Manuscript En07-B@1 · En07-B.tex · theorem · 45dff4a1

Theorem: Record-only quantum-return obstruction

Enter, forgetting, and surviving quantum registers › Record-only reconstruction · lines 690–748 · proof included or linked

En07-B@1#thm:classical

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Suppose a channel from the code to a returned quantum register factors through a classical record, with no surviving input-dependent quantum memory. Then its half-diamond distance from $id_{C}$ is at least

1 - \frac{1}{d} .

More explicitly, if

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

where each $σ_{j}$ is a density operator on $C_{out}$ , then

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

The bound is attained by computational dephasing.

Proof. The normalized Choi output is

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

It is a separable density operator across the reference–output cut. For density operators $α, β$ ,

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

Convexity gives $Tr (P_{Φ_{d}} J_{n} (Λ)) \leq 1/ d$ . Testing the Bell projector therefore gives

d_{⋄} (Λ, id_{C}) \geq d_{tr} (J_{n} (Λ), P_{Φ_{d}}) \geq 1 - \frac{1}{d} .

For sharpness, write computational dephasing as

Δ = \frac{1}{d} b = 0 \sum^{d - 1} Ad_{Z^{b}} .

The triangle inequality and the fact that channels have diamond norm one give

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

The preceding Bell lower bound gives equality.

Displayed formulas in this passage

unlabeled · line 696
$1 - \frac{1}{d} .$
unlabeled · line 700
$Λ (X) = j \sum Tr (F_{j} X) σ_{j}, F_{j} \geq 0, j \sum F_{j} = I_{C},$
unlabeled · line 706
$d_{⋄} (Λ, id_{C}) \geq 1 - \frac{1}{d} .$
unlabeled · line 714
$J_{n} (Λ) = \frac{1}{d} j \sum F_{j}^{T} \otimes σ_{j} .$
unlabeled · line 721
$Tr (P_{Φ_{d}} (α \otimes β)) = \frac{1}{d} Tr (α^{T} β) \leq \frac{1}{d} .$
unlabeled · line 730
$d_{⋄} (Λ, id_{C}) \geq d_{tr} (J_{n} (Λ), P_{Φ_{d}}) \geq 1 - \frac{1}{d} .$
unlabeled · line 738
$Δ = \frac{1}{d} b = 0 \sum^{d - 1} Ad_{Z^{b}} .$
unlabeled · line 743
$∥ id_{C} - Δ ∥_{⋄} \leq 2 (1 - \frac{1}{d}) .$

Theorem: Record-only quantum-return obstruction

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