Manuscript X-J-ST@1 · X-J-ST.tex · lemma · 8f0d1a36

Lemma: Lipschitz amplitude preparation

Errors, finite composition, locality, and transport › Input precision and robust positive margins · lines 1691–1711 · proof included or linked

X-J-ST@1#lem:lipschitz

For all real $z, z^{'}$ ,

d_{tr} (τ_{σ} (z), τ_{σ} (z^{'})) \leq \frac{∣ z - z ^{'} ∣}{2 σ} .

Proof. The Bloch vector is $(s (z), 0, t (z))$ , whose derivative has Euclidean norm

\frac{1}{σ} s (z)^{2} t (z)^{2} + s (z)^{4} = \frac{s ( z )}{σ} \leq \frac{1}{σ} .

Trace distance of qubit states is half the Euclidean distance of their Bloch vectors. Integrate the derivative bound.

Displayed formulas in this passage

unlabeled · line 1694
$d_{tr} (τ_{σ} (z), τ_{σ} (z^{'})) \leq \frac{∣ z - z ^{'} ∣}{2 σ} .$
unlabeled · line 1703
$\frac{1}{σ} s (z)^{2} t (z)^{2} + s (z)^{4} = \frac{s ( z )}{σ} \leq \frac{1}{σ} .$