AI 中文总结
该研究提出基于超立方体分解与SAT的框架，构造并验证了多体系统中多种大小的不可扩展乘积基，其小维实例可用于递归构造更大系统的相关基。
AI 中文摘要
不可扩展乘积基（UPB）是量子信息理论中的重要结构，应用于完全纠缠子空间、束缚纠缠和局域不可区分性。由于UPB的诸多性质和应用与其基数密切相关，研究UPB的核心问题之一是确定给定多体系统中是否存在指定大小的UPB。本文中，我们引入了一种基于N维超立方体分解的SAT辅助框架，定义了O_N-块分解，并证明了块到UPB的定理：每个O_N-块分解都可通过基于逐块傅里叶乘积基和全局终止态的构造诱导出一个UPB。随后，我们将此类分解的搜索编码为布尔可满足性（SAT）问题，并使用SAT求解器生成显式实例。在验证方面，我们还实现了一种基于局域正交图和不饱和子空间的UPB验证算法，该算法可用于判定任意有限个乘积态的集合是否构成UPB。利用该框架，我们在一些三体和四体系统中得到了若干大小的UPB，包括在C³⊗C³⊗C³中的大小13、14、…、23的UPB。此外，此处获得的小维实例可作为递归构造的种子UPB，从而在更大的多体系统中得到更多实例。
英文摘要
Unextendible product bases (UPBs) are important structures in quantum information theory, with applications to completely entangled subspaces, bound entanglement, and local indistinguishability. Since many properties and applications of UPBs are closely related to their cardinalities, one of the central problems in the study of UPBs is to determine whether UPBs of prescribed sizes exist in a given multipartite system. In this paper, we introduce a SAT-assisted framework based on decompositions of the \(N\)-dimensional hypercube. We define \(O_N\)-tile decompositions and prove a tile-to-UPB theorem: every \(O_N\)-tile decomposition induces a UPB through a construction based on tile-wise Fourier product bases and a global stopper state. We then encode the search for such decompositions as a Boolean satisfiability (SAT) problem and use SAT solvers to generate explicit instances. In terms of verification, we also implement a UPB verification algorithm based on local orthogonality graphs and unsaturated subspaces. The algorithm can be used to determine whether an arbitrary finite set of product states forms a UPB. Using this framework, we obtain UPBs of several sizes in some tripartite and quadripartite systems, including sizes \(13,14,\ldots,23\) in \(\mathbb C^3\otimes\mathbb C^3\otimes\mathbb C^3\). Moreover, the small-dimensional instances obtained here can serve as seed UPBs for recursive constructions, leading to further examples in larger multipartite systems.
Comments19 pages,2 figures
