that Eve's and Bob's
measurements are completely uncorrelated, so it is possible that Eve obtains
inconclusive results while Bob is quite sure about the state Alice prepared,
and vice versa. The average information Eve can get depends on Alice'spre-
pared amplitude
η<
1. She uses a beam splitter with reflectivity of 1
−
η
), giving
a mutual information of
I
AE
between Alice and Eve. It can be shown [20] that
the mutual information between Alice and Eve depends only on the effective
amplitude of the prepared nonorthogonal coherent states and is
independent
of the measurement outcomes
x
of her and of Bob:
α
and Eve's portion of the signal (in this case 1
−
η
2
1
log
1
1
1
1
I
AE
=
+
−
f
2
(α
,
η)
+
−
f
2
(α
,
η)
2
1
log
1
,
1
1
1
+
−
−
f
2
(α
,
η)
−
−
f
2
(α
,
η)
(5.5)
where
f
is the overlap of each pair of the four signal states. Alice and
Bob can determine the error probability and shared information
I
AB
(Equa-
tion (5.4)) for each single event
x
after the measurement. For a given channel
transmission
(α
,
η)
η
and state overlap governed by
|
α
|
, a lower limit for Bob's mea-
surement result
x
can be given, where
I
AB
>
I
AE
. On this basis, Bob decides on
