This paper unifiedly addresses two kinds of key quantum secure tasks, i.e., quantum versions of secret sharing (SS) and symmetric private information retrieval (SPIR) by using multi-target monotone span program (MMSP), which characterizes the classical linear protocols of SS and SPIR. SS has two quantum extensions; One is the classical-quantum (CQ) setting, in which the secret to be sent is classical information and the shares are quantum systems. The other is the quantum-quantum (QQ) setting, in which the secret to be sent is a quantum state and the shares are quantum systems. The relation between these quantum protocols and MMSP has not been studied sufficiently. We newly introduce the third setting, i.e., the entanglement-assisted (EA) setting, which is defined by modifying the CQ setting with allowing prior entanglement between the dealer and the end-user who recovers the secret by collecting the shares. Showing that the linear version of SS with the EA setting is directly linked to MMSP, we characterize linear quantum versions of SS with the CQ ad QQ settings via MMSP. Further, we introduce the EA setting of SPIR, which is shown to link to MMSP. In addition, we discuss the quantum version of maximum distance separable codes.

翻译：本文通过多目标单调张成程序（MMSP）统一研究了两种关键的量子安全任务——秘密共享（SS）和对称私密信息检索（SPIR）的量子版本，MMSP是经典线性SS和SPIR协议的表征工具。SS具有两种量子扩展形式：其一是经典-量子（CQ）场景，其中待发送的秘密为经典信息，而份额为量子系统；其二是量子-量子（QQ）场景，其中待发送的秘密为量子态，份额亦为量子系统。这些量子协议与MMSP之间的关系尚未得到充分研究。我们新引入了第三种场景——纠缠辅助（EA）场景，该场景通过允许发送方与最终通过收集份额恢复秘密的接收方之间存在预先纠缠态来修改CQ场景得到。通过证明EA场景下的线性SS与MMSP直接关联，我们利用MMSP刻画了CQ和QQ场景下的线性量子SS版本。进一步，我们引入了SPIR的EA场景，并证明其与MMSP相关。此外，我们还讨论了量子版本的最大距离可分码。