Cryptography Reference
In-Depth Information
8.4.3 Characterization
Quantum state sharing can be characterized in a similar manner to quantum
teleportation and other quantum information protocols concerned with quan-
tum state reconstruction. We characterize the quality of the state reconstruc-
tion using fidelity
, which measures the overlap between the
secret and reconstructed quantum states [22]. While the secret state can, in
general, be an arbitrary unknown state, we simplify the characterization by
assuming that the secret is a coherent state. Since coherent states span Hilbert
space, a demonstration of quantum state sharing with coherent states can be
directly extended to arbitrary quantum states in general. Assuming that all
fields involved have Gaussian statistics, the fidelity can be expressed in terms
of experimentally measurable parameters as
F
=
ψ
|
ρ
|
ψ
in
out
in
2
e
−
(
k
+
+
k
−
)/
4
V
out
)(
V
out
)
F
=
(
1
+
1
+
(8.13)
where we have defined
k
±
=
X
in
2
g
±
)
2
V
out
). The fidelity for the
(
1
−
/(
1
+
{
}
reconstruction protocols, since the reconstructed state is a squeezed version of
the secret state, the fid
e
lity is determined by inferring the unitary parametric
operation
1,2
}
reconstruction protocol can be determined directly. For the
{
2,3
}
and
{
1,3
X
infer
=
(
√
3
X
out
)
)
∓
1
δ
δ
on the reconstructed state. In the ideal case,
X
in
. Any one of the access structures sets can, in the ideal case of
perfect squeezing and at unitary gain, achieve perfect reconstruction of the
secret quantum state
X
infer
=
δ
δ
1; the corresponding adversary structure obtains no
information about the secret state
F
=
0.
The efficacy of the quantum state sharing scheme can be characterized
by determining the average fidelity over all access structure permutations.
It is relatively easy to show that for a general (
k, n
) threshold quantum state
sharing scheme without any entanglement resources, the maximum fidelity
averaged over all access structure permutations is
F
=
clas
avg
n
. This limit can
only be exceeded by using quantum resources. For the (2, 3) quantum state
sharing scheme this limit reduces to
F
=
k
/
clas
avg
3.
Quantum state sharing can also be characterized by measuring the signal
transfer to (
F
=
2
/
), the reconstructed state [23].
This measure provides additional information to the fidelity measure about
the efficacy of state reconstruction. Such analysis has been used to character-
ize quantum nondemolition [24] and quantum teleportation experiments [18].
Unlike the fidelity measure described above, bothT
T
), and the additional noise on (
V
T
V
are invariant to uni-
tary parametric transformations of the reconstructed state. Therefore, for the
T
and
and
V
measures, it is unnecessary to infer a unitary transform after the
{
2,3
}
and
reconstruction protocols to characterize the state reconstruction.
The signal transfer describes the signal-to-noise transfer between the se-
cret and reconstructed state for both quadratures
{
1,3
}
T
−
. It is expressed
in terms of the quadrature signal transfer coefficients
T
±
=
T
+
+
T
=
SNR
in
,
where SNR is the signal-to-noise ratio. The additional noise describes extra
quadrature noise on the reconstructed state
SNR
out
/
V
cv
V
cv
and is expressed in
terms of the conditional variances between the secret and reconstructed state
V
=
