Graph theoretic quantum contextuality and unextendible Product Bases

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-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.

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References in corpus (21)

• Quantum Nonlocality without Entanglement
• Unextendible Product Bases and Bound Entanglement
• A simple test for hidden variables in spin-1 system
• Experimentally testable state-independent quantum contextuality
• Unextendible Product Bases, Uncompletable Product Bases and Bound Entanglement
• Graph-Theoretic Approach to Quantum Correlations
• Local orthogonality as a multipartite principle for quantum correlations
• Simple explanation of the quantum violation of a fundamental inequality
• Universality of state-independent violation of correlation inequalities for noncontextual theories
• Evaluation of convex roof entanglement measures
• Understanding entanglement as resource: locally distinguishing unextendible product bases
• Proposed experiment for testing quantum contextuality with neutrons
• Limitations on separable measurements by convex optimization
• The exclusivity principle forbids sets of correlations larger than the quantum set
• Basic exclusivity graphs in quantum correlations
• Exploring the Local Orthogonality Principle
• Bell inequalities with no quantum violation and unextendible product bases
• Experimental observation of Hardy-like quantum contextuality
• Tight Bell inequalities with no quantum violation from qubit unextendible product bases
• Local approximation for perfect discrimination of quantum states
• Local approximation of multipartite quantum measurements