Unextendible product bases (UPBs) play an important role in quantum information theory. However, very little is known about UPBs in Hilbert space of local dimension more than three. In this paper, we study the UPBs in qutrit-ququad system and find that there only exist six, seven and eight-state UPBs. We completely characterize the six-state and seven-state UPBs. For eight-state UPBs, seven classes of UPBs are found. As auxiliary results, we study the distinguishability of qutrit-ququad UPBs by separable measurements, and find that there exists a UPB that cannot be distinguished.Unextendible product bases (UPBs) play an important role in quantum information theory. However, very little is known about UPBs in Hilbert space of local dimension more than three. In this paper, we study the UPBs in qutrit-ququad system and find that there only exist six, seven and eight-state UPBs. We completely characterize the six-state and seven-state UPBs. For eight-state UPBs, seven classes of UPBs are found. As auxiliary results, we study the distinguishability of qutrit-ququad UPBs by separable measurements, and find that there exists a UPB that cannot be distinguished.

Scientific ReportsJul 14, 201532 citations

Nonseparability

Bernard d’Espagnat

Springer eBooksJan 1, 19830 citations

Unextendible product bases from orthogonality graphs

Shi, Fei, Bai, Ge, Xiande Zhang +2 more

Unextendible product bases (UPBs) play a key role in the study of quantum entanglement and nonlocality. A famous open question is whether there exist genuinely unextendible product bases (GUPBs), namely multipartite product bases that are unextendible with respect to every possible bipartition. Here we shed light on this question by providing a characterization of UPBs and GUPBs in terms of orthogonality graphs. Building on this connection, we develop a method for constructing UPBs in low dimensions, and we derive a lower bound on the size of any GUPB, significantly improving over the state of the art. Moreover, we show that every minimal GUPB saturating our bound must be associated to regular graphs. Finally, we discuss a possible path towards the construction of a minimal GUPB in a tripartite system of minimal local dimension.Unextendible product bases (UPBs) play a key role in the study of quantum entanglement and nonlocality. A famous open question is whether there exist genuinely unextendible product bases (GUPBs), namely multipartite product bases that are unextendible with respect to every possible bipartition. Here we shed light on this question by providing a characterization of UPBs and GUPBs in terms of orthogonality graphs. Building on this connection, we develop a method for constructing UPBs in low dimensions, and we derive a lower bound on the size of any GUPB, significantly improving over the state of the art. Moreover, we show that every minimal GUPB saturating our bound must be associated to regular graphs. Finally, we discuss a possible path towards the construction of a minimal GUPB in a tripartite system of minimal local dimension.

arXiv:2303.02553Mar 5, 20230 citations

Time ordered products

WORLD SCIENTIFIC eBooksMar 30, 20170 citations

Graph-theoretic quantum contextuality and unextendible product bases

Gurvir Singh, Arvind

Unextendible product bases(UPBs) are central to the study of local distinguishability of orthogonal product states. While their connection to quantum nonlocality via Bell inequalities is well established, their link to quantum contextuality remains largely unexplored. We establish a graph theoretic connection between contextuality and UPBs. First, an equivalence between Klyachko-Can-Binicioğlu-Shumovsky (KCBS) vectors and the Pyramid UPB is shown and then by constructing a one parameter family of UPB vectors, a quantitative connection between `contextuality strength' and bound entanglement of states associated with the corresponding UPB is demonstrated. This equivalence is extended to generalized KCBS vectors and the GenPyramid UPB. A new class of minimal UPBs in $\mathbb{C}^3 \otimes \mathbb{C}^n$ is constructed using Lovász-optimal orthogonal representations (LOORs) of cycle graphs and their complements which we term the GenContextual UPB. Any minimal UPB in this dimension is shown to be graph-equivalent to the GenContextual UPB. We briefly discuss the distinguishability properties of GenContextual UPB. In the reverse direction, we observe that the constituent vectors of the QuadRes UPB are LOORs of Paley graphs. The structural properties of these graphs make them suitable candidates for constructing noncontextuality inequalities, thereby establishing a bidirectional connection between quantum contextuality and UPBs.Unextendible product bases(UPBs) are central to the study of local distinguishability of orthogonal product states. While their connection to quantum nonlocality via Bell inequalities is well established, their link to quantum contextuality remains largely unexplored. We establish a graph theoretic connection between contextuality and UPBs. First, an equivalence between Klyachko-Can-Binicioğlu-Shumovsky (KCBS) vectors and the Pyramid UPB is shown and then by constructing a one parameter family of UPB vectors, a quantitative connection between `contextuality strength' and bound entanglement of states associated with the corresponding UPB is demonstrated. This equivalence is extended to generalized KCBS vectors and the GenPyramid UPB. A new class of minimal UPBs in $\mathbb{C}^3 \otimes \mathbb{C}^n$ is constructed using Lovász-optimal orthogonal representations (LOORs) of cycle graphs and their complements which we term the GenContextual UPB. Any minimal UPB in this dimension is shown to be graph-equivalent to the GenContextual UPB. We briefly discuss the distinguishability properties of GenContextual UPB. In the reverse direction, we observe that the constituent vectors of the QuadRes UPB are LOORs of Paley graphs. The structural properties of these graphs make them suitable candidates for constructing noncontextuality inequalities, thereby establishing a bidirectional connection between quantum contextuality and UPBs.

arXiv:2510.26719Physical Review AApr 20, 20260 citations

\mathbb{C}^3 \otimes \mathbb{C}^n

is constructed using Lovász-optimal orthogonal representations (LOORs) of cycle graphs and their complements which we term the GenContextual UPB. Any minimal UPB in this dimension is shown to be graph-equivalent to the GenContextual UPB. We briefly discuss the distinguishability properties of GenContextual UPB. In the reverse direction, we observe that the constituent vectors of the QuadRes UPB are LOORs of Paley graphs. The structural properties of these graphs make them suitable candidates for constructing noncontextuality inequalities, thereby establishing a bidirectional connection between quantum contextuality and UPBs.