Proposition: Accessible distinguishability and heralding

For $X={\Sigma }_0-{\Sigma }_1$,

Every fixed event bias is at most this detector total variation. For specified distributions, the positive-difference bins attain it.

If an attempted law is

with $r_h$ the normalized successful detector law, then equal success $s_0=s_1=s$ gives

For unequal success, $\mathrm{TV}(p_0,p_1)\ge |s_0-s_1|$.

Proof. The measurement output is diagonal, giving the first identity. Positivity and trace preservation contract Hermitian trace norm. The positive signed entries sum to half the absolute sum. For heralding, compare the explicit rejection and success entries.