The resource theory of complexness references in real quantum networks
In plain wordsIn quantum networks restricted to real numbers, the shared reference that restores the power of complex numbers can be multiplied: local real operations and messages turn one exact copy into any number, and noisy copies, which no finite batch converts exactly, still distill at unbounded rate.
The result
As the paper states it, in briefThe resource theory in which the shared two-rebit reference of Weilenmann, Gisin and Sekatski is the unit. Version 1.0.4 proves that the unit copies itself exactly and corrects a false remark.
How this was checked
- Exact certificates
- Exact checks, including the replication lemma added in version 1.0.4.
- Lean
- A Lean 4 and Mathlib project for the algebraic core.
- Peer review
- Not yet.
Every check here is internal. How we check results.
Novelty and credit
- Novelty
- New, with named precedent: the unit is the two-rebit reference of Weilenmann, Gisin and Sekatski, the skeleton of the theory transfers from multipartite entanglement theory, and the frame-code states are Wootters’ family, in Wang and Ying’s unlockable class. New, in the paper’s words: the arbitrary-code frame reading and the graded classification.
Versionsfive, each with its own DOI
| Version | Date | DOI | Notes |
|---|---|---|---|
| 1.0.4 | 10.5281/zenodo.23457692 | Correction: A remark expected asymptotic equivalence at rate one; it is false, because the unit reference copies itself exactly. Version 1.0.4 adds the exact-replication lemma and corrects the remark. PDF hosted here. | |
| 1.0.3 | 10.5281/zenodo.21911918 | Correction: Packaging: the verification suite of version 1.0.2 had a machine-local import path and a missing module, so it did not run from a clean clone. Version 1.0.3 repairs it and scopes a footnote to Corollary 7.3 to the surviving sector. | |
| 1.0.2 | 10.5281/zenodo.21911717 | Correction: The general clause of Corollary 7.3 was false for arbitrary frame codes; explicit counterexamples are given. The corollary now states the exact alignment and disjointness condition, and the guarantee for the target pair is unaffected. | |
| 1.0.1 | 10.5281/zenodo.21910329 | Correction: Version 1.0.0 called the accounting of the auxiliary descent unit-neutral at best; the paper’s own protocol is nondemolition, so each descent is net positive and catalytic. The corollary is corrected; no other statement is affected. | |
| 1.0.0 | 10.5281/zenodo.21907063 |
Paper, code and links
papers/complexness-v1.0.4.pdf, version 1.0.4, 481,779 bytes.SHA-256 3f9404d5cdde4c7d73312514bf094fb0ee8d0560d4c570416f42560b2e09edc4
- DOI
10.5281/zenodo.23457692; all versions 10.5281/zenodo.21907062
Corrections
- Version 1.0.4. A remark expected asymptotic equivalence at rate one; it is false, because the unit reference copies itself exactly. Version 1.0.4 adds the exact-replication lemma and corrects the remark.doi:10.5281/zenodo.23457692
- Version 1.0.3. Packaging: the verification suite of version 1.0.2 had a machine-local import path and a missing module, so it did not run from a clean clone. Version 1.0.3 repairs it and scopes a footnote to Corollary 7.3 to the surviving sector.doi:10.5281/zenodo.21911918
- Version 1.0.2. The general clause of Corollary 7.3 was false for arbitrary frame codes; explicit counterexamples are given. The corollary now states the exact alignment and disjointness condition, and the guarantee for the target pair is unaffected.doi:10.5281/zenodo.21911717
- Version 1.0.1. Version 1.0.0 called the accounting of the auxiliary descent unit-neutral at best; the paper’s own protocol is nondemolition, so each descent is net positive and catalytic. The corollary is corrected; no other statement is affected.doi:10.5281/zenodo.21910329
All corrections, and what counts as one.
Cite
@misc{douglas2026complexness,
author = {Douglas, Seth},
title = {The resource theory of complexness references in real quantum networks},
year = {2026},
version = {1.0.4},
doi = {10.5281/zenodo.23457692},
url = {https://doi.org/10.5281/zenodo.23457692}
}