Theorem: Oai276-A finite certificate

For the frozen channel $\mathcal{A}={\mathcal{A}}_{1/2,1/4}$, define $p^{\star }$ to be the unique zero in $(469/1000,47/100)$ of

Then:

1. $p^{\star }$ is the unique global maximizer of the one-use Holevo objective;

2. its value has the exact finite enclosure

   $$L_{\mathrm{cert}}\le \chi (\mathcal{A})\le U_{\mathrm{cert}},\frac{412377}{10^6}<\chi (\mathcal{A})<\frac{412378}{10^6}\text{bits};$$

3. the equiprobable pair

   $${{\varphi }^{\star }}_{\pm }=\sqrt{1-p^{\star }}|0⟩\pm \sqrt{p^{\star }}|1⟩$$

   attains this one-use value;

4. the explicitly algebraic pair

   $${{\varphi }^0}_{\pm }=\sqrt{\frac{1061}{2000}}|0⟩\pm \sqrt{\frac{939}{2000}}|1⟩$$

   has Holevo information $F_{1/2,1/4}(p_0)$, with

   $$0\le \chi (\mathcal{A})-F_{1/2,1/4}(p_0)<\frac{1}{2000000\ln 2}<10^{-6}\text{bits}.$$

No arbitrary-partner additivity, repeated-use upper bound, or separate-measurement optimality is a hypothesis or conclusion of this certificate.

Proof. The series bounds and Lemma lem:arithmetic give $D(a)>0>D(b)$. Continuity and strict decrease of $D$ give exactly one zero in the stated interval. By Lemma lem:derivative and Proposition prop:optimizer, it is the unique global maximizer.

At $p_0$, Lemma lem:series gives $A_{-}\le f(p_0)\le A_{+}$. The positive lower bound for $A_{-}$ and (eq:ln2) imply

Also $0<D(p_0)<1/1000$. Therefore (eq:quadratic) gives

Division by $\ln 2\ge {\ell }_{-}$ proves the upper enclosure. The strict outer interval is Lemma lem:arithmetic.

Theorem thm:oneuse supplies the exact maximizing pair and identifies its value with $\chi (\mathcal{A})$. The same quadratic bound gives (eq:nearoptimal). Finally, $\ln 2>1/2$, for example from the first term ${\ell }_{-}\ge 2/3$, proving the last inequality.