We consider a secret sharing setting with a monotone access structure involving a control node and $L$ users, connected via a classical-quantum broadcast channel whose input is controlled by the control node, referred to as the dealer. Unlike traditional secret sharing settings, where the dealer fully controls the shares given to each user, in our model, the dealer encodes the secret for transmission over the broadcast channel. This means that the shares received by users are perturbed by the channel and are not fully controlled by the dealer. Our main results are achievable one-shot secret sharing rates, as well as converse bounds for arbitrary monotone access structures. We further derive second-order and asymptotic achievable rates for arbitrary monotone access structures. In the special case where all shares are required to recover the secret, we show that our result coincides with the existing secret sharing capacity over classical channels.

翻译：我们考虑一种秘密共享场景，该场景具有包含一个控制节点和$L$个用户的单调访问结构，这些节点通过一个经典-量子广播信道连接，其输入由控制节点（称为分发者）控制。与传统秘密共享场景中分发者完全控制给予每个用户的份额不同，在我们的模型中，分发者对秘密进行编码以通过广播信道进行传输。这意味着用户接收到的份额会受到信道的扰动，且不完全受分发者控制。我们的主要成果是实现了任意单调访问结构下的可达一次性秘密共享速率，以及相应的逆界。此外，我们针对任意单调访问结构推导了二阶和渐近可达速率。在需要所有份额才能恢复秘密的特殊情况下，我们证明了我们的结果与经典信道上现有的秘密共享容量一致。