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Graph theoretic quantum contextuality and unextendible Product Bases

Gurvir Singh, Arvind

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.

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 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.

Paper Structure

This paper contains 4 theorems, 22 equations, 1 figure, 2 tables.

Key Result

Theorem 1

The product of LOORs of odd cycle graphs (loor_cn) and their complements (loor_bar_cn) given by vectors, $|\phi_j\rangle= |u_j\rangle \otimes |v_j\rangle$ constitute a class of minimal UPBs in $\mathbb{C}^3 \otimes \mathbb{C}^{n-2}$ for odd $n$.

Figures (1)

  • Figure 1: Orthogonality graph of UPBs in $\mathbb{C}^3 \otimes \mathbb{C}^3$. The KCBS vectors also exhibit the same pentagonal ($C_5$) orthogonality structure. Vertices label the five product states $|\alpha_i\rangle$ and edges encode orthogonality. Thick edges (Party 1) form a pentagon $C_5$ with cycle $0\!\to\!2\!\to\!4\!\to\!1\!\to\!3\!\to\!0$. The overall graph is the complete graph $K_5 = C_5 \cup \overline{C}_5$, with Party 1 forming the pentagon ($C_5$) and Party 2 its complement $\overline{C}_5$. Note that $C_5 \cong \overline{C}_5$.

Theorems & Definitions (9)

  • Theorem 1
  • proof
  • Lemma 1: Sufficient and Necessary Conditions for a UPB Fei2023
  • Definition 1: Graph Equivalence of UPBs Lovasz2009
  • Lemma 2
  • proof
  • Theorem 2
  • proof
  • Definition 2: Quadric Residues DiVincenzo2003