Proposition: Optimizer structure and endpoints

For $0\le \gamma <1$, the function $F_{\gamma ,\nu }$ is strictly concave, has a unique maximizer in $(0,1)$, and satisfies

At $\gamma =1$, the channel is a replacement channel, $F_{\gamma ,\nu }\equiv 0$, and every $p$ maximizes.

The one-use minimum output entropy is

Proof. When $\gamma <1$, the first term of $F$ is strictly concave as binary entropy composed with an injective affine map. The second term is convex by Lemma lem:convex. Thus their difference is strictly concave.

At $p=0,1$, the pure input is a computational-basis vector, its output is diagonal, and the two entropy terms coincide. For $0<p<1$, the output determinant satisfies

The determinant entropy is strictly increasing in the interior, so $F_{\gamma ,\nu }(p)>0$. The unique maximum is therefore interior.

At $\gamma =1$, (eq:channel) is independent of the input. For the minimum entropy, a pure input suffices by concavity. The completed square for $v$, its minimum at $p=\nu$, and monotonicity of $g_2$ prove the formula.