Manuscript Em05-B@1 · Em05-B-v1.tex · lemma · 6c050f5b

Lemma: Replacement, separability, and Pauli witnesses

Notation, tasks, and elementary witnesses · lines 363–404 · proof included or linked

Em05-B@1#lem:witnesses, Em05-B@1#eq:PauliComparison

For a decoded replacement qubit channel,

d_{⋄} (R_{τ}, id_{M}) \geq \frac{3}{4} .

For an entanglement-breaking qubit channel $Λ$ ,

F_{e} (Λ) \leq \frac{1}{2}, d_{⋄} (Λ, id_{M}) \geq \frac{1}{2} .

Consequently an entanglement-breaking accessible channel obeys $ε_{rec} (N) \geq 1/2$ .

For qubit Pauli channels, with $P \in {I, X, Y, Z}$ ,

Φ = P \sum a_{P} Ad_{P}, Ψ = P \sum b_{P} Ad_{P} ⟹ d_{⋄} (Φ, Ψ) = \frac{1}{2} P \sum ∣ a_{P} - b_{P} ∣.

In particular, $d_{⋄} (Φ, id_{M}) = 1 - a_{I}$ .

Proof. A replacement channel has Bell output $I_{Q} /2 \otimes τ$ , with Bell overlap $1/4$ . A normalized product vector satisfies

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

by Cauchy–Schwarz. Convexity extends this bound to all separable states. Local decoding preserves separability. Equation (eq:BellWitness) proves the lower bounds.

For (eq:PauliComparison), the channel triangle inequality and $∥ Ad_{P} ∥_{⋄} = 1$ give the upper bound. The four Pauli outputs on $∣Φ ⟩$ are orthogonal Bell projectors; their trace distance gives the matching lower bound.

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Em05-B@1#eq:PauliComparison · line 379
$Φ = P \sum a_{P} Ad_{P}, Ψ = P \sum b_{P} Ad_{P} ⟹ d_{⋄} (Φ, Ψ) = \frac{1}{2} P \sum ∣ a_{P} - b_{P} ∣.$
unlabeled · line 366
$d_{⋄} (R_{τ}, id_{M}) \geq \frac{3}{4} .$
unlabeled · line 370
$F_{e} (Λ) \leq \frac{1}{2}, d_{⋄} (Λ, id_{M}) \geq \frac{1}{2} .$
unlabeled · line 393
$∣ ⟨ Φ∣ a \otimes b ⟩ ∣^{2} \leq \frac{1}{2}$

Lemma: Replacement, separability, and Pauli witnesses

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