Free operations.
Natural choices for free operation classes are those that cannot increase holonomy-defined coherence. Candidate classes, adapted to InterTop, include:
Incoherent operations (IO): Kraus operators {Kn}\{K_n\} such that each KnKn/Tr[KnKn]IK_n\rho K_n^\dagger/\mathrm{Tr}[K_n\rho K_n^\dagger]\in\mathcal I whenever I\rho\in\mathcal I. Operationally, IO are transformations implementable without creating holonomy (no engineered informational connection).
Holonomy-noncreating thermal operations (H-TO): physically motivated class that permits manipulations of populations and classical mixing but forbids any operation that produces a deterministic synthetic phase (no new holonomy loops). H-TO is useful when baths and drives are physically constrained.
Which class is most appropriate depends on the experimental platform; both IO and H-TO serve as useful bounds. The essential requirement is that free operations cannot reduce holonomy variance in a way that increases deterministic holonomy-weighted overlaps without external holonomy engineering.
2. Monotonicity of coherence measures under free operations
Standard coherence monotones satisfy monotonicity under the accepted free operation classes. With the mappings introduced in Sec. V.A, these monotonicities translate into geometric statements:
1\ell_1-norm monotonicity under IO:
If \Lambda is an incoherent operation, then
C1([])C1().C_{\ell_1}(\Lambda[\rho]) \le C_{\ell_1}(\rho).
Mapped to InterTop, this says: any operation that does not create informational holonomy cannot increase the integrated holonomy-weighted overlaps ijOije12Varij\sum_{i\neq j} |\mathcal O_{ij}|e^{-\tfrac12\mathrm{Var}_{ij}}. Operationally, without actively engineering A\mathcal A or reducing holonomy variance using external control, you cannot increase the interferometric amplitude resource.
Relative-entropy monotonicity:
For free \Lambda in IO (or strictly incoherent operations, SIO),
Crel([])Crel().C_{\mathrm{rel}}(\Lambda[\rho]) \le C_{\mathrm{rel}}(\rho).
In InterTop terms: incoherent manipulations cannot reduce holonomy entropy deficit HmaxHholoH_{\rm max}-H_{\rm holo} (cannot sharpen the holonomy distribution) without active holonomy design.
These monotonicities give operational constraints that experiments can test: if an allowed free operation (e.g., passive filtering or population redistribution consistent with IO) appears to increase measured C1C_{\ell_1} or CrelC_{\mathrm{rel}} beyond statistical error, then either (i) the operation implemented an implicit holonomy-engineering step (e.g., modified A\mathcal A through coupling changes) or (ii) the adopted node basis was not the correct pointer basis.
3. Geometric interpretations of standard monotones
Using the mappings of Sec. V.A we express monotones in geometric language and derive inequalities that are experimentally meaningful.
1\ell_1-norm as integrated signed overlap.
From (M1) for a normalized state in the node basis,
C1()ijOije12Varij.C_{\ell_1}(\rho) \approx \sum_{i\neq j} \big|\mathcal O_{ij}\big| e^{-\tfrac12\mathrm{Var}_{ij}}.
Interpretation: C1C_{\ell_1} measures the total coherent perimeter of the informational manifold---the summed magnitude of pairwise holonomy-weighted overlaps. It is sensitive to both deterministic holonomy (via phases in Oij\mathcal O_{ij}) and stochastic holonomy variance.
Relative entropy as holonomy information.
Using (M2),
