Manuscript VIII-ST@1 · VIII-ST.tex · proposition · fe979f3b

Proposition: Complete observation criterion

Named access, residual instruments, and recovery boundaries › Record equality versus residual equality · lines 1148–1170 · proof included or linked

VIII-ST@1#prop:residual-criterion

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The record laws agree on every input exactly when

T^{*} (I_{w}^{*} (I)) = R_{w} .

Agreement on a declared recorded residual algebra $O$ holds exactly when

z : q (z) = w \sum P_{z} T^{*} (B) P_{z} = T^{*} (I_{w}^{*} (B)), B \in O .

These linear-map identities remain true with untouched references.

Proof. Pair each recorded block with $I$ , or with $B$ , and move the maps to the input by their adjoints. States separate Hermitian operators, and Hermitian elements span the matrix algebra. Reference extension tensors the resulting identities with the reference identity.

Displayed formulas in this passage

unlabeled · line 1151
$T^{*} (I_{w}^{*} (I)) = R_{w} .$
unlabeled · line 1156
$z : q (z) = w \sum P_{z} T^{*} (B) P_{z} = T^{*} (I_{w}^{*} (B)), B \in O .$

Proposition: Complete observation criterion

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