Manuscript Oai290-A@1 · Oai290-A.tex · section · 7c6e104a

Rank is not a stable approximate error parameter

Analytic examples and separation tests › Rank is not a stable approximate error parameter · lines 1745–1762

For $p \in [0, 1]$ , let $A = C \oplus C ∣ e ⟩$ and $B = C ∣ v ⟩ \oplus C$ , and define the flagged-erasure isometry

V_{p} ∣ ψ ⟩ = 1 - p ∣ ψ ⟩_{A} ∣ v ⟩_{B} + p ∣ e ⟩_{A} ∣ ψ ⟩_{B} .

Then

N_{p} (X) = (1 - p) X \oplus p Tr (X) ∣ e ⟩ ⟨ e ∣,

C_{p} (X) = (1 - p) Tr (X) ∣ v ⟩ ⟨ v ∣ \oplus pX .

Displayed formulas in this passage

unlabeled · line 1750
$V_{p} ∣ ψ ⟩ = 1 - p ∣ ψ ⟩_{A} ∣ v ⟩_{B} + p ∣ e ⟩_{A} ∣ ψ ⟩_{B} .$
unlabeled · line 1756
$N_{p} (X) = (1 - p) X \oplus p Tr (X) ∣ e ⟩ ⟨ e ∣,$
unlabeled · line 1759
$C_{p} (X) = (1 - p) Tr (X) ∣ v ⟩ ⟨ v ∣ \oplus pX .$