Then ijOije12Var(ij)\rho_{ij}\propto \mathcal{O}_{ij}\,e^{-\tfrac12\mathrm{Var}(\delta\phi_{ij})}, up to normalization and population factors. This explicit dependence shows that off-diagonal entries are holonomy-weighted amplitude overlaps suppressed by holonomy variance.
3. Mapping C1C_{\ell_1} to integrated holonomy overlaps
Using the representation above, a natural identification for the 1\ell_1-coherence is
C1()ijOije12Var(ij)=ijd(x)AiAjeiije12Var(ij).(M1)\boxed{\; C_{\ell_1}(\rho) \;\approx\; \sum_{i\neq j} \big|\mathcal{O}_{ij}\big|\; e^{-\tfrac12\mathrm{Var}(\delta\phi_{ij})} \;=\; \sum_{i\neq j} \left|\int d\mu(x)\,A_iA_j\,e^{i\Delta\theta_{ij}}\right| e^{-\tfrac12\mathrm{Var}(\delta\phi_{ij})}. \;} \tag{M1}
Interpretation and operational use:
The term Oij|\mathcal{O}_{ij}| is the geometric amplitude overlap between nodes i,ji,j and depends on spatial overlap of their support and on deterministic holonomy phases.
The factor e12Vare^{-\tfrac12\mathrm{Var}} encodes loss of coherence due to holonomy fluctuations (motion, slips, leakage).
Equation (M1) gives a direct recipe to compute (or estimate) C1C_{\ell_1} from interference experiments: measure pairwise fringe contrasts and phases to reconstruct Oij|\mathcal{O}_{ij}| and independently estimate Var(ij)\mathrm{Var}(\delta\phi_{ij}) from noise statistics.
A practical estimator for C1C_{\ell_1} in two-node interferometers reduces to
C12A1A2e12Var()=(A12+A22)V,C_{\ell_1} \;\propto\; 2|A_1A_2|\, e^{-\tfrac12\mathrm{Var}(\delta\phi)} \;=\; (A_1^2+A_2^2)\,V,
since visibility V=2A1A2A12+A22e12VarV=\frac{2|A_1A_2|}{A_1^2+A_2^2} e^{-\tfrac12\mathrm{Var}}. Thus measured VV and populations immediately deliver an 1\ell_1-estimate for the two-node subspace.
4. Mapping relative entropy of coherence to holonomy (topological) entropy
Relative entropy of coherence quantifies how far \rho is from its incoherent (diagonal) version. In InterTop the natural informational degrees of freedom are the holonomy values associated to loops or to node--node phase distributions. We propose the following operational mapping:
Form the empirical holonomy phase distribution Pij()P_{ij}(\phi) for each node pair (or a global holonomy distribution P()P(\phi) obtained by pooling many loops/probes). This distribution is experimentally accessible by repeated phase measurements across parameter settings or time.
Define the holonomy (Shannon) entropy
Hholo=dP()logP().H_{\rm holo} \;=\; -\int d\phi\; P(\phi)\,\log P(\phi).
