Quantum State Sharing - Quantum Communications and Cryptography - page 168

tapping private encrypted communications but still allowing some
level of protection against unauthorized tapping [7]. To achieve this,
the key used by the private parties to communicate would be shared
between two governmental escrow agencies. Both agencies would
then need to be contacted independently in order to obtain the cryp-
tographic key to tap the communications, thus providing a basic level
of protection to the citizens.

• Electronic voting. The privacy of an electronic voting procedure can be

ensured by secret sharing [6]. Here is a short explanation on how to
proceed. First, the voters need to split their vote into k shares and give
them to a group of trusted tallying authorities (who can also verify
the identity and uniqueness of the voter). At this point no tallying
authority knows what has been voted. These tallying authorities will
then count their partial votes independently and send their results to a
central authority. Finally, the central authority obtains the final result
by concatenating all the partial results together.

8.3 Translating Secret Sharing
to the Quantum Domain
As we have seen in the previous section, secret sharing is an important crypto-
graphic protocol designed to distribute secret information to n players, where
certain subsets, the access structure , can be trusted, and all other subsets, the
adversary structure , cannot be trusted.
In quantum information processing, the objective in a multiplayer system
is not to distribute information but rather to distribute quantum states to
the players; hence we employ the term quantum state sharing to describe a
quantum version of secret sharing. Now the dealer distributes a pure quantum
state, or alternatively a mixed state density operator, rather than some classical
information to the players. Nevertheless, the properties of classical secret
sharing as seen in the previous section are still applicable, except that, as a
result of the no-cloning theorem [8], a majority of the players must collaborate
to extract the state. This imposes the limitation n

≤ 2 k − 1 on quantum state

sharing.
Our employment of the term quantum state sharing follows the use of the
term quantum secret sharing by Cleve et al. [9], which corresponds to the quan-
tum version of secret sharing in cryptography developed by Shamir [1] and
Blakley [4]. However, we prefer to use the term quantum state sharing to quan-
tum secret sharing , as the latter term has been employed for another purpose:
protected dissemination of quantum states between completely trustworthy
parties in a hostile environment [2,10, 11]. These schemes correspond to our
( n, n

)
quantum state sharing. Hillery et al. [2] were the first to propose such a
scheme using discrete variable GHZ states. They were followed by Karlsson

)

threshold quantum state sharing, which is a trivial case of general ( k, n