Cryptography Reference
In-Depth Information
Equation (1.3). The particles fly apart along the z-axis toward the two le-
gitimate users of the channel, Alice and Bob, who, after the particles have
separated, perform measurements and register spin components along one
of three directions, given by unit vectors
a
i
and
b
j
(
i, j
=
1
,
2
,
3
)
, respectively,
a
i
and
b
j
vectors lie in the
x
-
y
plane,
perpendicular to the trajectory of the particles, and are characterized by az-
imuthal angles:
for Alice and Bob. For simplicity, both
1
2
1
3
1
1
1
2
1
3
3
. Su-
perscripts
a
and
b
refer to Alice's and Bob's analyzers, respectively, and the
angle is measured from the vertical
x
-axis. The users choose the orientation
of the analyzers randomly and independently for each pair of incoming par-
ticles. Each measurement can yield two results,
φ
=
0
,
φ
=
4
π
,
φ
=
2
π
and
φ
=
4
π
,
φ
=
2
π
,
φ
=
4
π
+
1 (spin up) and
−
1 (spin
down) and can reveal one bit of information.
After the transmission has taken place, Alice and Bob can announce in
public the orientations of the analyzers they have chosen for each particular
measurement and divide the measurements into two separate groups: a first
group for which they used different orientations of the analyzers and a sec-
ond group for which they used the same orientation of the analyzers. They
discard all measurements in which either or both of them failed to register a
particle at all. Subsequently Alice and Bob can reveal publicly the results they
obtained, but within the first group of measurements only. This allows them
to establish the value of
, which if the partic
l
es were not directly or indi-
rectly “disturbed” should be very close to
Q
2
√
2. This assures the legitimate
users that the results they obtained within the second group of measure-
ments are anticorrelated and can be converted into a secret string of bits —
the key.
An eavesdropper, Eve, cannot elicit any information from the particles
while in transit from the source to the legitimate users, simply because there
is no information encoded there. The information “comes into being” only
after the legitimate users perform measurements and communicate in public
afterwards. Eve may try to substitute her own prepared data for Alice and
Bob to misguide them, but as she does not know which orientation of the
analyzers will be chosen for a given pair of particles, there is no good strategy
to escape being detected. In this case her intervention will be equivalent to
introducing elements of
physical reality
to the spin components and will lower
−
Q
below its “quantum” value.
1.7.2 Prepare and Measure Protocols
Instead of tuning into an external source of entangled particles, Alice and
Bob may also rely on the Heisenberg uncertainty principle. Suppose a spin
2
particle is prepared in one of the four states, say spin up and down along the
vertical
x
-axis (
| ↑
,
| ↓
) and spin up and down along the horizontal
y
-axis
(
|→
,
|←
). Then the two
x
states
| ↑
and
| ↓
can be distinguished by one
measurement and the two
y
states
by another measurement.
The measurement that can distinguish between the two
x
states will give a
completely random outcome, when applied to distinguish between the two
y
|→
and
|←
