Graph Structured Operator Inequalities and Tsirelson-Type Bounds
James Tian
TL;DR
The paper addresses bounding the operator norm of bipartite tensor sums $B=\sum_i x_i\otimes y_i$ where $x_i,y_i$ are self-adjoint contractions, connecting Tsirelson-type bounds to a graph-based noncommutativity structure. It develops a dimension-free analytic framework by expanding $B^2$ into a diagonal part plus commutator/anticommutator terms, and then bounding these terms via norms of $[x_i,x_j]$ and \{x_i,x_j\}. The main contributions include exact complete-graph bounds and graph-aware extensions to sparse interaction patterns, plus weighted versions $B_c=\sum c_i x_i\otimes y_i$ with sharp bounds involving a graph-dependent constant $C(G)$ and edge-mass terms $\phi_{ij}$. The results provide quick analytic estimates that complement semidefinite and numerical methods for quantum correlations and network nonlocality, and they unify universal and graph-local inequalities with explicit dependence on commutator/anticommutator structure.
Abstract
We establish operator norm bounds for bipartite tensor sums of self-adjoint contractions. The inequalities generalize the analytic structure underlying the Tsirelson and CHSH bounds, giving dimension-free estimates expressed through commutator and anticommutator norms. A graph based formulation captures sparse interaction patterns via constants depending only on graph connectivity. The results link analytic operator inequalities with quantum information settings such as Bell correlations and network nonlocality, offering closed-form estimates that complement semidefinite and numerical methods.