Undecidable projection costs for quantum channels with finite tracial ancillas
Citable version that has not completed AIRR screening. Not peer reviewed. Screening record
We transfer a quantitative unitary-relation construction to a quadratic optimization problem for quantum channels. The input is a real rational orthogonal projection on the Choi space, together with an explicit rational bistochastic Choi state supported exactly on its kernel. The minimum cost over all bistochastic channels is therefore zero. Nevertheless, minimizing the same cost over channels implemented by a unitary interaction with an unrestricted finite tracial ancilla is undecidable at one fixed additive precision. The objective is a projective measurement probability, not a determinant. Its equivalent matrix formulation is the squared distance between a unitary and an amplified linear subspace. This distance at fixed dimension is the Kraus-space quantity studied by Yu, Duan and Xu; we examine its stabilization under a completely depolarizing ancillary channel. We prove the transfer with explicit constants, derive the absence of a recursive uniform ancilla bound, and give the exact answer when the subspace projection is completely positive. The source hardness and the relation compiler are inherited results; no general novelty claim about noncomputable tracial optimization is made.
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Cite this paper
Lluis Eriksson (2026). Undecidable projection costs for quantum channels with finite tracial ancillas. AIRR.SCIENCE, ARR-2026-7WNWKBJVQ28KZAKA, v1. https://airr.science/papers/ARR-2026-7WNWKBJVQ28KZAKA/versions/v1/ Research preprint; ARR-2026-7WNWKBJVQ28KZAKA v1. Not peer reviewed by AIRR. Working paper; not admitted to the AIRR accepted collection.
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Version history
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Verification record
- Frontier-model screening
- Not assessed
- Source integrity
- Pass
- Bibliographic integrity
- Pass
- Reproducibility
- Pass
- Lean 4
- Not applicable
Recorded under ARR-VERIFY-1.0. AIRR verification and screening are not peer review.
AI assistance statement
Substantial OpenAI Codex assistance with research, proofs, writing and computation. Exact-version review requested from GPT-6 Astra at xhigh effort in a new agent without inherited conversation. Prior reports and scores were withheld. Ordinary quality 8.2/10 and AIRR importance 3.6/10 are distinct. Prior model-family participation in construction is declared. Dispatch controls are recorded; effective runtime identity and saved-memory controls were not separately attested. No certified blind review, independent human review or formal verification is claimed.
Frontier-model screening
Status: not_assessed. Any listed reports correspond to this exact version under ARR-SCREEN-1.0; no absent assessment is represented as a pass.
Model assessments and review context
- Distinct models
- 1
- Range
- 3.60–3.60
- Method
- bounded capability-weighted mean · exact version only
Limited evidence · 0/1 benchmark matches. This is an evidence description, not a probability of correctness.
Only one distinct model; One provider; shared blind spots possible; Fresh blinded review context not documented for every model; Benchmark match incomplete
OpenAI · gpt-6-astra · involved in manuscript · review context: unknown3.60
Recommendation: Accept · Material objections: 0
The exact eight-page manuscript gives a valid-looking, quantitatively explicit transfer from the supplied relation compiler to projection-cost optimization over channels with unrestricted finite tracial ancillas. I found no unresolved material objection and recommend acceptance as a focused theoretical contribution. The analytic proof, not the finite diagnostics, supports undecidability. The strongest novelty candidate is the simultaneous rational projection, fixed gap and exact bistochastic support certificate; general noncomputability and the distance quantity are established background. This reviewer model family was involved in the manuscript. The operator reports requesting gpt-6-astra with xhigh effort and fork_turns=none; runtime identity and context isolation were not independently attested. No human, certified blind or proof-assistant review is claimed.
- Correctness confidence
- Theorem 1 follows from the inspected compiler interface and the checked quadratic transfer; Choi normalization, singular polar factors, the fixed gap and quantifiers are consistent. No material flaw found. Foundational source hardness is cited, not independently re-proved.
- Rigor
- The proof accounts for disjoint gate occurrences, singleton weights, arbitrary starting unitaries, nonattained infima and finite-ancilla convex closures. The distinction between a positive projection matrix and a completely positive subspace map is correct.
- Novelty
- The plausible new combination is a single rational projection cost, a constant undecidability gap and an explicit rational zero-cost bistochastic certificate. Kraus-space distance, factorizable-channel closures, source hardness and most compiler machinery are inherited; targeted primary-source checking does not certify worldwide priority.
- Significance
- A substantive specialist restriction of known noncomputable tracial optimization to an operational channel test. Its substantial dependence on the earlier compiler, variable system dimension and extremely small gap limit claims of broad or practical impact; no field-shaping or historic conclusion is supported.
- Reproducibility
- All 17 manifest entries match and the canonical PDF has eight pages. Both manuscript and both dependency verifiers pass in writable copies, reproducing their JSON files byte-for-byte; 72 additional direct rounding checks pass. There is no end-to-end machine-to-game implementation, global optimization certificate or proof-assistant verification.
Strengths
- Theorem 1 provides a single real rational projective test with a displayed rational bistochastic zero-cost state supported exactly on its kernel.
- Lemma 3 establishes both quadratic transfers with explicit constants and correctly controls repeated labels through disjoint gate occurrences.
- The Choi identity includes the correct ancillary m-squared factor, and the soundness proof handles arbitrary projected unitaries and singular polar extensions.
- Corollary 4 correctly enumerates all ancilla sizes below a proposed recursive bound; the amplification and semicomputability statements have the right quantifiers.
- The manuscript accurately credits prior distance and tracial-optimization results and states its limits on finite attainment, fixed dimension, membership and practical thresholds.
Weaknesses
- Most of the difficult construction is inherited from the supplied compiler, so the incremental contribution is narrower than the broad undecidability language might initially suggest.
- The package reproduces finite identities but does not implement the full machine-to-game map or export a small end-to-end exact channel instance from a source game.
- Individual lemma references for the imported compiler interface would improve traceability beyond the present reference to Sections 2–5.
- Worldwide priority remains unverified, and the result gives no fixed system dimension or useful practical ancilla/circuit resource estimate.
Potential errors
None reported.
Strong novelty candidates
- Constant-precision undecidability for a single rational orthogonal projection cost on Choi space, with an explicit rational bistochastic optimizer of the zero-valued semidefinite relaxation and exact support/kernel equality.
- The direct quadratic transfer from the balanced relation pencil to the stabilized Yu–Duan–Xu Kraus-space distance, including the uniform reverse rounding of an arbitrary projected unitary.
Unresolved material objections
None reported.
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Editorial disclosure
Author-authorized exact-version Working paper. Author is AIRR operator. Advisory AI assessment is retained with its provenance and limitations. This is not Accepted status or independent human peer review.