Manuscript Oai272-A@1 · Oai272-A.tex · proposition · a40828a0

Proposition: Sharp record-only quantum-recovery bound

Stage 2: what local resource and recovery certificates price › A readout obstruction is not a key obstruction · lines 1352–1380 · proof included or linked

Oai272-A@1#prop:record

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Let $dim Q = d \geq 2$ . If the only input-dependent accessible output is a classical measurement record, then every quantum decoder implements an entanglement-breaking channel $Λ$ , and

d_{⋄} (Λ, id_{Q}) \geq 1 - \frac{1}{d} .

Basis measurement followed by basis-state preparation attains the bound.

Proof. The decoded channel has the form

Λ (X) = z \sum Tr (E_{z} X) σ_{z} .

Its Bell-input output is separable. For normalized product states,

⟨ Φ_{d} ∣ α \otimes β ∣ Φ_{d} ⟩ = \frac{1}{d} Tr (α^{T} β) \leq \frac{1}{d} .

Convexity and the Bell-projector test give the lower bound. Computational dephasing is the average of $d$ phase-unitary conjugations; the triangle inequality gives the matching upper bound $1 - 1/ d$ .

Displayed formulas in this passage

unlabeled · line 1358
$d_{⋄} (Λ, id_{Q}) \geq 1 - \frac{1}{d} .$
unlabeled · line 1367
$Λ (X) = z \sum Tr (E_{z} X) σ_{z} .$
unlabeled · line 1372
$⟨ Φ_{d} ∣ α \otimes β ∣ Φ_{d} ⟩ = \frac{1}{d} Tr (α^{T} β) \leq \frac{1}{d} .$

Proposition: Sharp record-only quantum-recovery bound

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