Theorem: Privacy against the complete declared residual

If $d_{\diamond }(\mathcal{C},{\mathcal{T}}_{{\tau }_B})\le \zeta$, then

for every referenced input. For any two unreferenced inputs,

An observer restricted to $B$, with equal prior probabilities for these inputs, has discrimination success probability at most $1/2+\zeta$. Arbitrary processing of $B$ cannot increase these bounds.

Proof. Apply the diamond comparison to ${\rho }_{QL}$. Use the triangle inequality through ${\tau }_B$ for the second claim. For equally likely states, the best success probability is $(1+d_{\mathrm{tr}})/2$, giving the third claim. CPTP contraction proves the last assertion.