Manuscript VII-G@1 · VII-G.tex · example · 54f561e8

Example: Equal records, different future geometry

Implementation equivalence and represented readout › A counterexample using the same effects · lines 1065–1090

VII-G@1#ex:residual

For a qubit, consider two fair two-outcome instruments:

Ψ_{a} (X) = \frac{1}{2} X, Ψ_{a} (X) = \frac{1}{2} Δ_{Z} (X), a \in {0, 1},

where

Δ_{Z} (X) = P_{0} X P_{0} + P_{1} X P_{1} .

Both have effects $I /2$ . On input $∣ + ⟩ ⟨ + ∣$ , their normalized residuals are respectively $∣ + ⟩ ⟨ + ∣$ and $I /2$ , at trace distance $1/2$ .

If the next geometric output records an $X$ -basis measurement, the first implementation gives

p (a, +) = \frac{1}{2}, p (a, -) = 0,

whereas the second gives $p (a, \pm) = 1/4$ . The complete two-step geometric laws have total-variation distance $1/2$ .

Displayed formulas in this passage

unlabeled · line 1068
$Ψ_{a} (X) = \frac{1}{2} X, Ψ_{a} (X) = \frac{1}{2} Δ_{Z} (X), a \in {0, 1},$
unlabeled · line 1075
$Δ_{Z} (X) = P_{0} X P_{0} + P_{1} X P_{1} .$
unlabeled · line 1084
$p (a, +) = \frac{1}{2}, p (a, -) = 0,$