Cryptography Reference
In-Depth Information
It has been shown that any (
k, n
)
threshold quantum state sharing schemes
with
n
1)
scheme [9,14]. We can therefore restrict our analysis to the special case of
(
k,
2
k
≤
2
k
−
1 can be achieved by throwing away shares in a (
k,
2
k
−
1) threshold secret sharing without any loss of generality. We may
also restrict our analysis to any set of states that span Hilbert space, since an
arbitrary state can be constructed from a linear combination of such states.
Here we consider coherent states, since they span Hilbert space and can be
readily obtained and manipulated in an experimental setting. A thorough
analysis of (
k,
2
k
−
1) threshold quantum state sharing of coherent states can
be found in the papers of Tyc et al. [14,20]. To provide the clearest possible
analysis, and due to its relevance to the experimental work presented later in
this chapter, we restrict ourselves henceforth to the simplest nontrivial case,
(2, 3) threshold quantum state sharing.
−
8.4 Implementation of a (2,3) Quantum
State Sharing Scheme
In the continuous variable regime, it is convenient to represent the quan-
tum states using the Heisenberg picture of quantum mechanics. The analysis
presented in this chapter will be undertaken in this picture. In particular, we
consider states at the frequency sidebands of an electromagnetic field. A quan-
tum state is associated with the field annihilation operator
a
X
+
+
i X
−
)/
=
(
2,
X
±
are the amplitude (
where
+
) and phase (
−
) quadratures. These quadra-
X
±
X
±
components
tures are expanded into steady state
and fluctuating
δ
respectively, with
X
±
=
X
±
+
δ
X
±
. The variance of the quadrature operator
X
±
)
is given by
V
±
=
(δ
2
.
8.4.1 The Dealer Protocol
In the dealer protocol for the (2,3) quantum state sharing scheme proposed by
Tyc and Sanders, an entangled state is utilized to encode and distribute the se-
cret quantum state to the players. One way in which this entangled state may
be generated is by interfering two amplitude quadrature squeezed beams on
a 1:1 beam splitter (x:y beamsplitter with reflectivity
x
/(
x
+
y
)
and transmit-
tivity
y
2, as shown in Figure 8.4.
Amplitude squeezed beams are so called because they exhibit an amplitude
quadrature noise variance below the quantum noise limit, while correspond-
ingly the phase quadrature has fluctuations over the quantum noise limit. The
two output beams resulting from the interference of the amplitude squeezed
beams are entangled and can be expressed as
/(
x
+
y
)
with a relative phase shift of
π/
√
2
=
(
+
)/
a
EPR1
a
sqz1
i a
sqz2
(8.1)
√
2
a
EPR2
=
(
i a
sqz1
+
a
sqz2
)/
(8.2)
where
a
sqz1
and
a
sqz2
are the annihilation operators of the amplitude quadra-
ture squeezed beams. The signature of this form of entanglement is that
