Proposition: Complete fixed-table distance

For two Bernoulli tables $p,q$ on the same complete interface,

Thus certified rational approximations of the parameters give exactly the corresponding fixed-table channel bound.

Proof. After adjoining an arbitrary reference and dephasing the clock, the $j$-block of an input operator is $X_{jj}$. The two binary output blocks of the difference are $(p_j-q_j)X_{jj}$ and its negative. Hence their total trace norm is bounded by

Pinching the clock is an average of unitary conjugations, so ${\sum }_j{\Vert X_{jj}\Vert }_1\le {\Vert X\Vert }_1$. This proves the upper bound, including every reference. A basis-clock input at an index attaining the maximum gives the lower bound.