Manuscript I@2 · I.tex · theorem · 85631fa7

Theorem: Internal no-signaling of

Metatheory of the specified calculus › Internal no-signaling · lines 928–953 · proof included or linked

I@2#thm:internal-ns

Alice's terminal classical law is independent of Bob's local program and of the admissible interleaving.

Proof. Let $K_{a}$ and $L_{b}$ be their local path maps, including allocations and returned outputs. Disjoint maps commute with the canonical output identifications. The joint history weight is $∥ (K_{a} \otimes L_{b}) ψ ∥^{2}$ . Consequently

Pr (a) = b \sum ⟨ ψ ∣ K_{a}^{†} K_{a} \otimes L_{b}^{†} L_{b} ∣ ψ ⟩ = ⟨ ψ ∣ K_{a}^{†} K_{a} \otimes I ∣ ψ ⟩ = tr (K_{a}^{†} K_{a} ρ_{A}),

where the second line uses local completeness. Summing histories with the same reported output proves the claim. The calculation is independent of the interleaving.

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unlabeled · line 941
$Pr (a) = b \sum ⟨ ψ ∣ K_{a}^{†} K_{a} \otimes L_{b}^{†} L_{b} ∣ ψ ⟩ = ⟨ ψ ∣ K_{a}^{†} K_{a} \otimes I ∣ ψ ⟩ = tr (K_{a}^{†} K_{a} ρ_{A}),$

Theorem: Internal no-signaling of

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