Cryptography Reference
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et al. [10], who included entanglement as a resource and also speculated
on the possibility of general (
k, n
) threshold quantum secret sharing. Tittel
et al. [11] experimentally realized the (2,2) threshold quantum secret sharing
scheme in an elegant experiment involving energy time entangled pseudo-
GHZ states. In quantum secret sharing, security against eavesdropper attacks
is of paramount importance. Ultimately such techniques offer no security
advantage over classical secret sharing used in conjunction with quantum
cryptography. Quantum state sharing, on the other hand, is concerned with
situations in which the players in the protocol are not themselves completely
trustworthy. Such a scheme was proposed by Cleve et al. [9], who gave the
full theory for (
k, n
) threshold quantum state sharing and provided a detailed
analysis of the (2,3) threshold quantum state sharing case.
Threshold quantum state sharing, in the spirit of Cleve et al., is interest-
ing for numerous applications. One illustrative example is the distribution
of quantum money. One of the first motivations for quantum information
theory was Wiesner's suggestion that quantum money could be employed,
which was impervious to counterfeiting [12]. Nowadays we have an interest
in quantum resources, such as a supply of ebits, or Bell states, but in a sense
we can think of this as quantum money since money is a representation of
resources. Quantum state sharing allows the distribution of quantum money
or quantum resources to multiple players who have to collaborate in prede-
termined ways (the access structure) in order to use, or spend, this resource.
Quantum state sharing allows the quantum money to be locked in a vault until
an access structure set with sufficient numbers of keys accesses the vault and
removes the quantum money. Other, and probably more crucial, applications
of quantum state sharing will be discussed in Section 8.6.
The Cleve et al. result is important as it provides the general protocol for
threshold quantum state sharing. The drawback is that it is hard to imple-
ment. Even the simplest nontrivial case, namely the (2,3) threshold quantum
state sharing scheme, is difficult because it relies on having three qutrits avail-
able and the capability of universal transformations on these qutrits. Whereas
qutrits and higher order qudits are hard to create and manipulate [13], quan-
tum state sharing with continuous variables is feasible as shown by Tyc and
Sanders [14], who developed continuous variable (
k, n
threshold quantum
state sharing and showed explicitly how to realize the (2,3) threshold quantum
state sharing special case. Their scheme utilized Einstein-Podolsky-Rosen
(EPR) entanglement [14]. This entanglement is an experimentally accessible
quantum resource [15,16] used in quantum information experiments such as
continuous variable quantum teleportation [17,18]. This scheme was adapted
to a practical scenario by Lance et al. [19], the details of which will be dis-
cussed in Sections 8.4 and 8.5. Moreover, Tyc et al. [20] demonstrated that,
in the general (
k, n
)
threshold quantum state sharing case, the players never
need more than a single EPR-entangled pair, which is an important cost saving
for implementation of quantum state sharing protocols.
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