Manuscript Oai280-B@1 · Oai280-B.tex · lemma · 6d92ab75

Lemma: Exact Pauli conversion

Feasibility, binomial inversion, and channel structure › Independently established Pauli structure · lines 1707–1727 · proof included or linked

Oai280-B@1#lem:pauli

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For a qubit Pauli channel

Λ = P \in {I, X, Y, Z} \sum π_{P} Ad_{P}, π_{P} \geq 0, P \sum π_{P} = 1,

one has

F (Λ) = π_{I}, d_{⋄} (Λ, id_{2}) = 1 - π_{I} = 1 - F (Λ) .

Proof. The four Pauli actions on a Bell state give four orthogonal Bell states. The Bell overlap is therefore $π_{I}$ , and the Bell-projector test gives error at least $1 - π_{I}$ . The triangle inequality applied to the nonidentity mixture gives the matching upper bound.

Displayed formulas in this passage

unlabeled · line 1710
$Λ = P \in {I, X, Y, Z} \sum π_{P} Ad_{P}, π_{P} \geq 0, P \sum π_{P} = 1,$
unlabeled · line 1715
$F (Λ) = π_{I}, d_{⋄} (Λ, id_{2}) = 1 - π_{I} = 1 - F (Λ) .$

Nearby in this section

section: Independently established Pauli structure
proposition: Proposition: Physical logical twirling defines a Pauli task
section: Independently established Pauli structure