Theorem: One-use Holevo formula and phase-pair attainment

For all $\gamma ,\nu \in [0,1]$,

At every maximizing $p$, the equiprobable pair

attains the maximum.

For every $n$, its independent product ensemble has Holevo information $n\chi ({\mathcal{A}}_{\gamma ,\nu })$. This last assertion is an attained lower bound on $\chi ({{\mathcal{A}}_{\gamma ,\nu }}^{\otimes n})$, not an upper bound on entangled block signals.

Proof. Take any ensemble $\{q_a,{\rho }_a\}$, with populations $p_a$, and put $\overline{p}={\sum }_a{q}_a{p}_a$. The average output has population $t(\overline{p})$. Lemma lem:entropy gives

Lemma lem:determinant and convexity give

Subtracting proves the upper bound by $F_{\gamma ,\nu }(\overline{p})$. Continuity on $[0,1]$ gives existence of a maximizing $p$.

For the displayed pair, the opposite coherences cancel in the average. Its average output is diagonal with population $t(p)$, and each individual output has entropy $e(p)$. Its Holevo information is exactly $F_{\gamma ,\nu }(p)$.

For products, both the average output and each conditional output are tensor products. Entropy additivity gives $n{F}_{\gamma ,\nu }(p)$. No unrestricted tensor-power upper bound has been proved in this argument.