Manuscript XVII-A@1 · XVII-A.tex · lemma · 711d7ee4

Lemma: Indistinguishable-input obstruction

The reconstructed record does not recover a second observable · lines 908–934 · proof included or linked

XVII-A@1#lem:indistinguishable

Let $C$ be a channel and let $J$ be an effect. If

C (ρ_{0}) = C (ρ_{1}), ∣ Tr (J ρ_{0}) - Tr (J ρ_{1})∣ = Δ,

then every scalar estimator $h$ depending only on the output of $C$ has worst-case expectation error at least $Δ/2$ :

ρ sup ∣ h (C (ρ)) - Tr (J ρ)∣ \geq \frac{Δ}{2} .

The assertion does not require $h$ to be linear.

Proof. The estimator assigns the same value $s$ to the two equal outputs. For the two target values $j_{0}, j_{1}$ ,

Δ = ∣ j_{0} - j_{1} ∣ \leq ∣ j_{0} - s ∣ + ∣ s - j_{1} ∣ .

At least one error is at least $Δ/2$ .

Displayed formulas in this passage

unlabeled · line 911
$C (ρ_{0}) = C (ρ_{1}), ∣ Tr (J ρ_{0}) - Tr (J ρ_{1})∣ = Δ,$
unlabeled · line 918
$ρ sup ∣ h (C (ρ)) - Tr (J ρ)∣ \geq \frac{Δ}{2} .$
unlabeled · line 929
$Δ = ∣ j_{0} - j_{1} ∣ \leq ∣ j_{0} - s ∣ + ∣ s - j_{1} ∣ .$