Manuscript Hawk04-A@1 · Hawk04-A.tex · example · 526092ba

Example: Equal marginals for two basis states

Reset, dephasing, and omitted purifiers › Restricted ensembles can hide information in correlations · lines 656–673

Define

V ∣ 0 ⟩ = \frac{∣ 00 ⟩ + ∣ 11 ⟩}{2}, V ∣ 1 ⟩ = \frac{∣ 00 ⟩ - ∣ 11 ⟩}{2} .

The two basis inputs have identical maximally mixed marginals on both outputs, yet orthogonal complete outputs. However,

V \frac{∣ 0 ⟩ + ∣ 1 ⟩}{2} = ∣ 00 ⟩, V \frac{∣ 0 ⟩ - ∣ 1 ⟩}{2} = ∣ 11 ⟩ .

The reduced channels are therefore not constant on the full code. There is no contradiction with Corollary cor:nomasking.

Displayed formulas in this passage

unlabeled · line 658
$V ∣ 0 ⟩ = \frac{∣ 00 ⟩ + ∣ 11 ⟩}{2}, V ∣ 1 ⟩ = \frac{∣ 00 ⟩ - ∣ 11 ⟩}{2} .$
unlabeled · line 666
$V \frac{∣ 0 ⟩ + ∣ 1 ⟩}{2} = ∣ 00 ⟩, V \frac{∣ 0 ⟩ - ∣ 1 ⟩}{2} = ∣ 11 ⟩ .$