Definition: Decohered fine-graining certificates

An $\mathsf{FG}$-certificate for this law uses disjoint registers $S,E^{\prime },E^{\prime \prime }$, with $\left|S\right|=s$, and a state

such that:

1. the labels $r(\alpha )$ are pairwise distinct;

2. the companions $(k(\alpha ),e(\alpha ))$ are pairwise distinct;

3. the diagonal marginal on $S$ is the target law.

Its certificate cost is the number of qubits simultaneously hosting $S,E^{\prime },E^{\prime \prime }$. Let $C_{\mathsf{FG}}(s,\vec{m})$ be the minimum within this specified family.