Multipartite N–Q–S: Spectator Bounds, Product Lower Bounds, and Mixing in Instrument Networks

Summary:
This preprint develops a multipartite N-Q-S framework for sequential quantum instruments. We give (i) spectator bounds that decompose order effects into local terms plus a nonlocal residue with an explicit norm control, (ii) a product lower bound under Doeblin-type minorization that yields quantitative convergence guarantees, and (iii) network mixing rates that propagate along an interaction graph and imply exponential convergence of the look–return loop to a unique fixed point. Two interchangeable evaluation routes are provided: Route I (operator/diamond-norm based) and Route II (probabilistic/coupling based). We focus on genuinely multipartite regimes and network aspects; bipartite equality windows and monitored continuous-time limits are treated separately in a companion “Composite” paper.

What is included:

• Main manuscript PDF: Yoshida_NQS_Multi_2025-09-23.pdf

• Online Resource 1 (ESM) PDF: Online_Resource_1.pdf, containing full proofs, constants, and worked examples (including a compact three-body instance), plus technical lemmas and derivations.

Scope and limitations:

• Finite-dimensional instrument networks and operationally verifiable inequalities.

• We do not attempt to resolve foundational questions beyond the operational scope (e.g., full sector-selection or infinite-dimensional QDS theory).

Reproducibility:

• No external datasets are required. All figures use synthetic data generated from the definitions in the text. Pseudocode and proof details are provided in Online Resource 1.