Minimum Cardinalities of Multipartite Unextendible Product Bases

Abstract

In quantum information theory, the state space of a multipartite quantum system is modeled by a tensor product. In the tensor-product space $\C^{d_1}\otimes\cdots\otimes\C^{d_p}$, a nonzero vector is a \emph{product state} if it can be written as $\ket{\varphi_1}\otimes\cdots\otimes\ket{\varphi_p}$ with $\ket{\varphi_j}\in\C^{d_j}\setminus\{0\}$. An \emph{unextendible product basis} (UPB) is a finite family of pairwise orthogonal product states such that no nonzero product state is orthogonal to all of them. UPBs play a key role in investigating quantum entanglement and nonlocal phenomena. Finding a smallest UPB is a natural extremal problem: it asks how few pairwise orthogonal product states suffice to prevent any further product state from being added. The general minimum-size problem for UPBs has been studied for over two decades since the seminal work of Alon and Lovász. Let $f_m(d_1,\ldots,d_p)$ be the minimum cardinality of a UPB in the space, and let \(f_{LB}(d_1,\ldots,d_p)=1+\sum_{j=1}^{p}(d_j-1)\) be the natural lower bound. Alon and Lovász determined exactly when \(f_m\) attains the lower bound \(f_{LB}\), but the obstructed multipartite cases remained open in general. We prove a stabilization theorem: for every non-all-qubit system with $p\ge3$, whenever the natural lower bound $f_{LB}$ is unattainable, the true minimum is exactly $f_{LB}+1$. Precisely, if the number of even local dimensions $d_i$ is positive and even, and at least one local dimension is greater than two, then \(f_m(d_1,\ldots,d_p)=f_{LB}(d_1,\ldots,d_p)+1\). The proof is built on a unified graph-theoretic framework. Our result, together with earlier work, settles the minimum size problem for UPBs in all finite quantum systems.