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

One physical split instrument

Gaussian preparation and outcome-conditioned splitting › One physical split instrument · lines 1085–1123

En01-C@1#eq:splitlaw

Let $η > 0$ be fixed in physical coordinates and let

m_{z} (x) = (\frac{2}{π})^{1/4} e^{- (η x^{2} - z)^{2}}, M_{z} = m_{z} (x) e^{i ϕ (x)} .

Then

\int_{R} M_{z}^{†} M_{z} d z = 1.

For hypothesis $h$ ,

τ_{h} (z) = M_{z} ρ_{h}^{pre} M_{z}^{†}, p_{h} (z) = tr τ_{h} (z), s_{h} (W) = \int_{W} p_{h} (z) d z .

For a Gaussian position marginal,

p_{h} (z) = \frac{2}{π} \int_{R} e^{- 2 (η x^{2} - z)^{2}} \frac{e ^{- x^{2} / (2 v_{x, h})}}{2 π v _{x, h}} d x .

Thus the outcome law depends on position variance, not directly on purity or point-pair coherence.

The normalized likelihood is the classical representation

Z = η X^{2} + N, N \sim N (0, 1/4),

with $X$ distributed according to the input position density. It is the physical outcome law, not a claim that physical outcomes are the reference Wiener innovations.

Displayed formulas in this passage

En01-C@1#eq:splitlaw · line 1107
$p_{h} (z) = \frac{2}{π} \int_{R} e^{- 2 (η x^{2} - z)^{2}} \frac{e ^{- x^{2} / (2 v_{x, h})}}{2 π v _{x, h}} d x .$
unlabeled · line 1088
$m_{z} (x) = (\frac{2}{π})^{1/4} e^{- (η x^{2} - z)^{2}}, M_{z} = m_{z} (x) e^{i ϕ (x)} .$
unlabeled · line 1095
$\int_{R} M_{z}^{†} M_{z} d z = 1.$
unlabeled · line 1099
$τ_{h} (z) = M_{z} ρ_{h}^{pre} M_{z}^{†}, p_{h} (z) = tr τ_{h} (z), s_{h} (W) = \int_{W} p_{h} (z) d z .$
unlabeled · line 1117
$Z = η X^{2} + N, N \sim N (0, 1/4),$

One physical split instrument

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