The information rate is an important metric of the performance of a quantum-secret-sharing scheme.In this paper,we characterize the quantum access structures by means of the theory of hypergraph.Furthermore,we derive the optimal information rate and the construction of perfect quantum-secret-sharing schemes corresponding to the quantum access structures with at most four players which are given in terms of the relationship between certain access structures and hypergraphs.The exact values for the optimal information rate in 5 of the 13 access structures are computed and the relevant construction of perfect secret sharing schemes is discussed.At the same time,the upper bounds for the information rate of other 8 quantum access structures are computed.
In a secret sharing scheme, a secret s is divided into multiple shares and distributed among n participants. As more shares are collected, the mutual information between the secret and the shares increases monotonically. In the perfect case, the best known explicit lower bound on the share size is given by
k2k−1⋅H(s), where H(s) is the entropy of the secret and k is the largest integer such that n ≥ 2k+k−2. This paper considers the non-perfect case, in which each share reveals at most a
L1 fraction of the secret's entropy, for an integer L > 1. A prior result in this setting provides a lower bound of
kL2k−1⋅H(s), under the condition n ≥ 2k+k+L−3. We provide a new lower bound that approaches the perfect-case bound, namely
k2k−2+1/L⋅H(s), which holds under the condition n ≥ 2k+k+2L−4. The proof generalizes perfect-case techniques by increasing the number of participants by 2(L−1). These results refine our understanding of the trade-off between share size and information leakage in non-perfect secret sharing. In particular, we construct non-perfect access structures that nearly attain the perfect lower bound with fewer participants than previous non-perfect bounds, thus complementing and extending prior work.