Manuscript VIII-B@3 · VIII-B-v3.tex · proposition · 3aa6baa1

Proposition: Sharp record-only obstruction

Classical table decoding is not quantum recovery › Record-only recovery · lines 2216–2256 · proof included or linked

VIII-B@3#prop:classical-recovery, VIII-B@3#eq:record-recovery

Let $dim H_{Q} = d \geq 2$ . If a channel $E$ is measure-and-prepare and $D$ is any channel returning $Q$ , then

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

The bound is attained by basis readout followed by basis-state preparation. It does not apply when an authorized input-dependent quantum memory is also retained.

Proof. A measure-and-prepare map has the form

E (X) = z \sum Tr (F_{z} X) σ_{z}, F_{z} \geq 0, z \sum F_{z} = I .

Applying it and then $D$ to half of $∣ Φ_{d} ⟩ = d^{- 1/2} \sum_{i} ∣ i, i ⟩$ gives a separable reference–output state. For density operators $α, β$ ,

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

Convexity gives the same upper bound for the separable output. Testing the Bell projector yields trace distance at least $1 - 1/ d$ from the ideal output, proving (eq:record-recovery).

For sharpness, computational dephasing is

Δ_{d} = \frac{1}{d} b = 0 \sum^{d - 1} Ad_{Z_{d}^{b}}, Z_{d} = diag (1, e^{2 π i / d}, \dots, e^{2 π i (d - 1) / d}) .

The triangle inequality bounds $∥ id - Δ_{d} ∥_{⋄}$ by $2 (1 - 1/ d)$ . The Bell lower bound gives equality.

Displayed formulas in this passage

VIII-B@3#eq:record-recovery · line 2221
$d_{⋄} (D E, id_{Q}) \geq 1 - \frac{1}{d} .$
unlabeled · line 2232
$E (X) = z \sum Tr (F_{z} X) σ_{z}, F_{z} \geq 0, z \sum F_{z} = I .$
unlabeled · line 2239
$⟨ Φ_{d} ∣ α \otimes β ∣ Φ_{d} ⟩ = \frac{1}{d} Tr (α^{T} β) \leq \frac{1}{d} .$
unlabeled · line 2248
$Δ_{d} = \frac{1}{d} b = 0 \sum^{d - 1} Ad_{Z_{d}^{b}}, Z_{d} = diag (1, e^{2 π i / d}, \dots, e^{2 π i (d - 1) / d}) .$

Proposition: Sharp record-only obstruction

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