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

Lemma: Record marginal identity

Finite operational strategies and explicit requirements › Locality and record marginalization · lines 514–547 · proof included or linked

En02-A@1#lem:record-marginal, En02-A@1#eq:copy, En02-A@1#eq:record-marginal

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Suppose $n = ∣ L ∣$ , $s = ∣ S ∣$ , and $n + s \leq N$ . Define the copied state

C_{S} ψ = κ \sum ∣ κ ⟩_{S} ∣ ξ_{κ} ⟩_{B} ∣ κ ⟩_{R}, ψ = κ \sum ∣ κ ⟩_{S} ∣ ξ_{κ} ⟩_{B} .

Then

P^{S}_{ψ} = P^{R}_{C_{S} ψ}, ρ_{R} = κ \sum ∥ ξ_{κ} ∥^{2} ∣ κ ⟩ ⟨ κ ∣ .

Proof. Use the legal continuation $u = Enter (B)$ , retaining all its classical outputs. If $B$ is empty, use the unit continuation. The direct joint law has $S$ -marginal $P^{S}_{ψ}$ by composability. In the deferred program, the marginal law of $R$ equals its law before the remote readout of $B$ , by no-signaling. Record equality identifies these marginals. The density-operator formula follows by tracing the copied vector: the original computational labels in $S$ are orthogonal.

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En02-A@1#eq:copy · line 518
$C_{S} ψ = κ \sum ∣ κ ⟩_{S} ∣ ξ_{κ} ⟩_{B} ∣ κ ⟩_{R}, ψ = κ \sum ∣ κ ⟩_{S} ∣ ξ_{κ} ⟩_{B} .$
En02-A@1#eq:record-marginal · line 528
$P^{S}_{ψ} = P^{R}_{C_{S} ψ}, ρ_{R} = κ \sum ∥ ξ_{κ} ∥^{2} ∣ κ ⟩ ⟨ κ ∣ .$

Lemma: Record marginal identity

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