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

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.

References in corpus (29)

• Mixed-state entanglement and distillation: is there a ``bound'' entanglement in nature?
• Quantum Nonlocality without Entanglement
• Unextendible Product Bases and Bound Entanglement
• Unextendible Product Bases, Uncompletable Product Bases and Bound Entanglement
• Multipartite Nonlocality without Entanglement in Many Dimensions
• Distinguishability of complete and unextendible product bases
• On the dimension of subspaces with bounded Schmidt rank
• House of Graphs 2.0: a database of interesting graphs and more
• Generic local distinguishability and completely entangled subspaces
• The Minimum Size of Unextendible Product Bases in the Bipartite Case (and Some Multipartite Cases)
• A completely entangled subspace of maximal dimension
• The Structure of Qubit Unextendible Product Bases
• Genuinely entangled subspace with all-encompassing distillable entanglement across every bipartition
• From unextendible product bases to genuinely entangled subspaces
• Bell inequalities with no quantum violation and unextendible product bases
• Tight Bell inequalities with no quantum violation from qubit unextendible product bases
• Strong quantum nonlocality for unextendible product bases in heterogeneous systems
• Entanglement of genuinely entangled subspaces and states: exact, approximate, and numerical results
• Universal construction of genuinely entangled subspaces of any size
• Multiqubit UPB: The method of formally orthogonal matrices
• An approach to constructing genuinely entangled subspaces of maximal dimension
• Nonexistence of $n$-qubit unextendible product bases of size $2^n-5$
• Strong quantum nonlocality and unextendibility without entanglement in $N$-partite systems with odd $N$
• Unextendible and uncompletable product bases in every bipartition
• Fully non-positive-partial-transpose genuinely entangled subspaces
• A Complete Hierarchy of Linear Systems for Certifying Quantum Entanglement of Subspaces
• Negative result about the construction of genuinely entangled subspaces from unextendible product bases
• Unextendible and strongly uncompletable product bases
• X-arability of mixed quantum states