Manuscript Hawk04-C@1 · Hawk04-C-v1.tex · theorem · 94d46352

Theorem: Sharp classical-record recovery bound

Row 6: thermality, classical records, and information loss › The record-only obstruction · lines 1243–1310 · proof included or linked

Hawk04-C@1#thm:classical, Hawk04-C@1#eq:record-channel, Hawk04-C@1#eq:classical-bound

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Let a channel from $C$ retain only a finite classical record:

F (X) = x \sum Tr (F_{x} X) ∣ x ⟩ ⟨ x ∣_{K}, F_{x} \geq 0, x \sum F_{x} = I_{C} .

For every CPTP decoder $D : L (K) \to L (C)$ ,

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

The same bound holds for every measure-and-prepare channel. It is sharp: computational readout followed by computational-basis preparation attains it.

Proof. Write $σ_{x} = D (∣ x ⟩ ⟨ x ∣)$ . Then

D F (X) = x \sum Tr (F_{x} X) σ_{x} .

Its normalized Choi state is

ω_{QC} = \frac{1}{d} x \sum F_{x}^{T} \otimes σ_{x},

which is separable across $Q ∣ C$ . For normalized product vectors,

∣ ⟨ Φ_{d} ∣ a \otimes b ⟩ ∣^{2} \leq \frac{1}{d}

by Cauchy–Schwarz. Convexity gives $⟨ Φ_{d} ∣ ω_{QC} ∣ Φ_{d} ⟩ \leq 1/ d$ . Testing the Bell projection therefore gives

d_{tr} (ω_{QC}, ∣ Φ_{d} ⟩ ⟨ Φ_{d} ∣) \geq 1 - \frac{1}{d} .

This is one admissible referenced channel test, proving (eq:classical-bound). The argument already covers a general measure-and-prepare output.

For sharpness, let

Z_{d} = j = 0 \sum^{d - 1} e^{2 π ij / d} ∣ j ⟩ ⟨ j ∣ .

Computational dephasing satisfies

Δ_{d} = \frac{1}{d} k = 0 \sum^{d - 1} Ad_{Z_{d}^{k}} .

Hence

∥ id_{C} - Δ_{d} ∥_{⋄} \leq \frac{1}{d} k = 1 \sum^{d - 1} id_{C} - Ad_{Z_{d}^{k}}_{⋄} \leq 2 (1 - \frac{1}{d}) .

The lower bound already proved gives equality after dividing by two.

Displayed formulas in this passage

Hawk04-C@1#eq:record-channel · line 1246
$F (X) = x \sum Tr (F_{x} X) ∣ x ⟩ ⟨ x ∣_{K}, F_{x} \geq 0, x \sum F_{x} = I_{C} .$
Hawk04-C@1#eq:classical-bound · line 1255
$d_{⋄} (D F, id_{C}) \geq 1 - \frac{1}{d} .$
unlabeled · line 1267
$D F (X) = x \sum Tr (F_{x} X) σ_{x} .$
unlabeled · line 1271
$ω_{QC} = \frac{1}{d} x \sum F_{x}^{T} \otimes σ_{x},$
unlabeled · line 1278
$∣ ⟨ Φ_{d} ∣ a \otimes b ⟩ ∣^{2} \leq \frac{1}{d}$
unlabeled · line 1285
$d_{tr} (ω_{QC}, ∣ Φ_{d} ⟩ ⟨ Φ_{d} ∣) \geq 1 - \frac{1}{d} .$
unlabeled · line 1293
$Z_{d} = j = 0 \sum^{d - 1} e^{2 π ij / d} ∣ j ⟩ ⟨ j ∣ .$
unlabeled · line 1297
$Δ_{d} = \frac{1}{d} k = 0 \sum^{d - 1} Ad_{Z_{d}^{k}} .$
unlabeled · line 1301
$∥ id_{C} - Δ_{d} ∥_{⋄} \leq \frac{1}{d} k = 1 \sum^{d - 1} id_{C} - Ad_{Z_{d}^{k}}_{⋄} \leq 2 (1 - \frac{1}{d}) .$

Nearby in this section

section: The record-only obstruction