Manuscript En01-C@1 · En01-C.tex · proposition · 00a2161c

Proposition: Point-pair filtering identity

Gaussian preparation and outcome-conditioned splitting › One physical split instrument · lines 1124–1141 · proof included or linked

En01-C@1#prop:filter

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For any continuous positive kernel and a diagonal filter $g_{z} (x)$ , the normalized point-pair coherence magnitude is unchanged wherever $g_{z} (x) g_{z} (y) \neq = 0$ and both input diagonal entries are positive:

\frac{∣ τ _{z} ( x , y ) ∣}{τ _{z} ( x , x ) τ _{z} ( y , y )} = \frac{∣ ρ ( x , y ) ∣}{ρ ( x , x ) ρ ( y , y )} .

The identity is unaffected by normalization of the selected state.

Proof. The numerator acquires $∣ g_{z} (x) g_{z} (y) ∣$ , and the denominator acquires the same factor. A common trace normalization also cancels.

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unlabeled · line 1129
$\frac{∣ τ _{z} ( x , y ) ∣}{τ _{z} ( x , x ) τ _{z} ( y , y )} = \frac{∣ ρ ( x , y ) ∣}{ρ ( x , x ) ρ ( y , y )} .$

Proposition: Point-pair filtering identity

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