Manuscript X-Enter@1 · X-Enter.tex · proposition · 484b71bc

Proposition: Exact-decoder obstruction

Retained diagnostic branches and exact decoding › No exact decoder of the convolution witness pair · lines 1256–1278 · proof included or linked

X-Enter@1#prop:decoder

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No program in $P_{4}$ with a fixed classical decoder returns $0$ surely at $a = 0$ and $c_{ε}$ surely at $a = c_{ε}$ under encoding $ι_{1}$ .

Proof. Let $E$ be the sum of effects for outcomes decoded as $c_{ε}$ . Then $0 \leq E \leq I$ , and exact decoding would require

⟨ ι_{1} (0)∣ E ∣ ι_{1} (0)⟩ = 0, ⟨ ι_{1} (c_{ε})∣ E ∣ ι_{1} (c_{ε})⟩ = 1.

Positivity implies

E ι_{1} (0) = 0, E ι_{1} (c_{ε}) = ι_{1} (c_{ε}) .

Since $E$ is Hermitian, their inner product would vanish. Both vectors have strictly positive real components, so their inner product is positive, a contradiction.

Displayed formulas in this passage

unlabeled · line 1266
$⟨ ι_{1} (0)∣ E ∣ ι_{1} (0)⟩ = 0, ⟨ ι_{1} (c_{ε})∣ E ∣ ι_{1} (c_{ε})⟩ = 1.$
unlabeled · line 1271
$E ι_{1} (0) = 0, E ι_{1} (c_{ε}) = ι_{1} (c_{ε}) .$

Nearby in this section

section: No exact decoder of the convolution witness pair