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

Theorem: Projective update

Born laws and destructive projective residuals › Residual identification by future tests · lines 1764–1836 · proof included or linked

I@2#thm:update

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Assume the Born outcome law, $(C)^{⋆}$ , and the continuation instances of $(R)^{⋆}$ . If

ψ = κ \sum ∣ κ ⟩_{S} ∣ ξ_{κ} ⟩_{B}, p_{κ} = ∥ ξ_{κ} ∥^{2} > 0,

then

U^{S}_{ψ} (κ) ≐ \frac{ξ _{κ}}{∥ ξ _{κ} ∥} .

Proof. Fix a positive branch $κ$ . Choose unit representatives

u_{κ} = U^{S}_{ψ} (κ), φ_{κ} = ξ_{κ} / ∥ ξ_{κ} ∥ .

If $B = \emptyset$ , its Hilbert space is one dimensional and the claim is immediate.

For a unitary $V$ on $B$ , use

t_{V} = let x = Enter (q_{S}) in ⟨ Enter (V (q_{B})), x ⟩ .

By the unknown strategy's transition rule and state sufficiency, the joint source weight for $(λ, κ)$ is

p_{κ} P^{B}_{V u_{κ}} (λ) .

This does not use a Born update.

The deferred vector after copying and applying $V$ , before either readout, is

Ψ^{'}_{V} = j \sum ∣ j ⟩_{S} (V ξ_{j})_{B} ∣ j ⟩_{R} .

The target reads $B$ , then $R$ , and returns $S$ . Composability equates its joint law to joint $B \cup R$ readout. The established Born outcome law gives joint weight

∣ ⟨ λ ∣ V ∣ ξ_{κ} ⟩ ∣^{2} .

Records therefore imply

P^{B}_{V u_{κ}} (λ) = ∣ ⟨ λ ∣ V ∣ φ_{κ} ⟩ ∣^{2} .

Using the established outcome law on the left,

∣ ⟨ λ ∣ V ∣ u_{κ} ⟩ ∣^{2} = ∣ ⟨ λ ∣ V ∣ φ_{κ} ⟩ ∣^{2}

for every $V, λ$ .

Choose $V φ_{κ} = ∣ 0^{∣ B ∣} ⟩$ . Then

⟨ 0^{∣ B ∣} ∣ V ∣ u_{κ} ⟩^{2} = 1,

so the unit rays agree.

Displayed formulas in this passage

unlabeled · line 1769
$ψ = κ \sum ∣ κ ⟩_{S} ∣ ξ_{κ} ⟩_{B}, p_{κ} = ∥ ξ_{κ} ∥^{2} > 0,$
unlabeled · line 1775
$U^{S}_{ψ} (κ) ≐ \frac{ξ _{κ}}{∥ ξ _{κ} ∥} .$
unlabeled · line 1784
$u_{κ} = U^{S}_{ψ} (κ), φ_{κ} = ξ_{κ} / ∥ ξ_{κ} ∥ .$
unlabeled · line 1793
$t_{V} = let x = Enter (q_{S}) in ⟨ Enter (V (q_{B})), x ⟩ .$
unlabeled · line 1800
$p_{κ} P^{B}_{V u_{κ}} (λ) .$
unlabeled · line 1807
$Ψ^{'}_{V} = j \sum ∣ j ⟩_{S} (V ξ_{j})_{B} ∣ j ⟩_{R} .$
unlabeled · line 1814
$∣ ⟨ λ ∣ V ∣ ξ_{κ} ⟩ ∣^{2} .$
unlabeled · line 1818
$P^{B}_{V u_{κ}} (λ) = ∣ ⟨ λ ∣ V ∣ φ_{κ} ⟩ ∣^{2} .$
unlabeled · line 1823
$∣ ⟨ λ ∣ V ∣ u_{κ} ⟩ ∣^{2} = ∣ ⟨ λ ∣ V ∣ φ_{κ} ⟩ ∣^{2}$
unlabeled · line 1832
$⟨ 0^{∣ B ∣} ∣ V ∣ u_{κ} ⟩^{2} = 1,$

Theorem: Projective update

Displayed formulas in this passage

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