Manuscript X-L-ST@1 · X-L-ST.tex · theorem · 20a32a22

Theorem: Complete finite composition

Finite composition, causal transport, and error accounting › Complete adaptive histories · lines 1762–1785 · proof included or linked

X-L-ST@1#thm:compose, X-L-ST@1#eq:history

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The complete leaf-history channel is

C (X) = a leaf ⨁ μ \sum K_{a, μ} X K_{a, μ}^{†}, a, μ \sum K_{a, μ}^{†} K_{a, μ} = I,

where $K_{a, μ}$ is the compatible ordered product along that refined history. All failures and reference extensions are included.

Proof. At each node, pull the completeness identity of its child continuations back through its Kraus maps. Backward induction over the finite tree gives the identity in (eq:history). Tensor every map with $I_{E}$ for the reference statement.

Displayed formulas in this passage

X-L-ST@1#eq:history · line 1765
$C (X) = a leaf ⨁ μ \sum K_{a, μ} X K_{a, μ}^{†}, a, μ \sum K_{a, μ}^{†} K_{a, μ} = I,$

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section: Complete adaptive histories
section: Complete adaptive histories