Proposition: Visibility and preparation error

Computational destructive readout of $\tau (a)$ has probability $p(a)$ for code $1$. For every $a,b\in \mathbb{R}$,

A fixed subsequent complete channel cannot enlarge this bound, including with an arbitrary independent input–reference state.

Proof. The probability is the squared second amplitude. The Bloch vector is

Its derivative has Euclidean norm $\mathrm{sech}(a/{\sigma }_{\mathrm{e}})/{\sigma }_{\mathrm{e}}\le 1/{\sigma }_{\mathrm{e}}$. Qubit trace distance is half the Euclidean distance between Bloch vectors, proving (eq:encodingerror). Tensoring with a state preserves trace norm, and channels contract trace distance.