Manuscript XIX-D@1 · XIX-D.tex · proposition · 1f47e33b

Proposition: Identical records for all inputs, different residuals

Finite instruments and continuum quantum non-determination › Record agreement can conceal different residual channels · lines 1015–1045 · proof included or linked

XIX-D@1#prop:record

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There are two complete recorded channels with identical classical record laws on every input but different residual outputs on one qubit input.

Proof. Let $P_{0} = ∣ 0 ⟩ ⟨ 0 ∣$ , $P_{1} = ∣ 1 ⟩ ⟨ 1 ∣$ , and define

D (ρ) = P_{0} ρ P_{0} + P_{1} ρ P_{1} .

Use one record symbol $*$ and the channels

G (ρ) = ∣ * ⟩ ⟨ * ∣_{R} \otimes ρ, F (ρ) = ∣ * ⟩ ⟨ * ∣_{R} \otimes D (ρ) .

Both always report $*$ . For $ρ = ∣ + ⟩ ⟨ + ∣$ , where $∣ + ⟩ = (∣ 0 ⟩ + ∣ 1 ⟩) / 2$ ,

D (ρ) = I_{2} /2.

The residual difference is $X /2$ , with eigenvalues $1/2, - 1/2$ . Therefore

d_{tr} (G (ρ), F (ρ)) = \frac{1}{2} ∥ G (ρ) - F (ρ) ∥_{1} = \frac{1}{2} .

Displayed formulas in this passage

unlabeled · line 1024
$D (ρ) = P_{0} ρ P_{0} + P_{1} ρ P_{1} .$
unlabeled · line 1028
$G (ρ) = ∣ * ⟩ ⟨ * ∣_{R} \otimes ρ, F (ρ) = ∣ * ⟩ ⟨ * ∣_{R} \otimes D (ρ) .$
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$D (ρ) = I_{2} /2.$
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$d_{tr} (G (ρ), F (ρ)) = \frac{1}{2} ∥ G (ρ) - F (ρ) ∥_{1} = \frac{1}{2} .$

Proposition: Identical records for all inputs, different residuals

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