Manuscript Hawk04-A@1 · Hawk04-A.tex · lemma · 526092ba

Lemma: Isometry comparison

Quantitative channel certificates › Normalization checks · lines 757–772 · proof included or linked

Hawk04-A@1#lem:isometry

For isometries $V_{1}, V_{2} : C \to R \otimes B$ ,

d_{⋄} (V_{1}, V_{2}) \leq ∥ V_{1} - V_{2} ∥ .

The same bound holds for either reduced channel.

Proof. For a pure referenced input, the two output vectors differ by at most $∥ V_{1} - V_{2} ∥$ . Their pure-state trace distance is no larger than that vector distance. Convexity handles mixed inputs, and partial trace is contractive.

Displayed formulas in this passage

unlabeled · line 760
$d_{⋄} (V_{1}, V_{2}) \leq ∥ V_{1} - V_{2} ∥ .$

Nearby in this section

lemma: Lemma: Constant-output recovery benchmark
section: Normalization checks
lemma: Lemma: Choi and matrix-unit bounds
section: Normalization checks