Manuscript En07-C@1 · En07-C.tex · proposition · 3f7e9ebb

Proposition: Featureless-accessible-output lower bound

Featureless channels and the thermal-state inference gap · lines 839–865 · proof included or linked

En07-C@1#prop:featureless, En07-C@1#eq:featureless-hypothesis, En07-C@1#eq:featureless-bound

Suppose

d_{⋄} (N, T_{ω}) \leq η .

Every deterministic decoder obeys

d_{⋄} (DN, id_{C}) \geq max {0, q_{d} - η} .

This is a recovery lower bound.

Proof. Contraction gives $d_{⋄} (DN, D T_{ω}) \leq η$ . The triangle inequality and the replacement-channel lower bound in Lemma lem:constant-error give

d_{⋄} (DN, id_{C}) \geq q_{d} - η .

Distances are nonnegative.

Displayed formulas in this passage

En07-C@1#eq:featureless-hypothesis · line 842
$d_{⋄} (N, T_{ω}) \leq η .$
En07-C@1#eq:featureless-bound · line 847
$d_{⋄} (DN, id_{C}) \geq max {0, q_{d} - η} .$
unlabeled · line 861
$d_{⋄} (DN, id_{C}) \geq q_{d} - η .$

Nearby in this section

lemma: Lemma: Optimal error of a replacement channel
section: Featureless channels and the thermal-state inference gap