This is an expository reference on the mathematical structure of quantum entanglement.It contains no new results. Part I treats the nite-dimensional theory, where entanglement isdened by the failure of a convex tensor decomposition and is computable in principle. PartII treats the innite-dimensional and algebraic setting. For sharp local regions in relativisticquantum eld theory, the tensor-product and reduced-density-matrix framework generally fails:under the usual phase-space and scaling assumptions, local observable algebras are typicallyhypernite factors of type III1, with no intrinsic normal seminite trace, no intrinsic localdensity matrix, and no reduced-state von Neumann entropy. Part III develops the structuresthat replace that framework TomitaTakesaki modular theory, Araki relative entropy, and thecrossed-product construction. The continuous core of a type III1 factor is type II∞ and carriesa seminite trace; in the gravitationally dressed de Sitter observer construction, the resultingobservable algebra is type II1. Statements are proved when the proof is short and otherwiseattributed precisely. The intended use is as a denitional reference: every term is pinned, andevery substantive claim is either proved here or tied to a cited result with its hypotheses stated.