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

Proposition: Flagged-cutoff check

Disorder, omitted residuals, and cutoff discipline › Discarded probability is not a coherent cutoff error · lines 1697–1720 · proof included or linked

Hawk02-E@1#prop:cutoff-check

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Let

∣ + ⟩ = \frac{∣ 0 ⟩ + ∣ 1 ⟩}{2} .

Replacing the rejected $∣ 1 ⟩$ component by a distinct failure flag gives

σ = \frac{1}{2} ∣ 0 ⟩ ⟨ 0 ∣ + \frac{1}{2} ∣ ⊥ ⟩ ⟨ ⊥ ∣ .

In the common three-dimensional output space,

d_{tr} (∣ + ⟩ ⟨ + ∣, σ) = \frac{5 + 1}{4} > \frac{1}{2} .

Proof. The nonzero two-dimensional block of $∣ + ⟩ ⟨ + ∣ - σ$ has eigenvalues $(1 + 5) /4$ and $(1 - 5) /4$ . The failure block has eigenvalue $- 1/2$ . Taking half the sum of their absolute values proves the result.

Displayed formulas in this passage

unlabeled · line 1700
$∣ + ⟩ = \frac{∣ 0 ⟩ + ∣ 1 ⟩}{2} .$
unlabeled · line 1705
$σ = \frac{1}{2} ∣ 0 ⟩ ⟨ 0 ∣ + \frac{1}{2} ∣ ⊥ ⟩ ⟨ ⊥ ∣ .$
unlabeled · line 1709
$d_{tr} (∣ + ⟩ ⟨ + ∣, σ) = \frac{5 + 1}{4} > \frac{1}{2} .$

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section: Discarded probability is not a coherent cutoff error