Manuscript Em03-H@8 · Em03-H-v8.tex · theorem · 91e9e3bd

Theorem: Complete finite channel interface

Frozen microscopic model and complete preparation › Code and channels · lines 488–518 · proof included or linked

Em03-H@8#thm:channels

The finite path has a unique unitary propagator. For

W^{0} x = V x \otimes ∣ \emptyset ⟩_{R}, Z_{A, t} = R_{A} U (t) W^{0},

the accessible channel and complete complement are the two partial traces of $Ad_{Z_{A, t}}$ .

If $σ_{K} = \sum_{a} p_{a} ∣ a_{σ} ⟩ ⟨ a_{σ} ∣$ , define

∣ Ξ_{i} ⟩ = a \sum p_{a} Z_{A, t} ∣ i, a_{σ} ⟩ \otimes ∣ a ⟩_{G} .

With $E = B (R ∖ A) G$ ,

N^{σ}_{A, t} (E_{ij}) = Tr_{E} ∣ Ξ_{i} ⟩ ⟨ Ξ_{j} ∣, C^{σ}_{A, t} (E_{ij}) = Tr_{A} ∣ Ξ_{i} ⟩ ⟨ Ξ_{j} ∣ .

These formulas determine every referenced input.

Proof. Finite-dimensional integral evolution gives existence and uniqueness; self-adjointness gives unitarity. Preparation, evolution, regrouping, and purification form an isometry. Its partial traces are CPTP. Matrix-unit expansion gives the formulas and their reference extensions.

Displayed formulas in this passage

unlabeled · line 492
$W^{0} x = V x \otimes ∣ \emptyset ⟩_{R}, Z_{A, t} = R_{A} U (t) W^{0},$
unlabeled · line 500
$∣ Ξ_{i} ⟩ = a \sum p_{a} Z_{A, t} ∣ i, a_{σ} ⟩ \otimes ∣ a ⟩_{G} .$
unlabeled · line 505
$N^{σ}_{A, t} (E_{ij}) = Tr_{E} ∣ Ξ_{i} ⟩ ⟨ Ξ_{j} ∣, C^{σ}_{A, t} (E_{ij}) = Tr_{A} ∣ Ξ_{i} ⟩ ⟨ Ξ_{j} ∣ .$

Theorem: Complete finite channel interface

Displayed formulas in this passage

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