Manuscript Em04-A@2 · Em04-A-v2.tex · theorem · f5b28abc

Theorem: Instrument consistency and no-signaling

Finite local instruments and typed records · lines 942–979 · proof included or linked

Em04-A@2#thm:instruments

For a density matrix $ρ$ :

$p_{a} = Tr Φ_{a} (ρ) = Tr (E_{a} ρ) \geq 0$ , and $\sum_{a} p_{a} = 1$ .
If $p_{a} > 0$ , the conditional state is $Φ_{a} (ρ) / p_{a}$ . If $p_{a} = 0$ , then $Φ_{a} (ρ) = 0$ ; no normalized branch is assigned.
$Φ$ is CPTP.
Actual tensor-local instruments on disjoint factors commute as selective superoperators after identifying output types.
A remote trace-preserving channel leaves local outcome probabilities unchanged, including for entangled inputs.

Proof. Kraus positivity, cyclicity, and completeness prove (i). A positive matrix of trace zero is zero, proving (ii). The record Kraus operators $∣ a ⟩_{R} \otimes M_{a μ}$ are complete, proving (iii). For separate factors, composition in either order gives the same Kraus operator $M_{a μ} \otimes K_{b ν}$ , proving (iv).

For a remote channel with $\sum_{b} K_{b}^{†} K_{b} = I_{B}$ ,

p^{'}_{a} = b \sum Tr [(E_{a} \otimes I) (I \otimes K_{b}) ρ (I \otimes K_{b}^{†})] = Tr [(E_{a} \otimes b \sum K_{b}^{†} K_{b}) ρ] = Tr [(E_{a} \otimes I) ρ] .

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$p^{'}_{a} = b \sum Tr [(E_{a} \otimes I) (I \otimes K_{b}) ρ (I \otimes K_{b}^{†})] = Tr [(E_{a} \otimes b \sum K_{b}^{†} K_{b}) ρ] = Tr [(E_{a} \otimes I) ρ] .$

Theorem: Instrument consistency and no-signaling

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