Manuscript X-K-ST@1 · X-K-ST.tex · definition · 93c39c0f

Definition: Complete node instrument

Finite interfaces, fresh clocks, and complete memory › Declared instruments and their Kraus refinements · lines 424–447

X-K-ST@1#def:node, X-K-ST@1#eq:nodecomplete

At a node $v$ , a finite outcome set $A_{v}$ and finite Kraus families

L_{v, a, μ} : H_{v} ⟶ H_{v a}

are supplied, with

a \in A_{v} \sum μ \sum L_{v, a, μ}^{†} L_{v, a, μ} = I_{H_{v}} .

The public outcome is $a$ , while $μ$ is an unobserved Kraus index unless separately reported.

The selective map is

Γ_{v, a} (X) = μ \sum L_{v, a, μ} X L_{v, a, μ}^{†},

and the recorded node channel appends $a$ to the public history.

Displayed formulas in this passage

X-K-ST@1#eq:nodecomplete · line 432
$a \in A_{v} \sum μ \sum L_{v, a, μ}^{†} L_{v, a, μ} = I_{H_{v}} .$
unlabeled · line 428
$L_{v, a, μ} : H_{v} ⟶ H_{v a}$
unlabeled · line 441
$Γ_{v, a} (X) = μ \sum L_{v, a, μ} X L_{v, a, μ}^{†},$