Manuscript VIII-A@3 · VIII-A-v3.tex · theorem · 2398a747

Theorem: Measure-and-prepare recovery bound

Recovery distinctions used only as interface safeguards › The sharp record-only obstruction · lines 2570–2626 · proof included or linked

VIII-A@3#A:thm:classical, VIII-A@3#A:eq:classical-bound

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Let $C$ have dimension $d$ , let

E (X) = j \sum Tr (F_{j} X) σ_{j}, F_{j} \geq 0, j \sum F_{j} = I_{C},

be a measure-and-prepare channel to an arbitrary finite output, and let $D$ be any CPTP decoder to $C$ . Then

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

Computational readout followed by computational-basis preparation attains the bound.

Proof. Let $∣ Φ_{d} ⟩ = d^{- 1/2} \sum_{i = 0}^{d - 1} ∣ i ⟩ ∣ i ⟩$ and $P_{Φ_{d}} = ∣ Φ_{d} ⟩ ⟨ Φ_{d} ∣$ . The normalized Choi output of the decoded channel is

J_{n} (D E) = \frac{1}{d} j \sum F_{j}^{T} \otimes D (σ_{j}) .

It is separable across reference and returned output. For normalized product vectors,

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

by Cauchy–Schwarz. Convexity gives the same overlap bound for every separable state. Testing the Bell projector gives

d_{tr} (J_{n} (D E), P_{Φ_{d}}) \geq 1 - \frac{1}{d} .

This is one admissible referenced channel test, proving A:eq:classical-bound.

For sharpness, put

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

Computational readout followed by basis preparation is $Δ_{d}$ . The triangle inequality and the fact that each channel has diamond norm one give

∥ id_{C} - Δ_{d} ∥_{⋄} \leq 2 (1 - 1/ d) .

The already proved Bell lower bound gives equality.

Displayed formulas in this passage

VIII-A@3#A:eq:classical-bound · line 2580
$d_{⋄} (D E, id_{C}) \geq 1 - \frac{1}{d} .$
unlabeled · line 2573
$E (X) = j \sum Tr (F_{j} X) σ_{j}, F_{j} \geq 0, j \sum F_{j} = I_{C},$
unlabeled · line 2593
$J_{n} (D E) = \frac{1}{d} j \sum F_{j}^{T} \otimes D (σ_{j}) .$
unlabeled · line 2600
$∣⟨ Φ_{d} ∣ a \otimes b ⟩ ∣^{2} \leq \frac{1}{d}$
unlabeled · line 2606
$d_{tr} (J_{n} (D E), P_{Φ_{d}}) \geq 1 - \frac{1}{d} .$
unlabeled · line 2614
$Z_{d} = j = 0 \sum^{d - 1} e^{2 π ij / d} ∣ j ⟩ ⟨ j ∣, Δ_{d} = \frac{1}{d} k = 0 \sum^{d - 1} Ad_{Z_{d}^{k}} .$
unlabeled · line 2621
$∥ id_{C} - Δ_{d} ∥_{⋄} \leq 2 (1 - 1/ d) .$

Nearby in this section

section: The sharp record-only obstruction