Progress in the study of the (non)existence of genuinely unextendible product bases

TL;DR

The study addresses whether genuinely unextendible product bases (GUPBs) exist and, in particular, whether a 13-element three-qutrit GUPB can occur. It develops a graph-theoretic program that maps UPBs to orthogonality graphs and uses forbidden induced subgraph (FIS) Characterizations together with faithful orthogonal representations (FOR) to prune candidates, combined with saturation tests on local-vector spanning properties. The main result shows that no 13-element three-qutrit GUPB exists, after ruling out all potential LOGs that admit FOR$(3)$ and satisfy span constraints; the paper also reports partial insights for larger bases and ququart subsystems. This approach clarifies the landscape of minimal GUPBs and highlights both the promise and computational limits of forbidden-subgraph techniques for resolving GUPB existence in higher dimensions. The findings connect combinatorial graph theory with quantum information constraints, offering a framework for future extensions to broader multipartite settings and higher local dimensions.$

Abstract

We investigate the open problem of the existence of genuinely unextendible product bases (GUPBs), that is, multipartite unextendible product bases (UPBs) which remain unextendible even with respect to biproduct vectors across all bipartitions of the parties. To this end, we exploit the well-known connection between UPBs and graph theory through orthogonality graphs and orthogonal representations, together with recent progress in this framework, and employ forbidden induced subgraph characterizations to single out the admissible local orthogonality graphs for GUPBs. Using this approach, we establish that GUPBs of size thirteen in three-qutrit systems-the smallest candidate GUPBs-do not exist. We further provide a partial characterization of graphs relevant to larger bases and systems with ququart subsystems.

Figures (22)

• Figure 1: (left) There is necessarily at least on repeating vector in a FOR(3) of the square graph (Lemma \ref{['4cykl']}) -- for $w_2$ and $w_4$ (green) or $w_1$ and $w_3$ (yellow), or both. (right) There is necessarily exactly one pair of equal vectors, for $w_2$ and $w_4$ (brown), in a FOR(3) of the diamond graph.
• Figure 2: (left) $C_5$ is the unique $4$-regular graph on $5$ vertices. Trivially, no FOR(3) exists for this graph (indicated by the cross mark). (right) The unique $4$-regular graph $D_6$ on $6$ vertices. This graph admits a FOR(3) (indicated by the checkmark) with necessarily repeating vectors for the colored vertices.
• Figure 3: Two non-isomorphic $4$-regular graphs on $7$ vertices: $D_{7,a}$ (left) and $D_{7,b}$ (right). $D_{7,a}$ admits a FOR(3) with the same vectors for $v_3, v_4,v_5$; there is no FOR(3) for $D_{7,b}$ as $H_5$ is its induced subgraph (highlighted in red).
• Figure 4: (left) $4$-clique $C_4$. Graphs with $4$-cliques obviously do not have FORs(3). (right) $C_4$ highlighted in red in $M_{4}$.
• Figure 5: (left) House graph $H_5$ --- a $5$-vertex graph without a FOR(3) [Lemma \ref{['grafy-bez-for']} (b)]. (right) $H_5$ highlighted in $M_{9048}$.

Theorems & Definitions (14)

• Definition 1
• Definition 2
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof