Graph theoretic quantum contextuality and unextendible Product Bases

TL;DR

This paper establishes a novel graph theoretic link between quantum contextuality and unextendible product bases (UPBs), revealing new insights into their structural properties and implications for quantum nonlocality and entanglement.

Contribution

It introduces a graph-based framework connecting UPBs and contextuality, including new classes of UPBs and their relation to graph representations and noncontextuality inequalities.

Findings

Equivalence between KCBS vectors and Pyramid UPB

Construction of a family of UPBs linking contextuality strength and bound entanglement

Introduction of GenContextual UPB using Lovász-optimal orthogonal representations

Abstract

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-Binicio\u{g}lu-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 Lov\'asz-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.