Automated Construction and Verification of Unextendible Product Bases
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.