Manuscript En02-A@1 · En02-A-v1.tex · lemma · bb1319d5

Lemma: Born inhabitation

Finite operational strategies and explicit requirements › The five capacity-restricted requirements · lines 428–470 · proof included or linked

En02-A@1#lem:born-inhabitation

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The strategy

P^{S}_{ψ} (κ) = b^{S}_{ψ} (κ), U^{S}_{ψ} (κ) = [ψ^{S}_{L, κ}]

belongs to $A_{N}$ for every $N$ .

Proof. Normalization follows from $\sum_{κ} (M^{S}_{κ})^{†} M^{S}_{κ} = I$ . Disjoint destructive components commute after identifying surviving factors, proving composability. Nullity and continuity follow directly from squared norms.

For a finite local program, compose allocation isometries, unitaries, and destructive measurement bras along each history. Backward induction on the outcome tree gives a complete family of path maps: $\sum_{h} K_{h}^{†} K_{h} = I$ . For bipartite histories the maps factor as $K_{a} \otimes L_{b}$ . Summing over the remote histories gives

b \sum ⟨ ψ ∣ K_{a}^{†} K_{a} \otimes L_{b}^{†} L_{b} ∣ ψ ⟩ = ⟨ ψ ∣ K_{a}^{†} K_{a} \otimes I ∣ ψ ⟩ .

This proves the specified no-signaling requirement.

For records write $ψ = \sum_{κ} ∣ κ ⟩_{S} ∣ ξ_{κ} ⟩_{B}$ . A continuation history $h$ , with path map $K_{h}$ , has joint weight $∥ K_{h} ξ_{κ} ∥^{2}$ in the direct program. The copied program has pre-readout vector

κ \sum ∣ κ ⟩_{S} K_{h} ∣ ξ_{κ} ⟩_{B} ∣ κ ⟩_{R},

and record readout gives the same weight. Summing histories proves record equality. All compared paths obey the capacity restriction.

Displayed formulas in this passage

unlabeled · line 431
$P^{S}_{ψ} (κ) = b^{S}_{ψ} (κ), U^{S}_{ψ} (κ) = [ψ^{S}_{L, κ}]$
unlabeled · line 451
$b \sum ⟨ ψ ∣ K_{a}^{†} K_{a} \otimes L_{b}^{†} L_{b} ∣ ψ ⟩ = ⟨ ψ ∣ K_{a}^{†} K_{a} \otimes I ∣ ψ ⟩ .$
unlabeled · line 464
$κ \sum ∣ κ ⟩_{S} K_{h} ∣ ξ_{κ} ⟩_{B} ∣ κ ⟩_{R},$

Lemma: Born inhabitation

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