Quantum Reference Frames Beyond Ideality: Complete-Order Quotients, Resource Lattices, and Certified Transformations

Максим Вячеславович Чурилов

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Quantum Reference Frames Beyond Ideality: Complete-Order Quotients, Resource Lattices, and Certified Transformations

М. В. Чурилов

2026-06-29

https://doi.org/10.24108/preprints-3115706

Quantum reference frame (QRF) transformations are usually presented as unitary or gauge-fixing changes of coordinates. That picture is exact for ideal frames but fails for finite clocks, gyroscopes, POVM-defined tokens, observers correlated with the targets they describe, and relativistic frames with quantum boost and clock degrees of freedom; the obstruction is loss of distinguishability. We model a non-ideal frame by a normal covariant perspective channel. A deterministic transformation A → B then exists exactly when the kernel of A's channel is contained in the kernel of B's channel, and is unique on the preparable image. Ancilla-stable transformations are complete-order quotient morphisms, while a laboratory implementation on a chosen output space is a separate Choi-Arveson extension problem. The common invariant content of a family of frames is a Heisenberg operator system, carrying an algebraic product only on its product-compatible core. For finite frames in the regular representation, concrete covariant degradability A → B is decided by a group-convolution linear program with explicit dual witnesses ; for Abelian groups this becomes a Fourier mode-support resource lattice. Our main result is that for every finite group this convolution linear program already equals the full diamond-norm deficiency: coherent, multiplicity-exploiting post-processings never outperform random translations, so the resource theory of frame degradation is entirely classical. The same quotient principle governs process tensors, protected relational sectors, entangled observers, relativistic covariance — the boosted-spin perspective being an explicit U(1) dephasing channel — and observer networks. Reproducible numerical certificates accompany every analytic claim.

Ссылка для цитирования:

Чурилов М. В. 2026. Quantum Reference Frames Beyond Ideality: Complete-Order Quotients, Resource Lattices, and Certified Transformations. PREPRINTS.RU. https://doi.org/10.24108/preprints-3115706

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