Manuscript Hawk02-E@1 · Hawk02-E.tex · proposition · 4841592b

Proposition: Qubit complete-residual test

Complete-residual certificates and decoder certificates › All qubit coherences · lines 1275–1316 · proof included or linked

Hawk02-E@1#prop:pauli

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Choose $σ_{B} = C_{t} (I_{C} /2)$ . Then

e_{J} (t) \leq \frac{1}{4} [∥ C_{t} (X) ∥_{1} + ∥ C_{t} (Y) ∥_{1} + ∥ C_{t} (Z) ∥_{1}] .

Hence

δ_{full} (t) \leq \frac{1}{2} [∥ C_{t} (X) ∥_{1} + ∥ C_{t} (Y) ∥_{1} + ∥ C_{t} (Z) ∥_{1}] .

The same statement holds for the finite approximation.

Proof. The Bell projector has the expansion

∣ Φ_{2} ⟩ ⟨ Φ_{2} ∣ = \frac{1}{4} (I \otimes I + X \otimes X - Y \otimes Y + Z \otimes Z) .

Subtracting $(I_{Q} /2) \otimes C_{t} (I_{C} /2)$ removes the identity term. Each Pauli matrix has trace norm two. The triangle inequality gives the first estimate, and Lemma lem:choi gives the second.

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unlabeled · line 1278
$e_{J} (t) \leq \frac{1}{4} [∥ C_{t} (X) ∥_{1} + ∥ C_{t} (Y) ∥_{1} + ∥ C_{t} (Z) ∥_{1}] .$
unlabeled · line 1289
$δ_{full} (t) \leq \frac{1}{2} [∥ C_{t} (X) ∥_{1} + ∥ C_{t} (Y) ∥_{1} + ∥ C_{t} (Z) ∥_{1}] .$
unlabeled · line 1304
$∣ Φ_{2} ⟩ ⟨ Φ_{2} ∣ = \frac{1}{4} (I \otimes I + X \otimes X - Y \otimes Y + Z \otimes Z) .$

Proposition: Qubit complete-residual test

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