The goal of the secret sharing is to share a message (quantum or classical) among $n$ parties in such a way, that only predetermined subsets of the participants can reconstruct it. Subsets of participants which can reconstruct the secret are called authorized. Any subset of participants which is not authorized is called unauthorized and cannot gain any information about the shared secret. In the classical secret sharing (i.e. the secret is a classical string), the only restriction is the monotonicity of authorized sets, that is, unauthorized set does not contain an authorized set. Thus access structure can be described as a list of minimal authorized sets. In quantum secret sharing (QSS), instead of sharing a classical information an unknown secret state $\ket\psi$ is shared. This imposes another restriction for access structures -- no cloning. No-cloning in this context translates to the following: If a subset of participants is authorized, it's complement is unauthorized. Otherwise two complementary sets would be able to reconstruct the secret and thus perform cloning of an unknown quantum state. No--cloning and monotonicity are the only restrictions for access structures in QSS.

The goal of this work is to explore the possibility of overcoming the no--cloning restriction by assuming that the distributing party holds $p$ copies of the state it wishes to share secretly. In other words, we will be seraching for QSS schemes, in which the distributing party encodes $\ket{\psi}^{\otimes p}$ into a n partite state in such a way, that every authorized set can reconstruct at least one secret state $\ket\psi$.