bibkey: marvian2014modes authors: Iman Marvian; Robert W. Spekkens year: 2014 title: ‘Modes of asymmetry: the application of harmonic analysis to symmetric quantum dynamics and quantum reference frames’ doi: 10.1103/PhysRevA.90.062110 url: https://arxiv.org/abs/1312.0680v2 claim: Time-translation covariant quantum maps preserve frequency modes; the mode-selection rule is an independent constraint on coherent thermal recovery. strata_touched:

• D5/S3/Quantum/Dynamics/HamiltonianEffectCompletionGenerator license: citation-only triage: anchor

Frequency modes and thermal recovery

Source scope

Physical Review A 90, 062110 (2014), author manuscript v2. Section II, equations (2.7)-(2.10), describes decomposition into symmetry modes and their preservation by symmetric processing. The rule concerns a specified symmetry representation on both the input and output; matching a stationary reference state alone does not imply this covariance.

Use in the computational-behavior volume

Sections 56-58 impose covariance on the same CPTP channel that restores two conditional Gibbs states. The resulting off-diagonal block obeys H0 X - X H1 = omega X. The frequency rule is prior art. The volume separately optimizes its trace over the thermally calibrated positive block, using noncommuting spectral projections, and treats finite-time covariance error.

For the two-label problem, each exact output energy has at most one partner at a fixed gap, so independent polar contractions assemble into one feasible map. That argument does not establish simultaneous saturation for three or more labels. Near-equal numerical gaps do not establish exact matching. The existing Lean commutator-orbit declaration supplies a dynamics interface; it does not prove the paper-level optimization or diamond-norm formulas.

The bibliography and the Section II mode formulas are the source basis. This note contains an original citation-only summary, without reproducing the article or claiming a thermal-operation implementation.