Classification of Small-size Quantum Secret Sharing Schemes using Uniform States

Abstract

We study the connection between quantum secret sharing (QSS) schemes and $k$-uniform states of qubits beyond the equivalence between threshold QSS schemes and AME states. Specifically, we show that $3$-uniformity is a necessary but not sufficient condition for constructing a $3$-homogeneous QSS scheme using states of qubits. To the best of our knowledge, this is the first result connecting \emph{non-threshold} QSS schemes with $k$-uniform states. As an application of our result, we classify QSS schemes for up to 7 players and provide explicit characterizations of their existence. Our results offer new insights into the role of $k$-uniform states in the design of QSS schemes (not necessarily threshold) and provide a foundation for future classifications of QSS schemes with more complex structures.

23 pages, 1 figure