Error-space geometry and positivity orders of encoded reduction maps
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We study tensor products of reduction maps after restriction to perfect-tensor encodings. An error-support expansion of the unnormalized Choi matrix accommodates multiple logical reference parties and arbitrary complex absolutely maximally entangled states. For one logical party, odd physical length n and 0<t<=1, averaging orthogonal error families over fixed-size site subsets gives a completely positive output complement under the sufficient condition d>=2n/t. The encoded map then has an exact Schmidt-rank sign criterion and optimal dimension-uniform positivity order floor(t^(-n)). At t=1/2, a separate estimate covers d>=2n. For two logical parties and four physical outputs, the half-parameter map is completely positive exactly in the admissible dimensions 2<=d<=12. In dimensions d>=13 it is not completely positive, but is k-positive at least through floor(2d(d-2)/(d-6)). Explicit polynomial-evaluation codes give failure at rank9d for every prime d>=43, establishing linear growth up to unmatched constants for those examples. Code existence, the Schmidt-overlap criterion and the reduction-map family are prior ingredients. The results retain substantive code-support restrictions and belong to the completely copositive class.
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Cite this paper
Lluis Eriksson (2026). Error-space geometry and positivity orders of encoded reduction maps. AIRR.SCIENCE, ARR-2026-5K8R1DT82Q95JSHY, v1. https://airr.science/papers/ARR-2026-5K8R1DT82Q95JSHY/versions/v1/ Research preprint; ARR-2026-5K8R1DT82Q95JSHY v1. Not peer reviewed by AIRR. Working paper; not admitted to the AIRR accepted collection.
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Native corrections and notes
- O1 · resolved · p.1 abstract; main.tex:22: Explicit 0<t<=1 aligns abstract with Theorem4; t=2 counterexample no longer applies. Separate t=1/2 domain preserved.
- O2 · resolved · p.2 attribution, p.9 references7-9; main.tex:39,398,403,408: Faithful comparison with CK0809.4909v1 Theorems1-2, BC1612.09598v1 section4.2 Theorem4.3, MOM2104.14058v5 Proposition3.1 and Theorem3.2; general method attributed and application-specific quantities distinguished.
- O3 · resolved · p.1 second contribution; main.tex:36: Maximal entanglement now quantified over each fixed E_i, explicitly not all superpositions in ranQ1; agrees with unchanged Theorem6.
- N1 · p.8 disclosure; main.tex:366: GPT-6 Astra xhigh is the requested/declared configuration; actual runtime is not independently attested here. No evidence of false identification was found.
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