Manuscript VII-E@1 · VII-E.tex · section · b713cf3c

Record agreement does not determine residuals

Record compatibility, residuals, and composition › Record agreement does not determine residuals · lines 1314–1332

VII-E@1#eq:fine-instrument, VII-E@1#eq:coarse-instrument

Now let the fine measurement be a PVM ${P_{z}}$ . Define

F (X) = w \sum ∣ w ⟩ ⟨ w ∣_{R} \otimes T z : q (z) = w \sum P_{z} X P_{z} .

For a coarse instrument with outcome maps $I_{w}$ , put

G (X) = w \sum ∣ w ⟩ ⟨ w ∣_{R} \otimes I_{w} (T (X)), E_{w} = I_{w}^{*} (I) .

These channels have the same record–residual output interface.

Displayed formulas in this passage

VII-E@1#eq:fine-instrument · line 1318
$F (X) = w \sum ∣ w ⟩ ⟨ w ∣_{R} \otimes T z : q (z) = w \sum P_{z} X P_{z} .$
VII-E@1#eq:coarse-instrument · line 1325
$G (X) = w \sum ∣ w ⟩ ⟨ w ∣_{R} \otimes I_{w} (T (X)), E_{w} = I_{w}^{*} (I) .$