Manuscript IX-A-ST@1 · IX-A-ST.tex · proposition · 6e99d1b2

Proposition: Sharp coarse Lüders compatibility

Residual compatibility, blindness, and recovery safeguards · lines 1682–1710 · proof included or linked

IX-A-ST@1#prop:luders

If $A_{w} (Y) = Q_{w} Y Q_{w}$ and $T^{*} (Q_{w}) = R_{w}$ for projections $Q_{w}, R_{w}$ , then

T^{*} (Q_{w} B Q_{w}) = R_{w} T^{*} (B) R_{w} .

Agreement with the aggregated fine residual instrument further requires, and is equivalent to,

P_{z} T^{*} (B) P_{z^{'}} = 0 (z \neq = z^{'}, q (z) = q (z^{'}) = w)

for the declared residual observables.

Proof. Write $T^{*} (B) = V^{†} (B \otimes I) V$ with $V$ an isometry obtained from a Kraus family. Expansion shows

((Q_{w} \otimes I) V - V R_{w})^{†} ((Q_{w} \otimes I) V - V R_{w}) = 0.

Thus $(Q_{w} \otimes I) V = V R_{w}$ , giving the first identity. Insert it into Proposition prop:compatibility; multiplication by distinct $P_{z}, P_{z^{'}}$ proves the remaining equivalence.

Displayed formulas in this passage

unlabeled · line 1686
$T^{*} (Q_{w} B Q_{w}) = R_{w} T^{*} (B) R_{w} .$
unlabeled · line 1691
$P_{z} T^{*} (B) P_{z^{'}} = 0 (z \neq = z^{'}, q (z) = q (z^{'}) = w)$
unlabeled · line 1702
$((Q_{w} \otimes I) V - V R_{w})^{†} ((Q_{w} \otimes I) V - V R_{w}) = 0.$

Proposition: Sharp coarse Lüders compatibility

Displayed formulas in this passage

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