发表机构
Stanford University（斯坦福大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究提出多副本量子秘密共享变体，证明不可克隆定理不再精确刻画可行性，并引入团路径障碍，在阈值下给出上下界，于k=2时匹配，为量子纠错提供新限制。
AI 中文摘要
秘密共享在密码学中无处不在。所有可能的访问结构在经典情况下都是可行的，并且在阈值访问结构的情况下，协议甚至非常高效。然而，当进入量子领域时，不可克隆定理表明许多访问结构是不可能的。事实上，不可克隆恰好刻画了可行性：对于阈值，当且仅当阈值 $t$ 严格大于 $n/2$ 时，量子秘密共享才是可能的。在这项工作中，我们提出了量子秘密共享（QSS）的一种变体，其中提供输入状态的两个或更多相同副本，但输出仅要求恢复一个副本。这一概念绕过了简单的单副本不可克隆障碍，尽管自然的 $k$ 副本推广仍然给出温和得多的障碍。对于使用 $k$ 个副本的阈值，不可克隆意味着当 $t\leq n/(k+1)$ 时，QSS 是不可能的。人们很容易假设不可克隆继续精确刻画多副本情况。然而，我们表明事实并非如此。我们给出了积极结果，表明多个副本允许略微超越单副本障碍：对于阈值，当 $t>(n-k+1)/2$ 时，我们构造 QSS。另一方面，我们提出了一种新颖的障碍，称为团路径障碍，它适用于任意访问结构。对于阈值，它表明当 $t\leq (n-1)/k$ 时，QSS 是不可能的。我们的上界和下界在 $k=2$ 时完全匹配。我们的两个结果都利用了与从访问结构导出的某些图的 $k$ 可着色性的联系。我们将 $k\geq 3$ 副本的差距闭合留作未来工作的一个引人入胜的方向。秘密共享与纠错密切相关，纠错也可以在多副本设置中考虑。我们的结果暗示了擦除信道量子纠错的类似障碍。
英文摘要
Secret sharing is ubiquitous throughout cryptography. All possible access structures are classically feasible, and in the case of threshold access structures, the protocols are even very efficient. However, when moving to the quantum setting, the no-cloning theorem shows that many access structures are impossible. In fact, no-cloning exactly characterizes feasibility: for thresholds, quantum secret sharing is possible if and only if the threshold $t$ is strictly more than $n/2$. In this work, we propose a variant of quantum secret sharing (QSS) where two or more identical copies of the input state are provided, but the output is only required to recover one copy. This notion circumvents the simple one-copy no-cloning obstruction, though the natural $k$-copy generalization still gives much milder obstructions. For thresholds using $k$ copies, no-cloning implies that QSS is impossible whenever $t\leq n/(k+1)$. It is tempting to hypothesize that no-cloning continues to exactly characterize the many-copy case. However, we show that this is not the case. We give positive results showing that multiple copies allow for going slightly beyond the single-copy obstruction: for thresholds, we construct QSS whenever $t>(n-k+1)/2$. On the other hand, we give a novel obstruction we call the Clique Path obstruction, which applies to arbitrary access structures. For thresholds, it shows that QSS is impossible whenever $t\leq (n-1)/k$. Our upper and lower bounds exactly match for $k=2$. Both our results leverage connections to the $k$-colorability of certain graphs derived from the access structure. We leave closing the gap for $k\geq 3$ copies as a fascinating direction for future work. Secret sharing is closely related to error correction, which can also be considered in the many-copy setting. Our results imply similar obstructions for quantum error correction for erasure channels.
