Proposition: A legitimate probabilistic certificate

Suppose a declared disorder distribution and a measurable family of residual states satisfy

For $0<p<1$,

On the complementary event,

Any decoder dependence on $\mathbf{J}$, and any realization information needed to implement it, must be declared.

Proof. Apply Markov's inequality to the nonnegative random trace distance, and then apply Theorem thm:certificate realization by realization.