= Equality in capacity data processing
{title2=$C_{g(Y)}=C_Y$}
For a finite <discrete memoryless channel> and a deterministic output map $T=g(Y)$, the <data processing inequality> gives $C_T\leq C_Y$. Equality holds exactly when an original capacity-achieving input law satisfies $I(X;Y\mid T)=0$. Sufficiency follows from the gap identity $I(X;Y)=I(X;T)+I(X;Y\mid T)$. For necessity, take an input law maximizing $I(X;T)$. If this maximum is $C_Y$, both inequalities $I(X;T)\leq I(X;Y)\leq C_Y$ are equalities at that law. <Injectivity> of $g$ on reachable outputs is a sufficient condition, since $T$ then determines $Y$; it is unnecessary if the discarded distinctions convey no further information about $X$ at some maximizing input law.
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