Manuscript Hawk04-B@2 · Hawk04-B.tex · section · ba3b0713

A coherent cutoff is not discarded probability

Additional channel debugging checks › A coherent cutoff is not discarded probability · lines 2511–2532

In a common three-dimensional output space, put

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

The difference has eigenvalues

\frac{1 + 5}{4}, \frac{1 - 5}{4}, - \frac{1}{2} .

Therefore

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

A discarded weight is not a generic coherent-channel allowance. The flagged-cutoff comparisons in CheungEn02E retain their commuting-state assumptions. The regulator used here instead compares complete prepared isometries.

Displayed formulas in this passage

unlabeled · line 2514
$∣ + ⟩ = \frac{∣ 0 ⟩ + ∣ 1 ⟩}{2}, σ = \frac{1}{2} ∣ 0 ⟩ ⟨ 0 ∣ + \frac{1}{2} ∣ ⊥ ⟩ ⟨ ⊥ ∣ .$
unlabeled · line 2519
$\frac{1 + 5}{4}, \frac{1 - 5}{4}, - \frac{1}{2} .$
unlabeled · line 2524
$d_{tr} (∣ + ⟩ ⟨ + ∣, σ) = \frac{5 + 1}{4} > \frac{1}{2} .$