Manuscript En03-A@1 · En03-A.tex · theorem · 5398614e

Theorem: Residual-state obstruction

Scale, records, and residual instruments › Residual instruments · lines 2475–2496 · proof included or linked

En03-A@1#thm:obstruction, En03-A@1#eq:obstruction-outputs, En03-A@1#eq:obstruction-norm

Exact record compatibility does not imply $F = G$ , even when $T = id$ .

Proof. Take one qubit, computational $P_{0}, P_{1}$ , one outcome $W = {*}$ , and the coarse identity instrument. Then $E_{*} = Π_{*} = I$ . For $ρ = ∣ + ⟩ ⟨ + ∣$ ,

G (ρ) = ∣ * ⟩ ⟨ * ∣_{R} \otimes ∣ + ⟩ ⟨ + ∣, F (ρ) = ∣ * ⟩ ⟨ * ∣_{R} \otimes I /2.

The residual difference is $X /2$ , with eigenvalues $\pm 1/2$ , so

∥ G (ρ) - F (ρ) ∥_{1} = 1.

The record error is zero, while residual trace distance is $1/2$ .

Displayed formulas in this passage

En03-A@1#eq:obstruction-outputs · line 2485
$G (ρ) = ∣ * ⟩ ⟨ * ∣_{R} \otimes ∣ + ⟩ ⟨ + ∣, F (ρ) = ∣ * ⟩ ⟨ * ∣_{R} \otimes I /2.$
En03-A@1#eq:obstruction-norm · line 2491
$∥ G (ρ) - F (ρ) ∥_{1} = 1.$

Theorem: Residual-state obstruction

Displayed formulas in this passage

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