bibkey: biamontebergholm2017tensornetworks authors: Jacob Biamonte; Ville Bergholm year: 2017 title: Quantum Tensor Networks in a Nutshell doi: 10.48550/arXiv.1708.00006 url: https://arxiv.org/abs/1708.00006v1 claim: Section 2.3 Connecting wires describes index contraction and distinguishes open tensors from fully contracted scalar networks, with vector and matrix examples in equations 7 and 8. strata_touched: [] license: citation-only triage: anchor

Typed contractions as a supplied mathematical operation

Primary text: arXiv v1 PDF, §2, item 3 “Connecting wires”, printed pages 4–5, equations (7)–(8). Connected indices are summed; open indices remain the output tensor. The following paragraph distinguishes vector spaces and duals and explains the preferred-basis inner-product identification used there.

The stable-relation-boundary volume §10 consumes finite contraction as an ingredient. It explicitly supplies port spaces, orthogonal representations, compatible pairings and transport intertwiners. Its same-source gluing condition comes from the existing FIB relational continuation volume §5. The tutorial does not identify unrelated port types, authorize combining incompatible source records, supply physical coherence, or guarantee efficient contraction of an arbitrary network.