A $((k,n))$ quantum threshold secret sharing (QTS) scheme is a quantum cryptographic protocol for sharing a quantum secret among $n$ parties such that the secret can be recovered by any $k$ or more parties while $k-1$ or fewer parties have no information about the secret. Despite extensive research on these schemes, there has been very little study on optimizing the quantum communication cost during recovery. Recently, we initiated the study of communication efficient quantum threshold secret sharing (CE-QTS) schemes. These schemes reduce the communication complexity in QTS schemes by accessing $d\geq k$ parties for recovery; here $d$ is fixed ahead of encoding the secret. In contrast to the standard QTS schemes which require $k$ qudits for recovering each qudit in the secret, these schemes have a lower communication cost of $\frac{d}{d-k+1}$ for $d>k$. In this paper, we further develop the theory of communication efficient quantum threshold schemes. Here, we propose universal CE-QTS schemes which reduce the communication cost for all $d\geq k$ simultaneously. We provide a framework based on ramp quantum secret sharing to construct CE-QTS and universal CE-QTS schemes. We give another construction for universal CE-QTS schemes based on Staircase codes. We derived a lower bound on communication complexity and show that our constructions are optimal. Finally, an information theoretic model is developed to analyse CE-QTS schemes and the lower bound on communication complexity is proved again using this model.

翻译：一个$((k,n))$量子门限秘密共享（QTS）方案是一种量子密码协议，用于在$n$个参与者之间共享量子秘密，使得任何$k$个或更多参与者能够恢复秘密，而$k-1$个或更少参与者对秘密一无所知。尽管对这些方案已有广泛研究，但在恢复过程中优化量子通信成本的研究却很少。最近，我们首次开展了通信高效的量子门限秘密共享（CE-QTS）方案的研究。这些方案通过访问$d\geq k$个参与者进行恢复来降低QTS方案的通信复杂度；此处$d$在编码秘密前就已固定。与标准QTS方案（要求每个秘密量子位需要$k$个量子位进行恢复）不同，这些方案在$d>k$时具有更低的通信成本，为$\frac{d}{d-k+1}$。本文进一步发展了通信高效的量子门限方案理论。在此，我们提出通用CE-QTS方案，能同时降低所有$d\geq k$情况下的通信成本。我们提供了一种基于斜坡量子秘密共享的框架来构造CE-QTS和通用CE-QTS方案，并给出了另一种基于阶梯码的通用CE-QTS构造。我们推导了通信复杂度的下界，并证明了我们的构造是最优的。最后，开发了一个信息论模型来分析CE-QTS方案，并利用该模型再次证明了通信复杂度的下界。