Recorded bin laws, loss and conditioning of the declared two-path detector

Adopt the observable and probability postulate P2 in its unsharp finite-outcome reading, the declared two-path preparation model The declared two-path preparation model and its ideal detection bins with path components ${\psi }_1,{\psi }_2\mathrm{\ne }0$, relative phase $\phi$ and weight $w$, and the declared recorded detector The declared recorded detector and sampling model of the two-path arrangement with finite label set $A$ satisfying $0\mathrm{\notin }A$, recorded weights $w_a$ and effects $F_a=M_{w_a}$, no-record effect $F_0=I-{\sum }_a{F}_a$, recorded probabilities $\nu (a)$ and recorded-event mass $\nu (A)={\sum }_{a\in A}\nu (a)$ (Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes). Assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The Axiom of Choice), the choice hypotheses of the cited trace-class, rank-one and square-root suppliers. Let ${\rho }_{\mathrm{c}\mathrm{o}\mathrm{h}}(\phi )$ and ${\rho }_{\mathrm{m}\mathrm{i}\mathrm{x}}(w)$ be the declared coherent and which-path preparations and put $N:=\mathrm{\parallel }{\psi }_1{\mathrm{\parallel }}^2+\mathrm{\parallel }{\psi }_2{\mathrm{\parallel }}^2$, $c:=\mathrm{Re}⟨{\psi }_1,{\psi }_2⟩$ and $w_{\ast }:=\mathrm{\parallel }{\psi }_1{\mathrm{\parallel }}^2\mathrm{/}N$. Then:

(a) Recorded coherent law. If the model is admissible at phase $\phi$, then for every label $a\in A$ ${\nu }_{\mathrm{c}\mathrm{o}\mathrm{h}(\phi )}(a)=\frac{{\displaystyle {\int }_{{\mathbb{R}}^3}{w}_a(x)\mathrm{\parallel }{\psi }_1(x)+e^{i\phi }{\psi }_2(x){\mathrm{\parallel }}_{{\mathbb{C}}^r}^{2}dx}}{\mathrm{\parallel }{\psi }_1+e^{i\phi }{\psi }_2{\mathrm{\parallel }}^2},$ and the numerator equals ${\int }_{{\mathbb{R}}^3}{w}_a(\mathrm{\parallel }{\psi }_1{\mathrm{\parallel }}^2+\mathrm{\parallel }{\psi }_2{\mathrm{\parallel }}^2)dx+2\mathrm{Re}(e^{-i\phi }{\int }_{{\mathbb{R}}^3}{w}_a⟨{\psi }_1,{\psi }_2{⟩}_{{\mathbb{C}}^r}dx)\mathrm{.}$

(b) Recorded which-path law. For the declared weight $w\in [0,1]$ and every $a\in A$, ${\nu }_{\mathrm{m}\mathrm{i}\mathrm{x}(w)}(a)=w\frac{{\displaystyle {\int }_{{\mathbb{R}}^3}{w}_a\mathrm{\parallel }{\psi }_1{\mathrm{\parallel }}^{2}dx}}{\mathrm{\parallel }{\psi }_1{\mathrm{\parallel }}^2}+(1-w)\frac{{\displaystyle {\int }_{{\mathbb{R}}^3}{w}_a\mathrm{\parallel }{\psi }_2{\mathrm{\parallel }}^{2}dx}}{\mathrm{\parallel }{\psi }_2{\mathrm{\parallel }}^2}\mathrm{.}$

(c) Loss. The no-record probability is $\nu (0)=1-\nu (A)=\mathrm{tr}(\rho F_0)\ge 0$, so $\nu (A)\le 1$. If ${\sum }_a{w}_a=1$ almost everywhere, then $F_0=0$ and $\nu (0)=0$ for every preparation; if $F_0\mathrm{\ne }0$, then there is a declared pure preparation with $\nu (0)>0$. When $\nu (A)>0$, the conditioned recorded law $\bar{\nu }(a)=\nu (a)\mathrm{/}\nu (A)$ is a probability measure on $A$.

(d) Ideal bins and coincidence. If $w_a={\mathbf{1}}_{B_a}$ for a declared ideal detector $(B_a{)}_{a=1}^m$, then $\nu (a)={\mu }_{\rho }^G(B_a)$ is the ideal bin probability of Coherent and which-path detection laws of the declared two-path model. Moreover, if the declared model is admissible at phase $0$ and if the almost-everywhere relation $N\mathrm{Re}⟨{\psi }_1(x),{\psi }_2(x){⟩}_{{\mathbb{C}}^r}=c(\mathrm{\parallel }{\psi }_1(x){\mathrm{\parallel }}_{{\mathbb{C}}^r}^2+\mathrm{\parallel }{\psi }_2(x){\mathrm{\parallel }}_{{\mathbb{C}}^r}^2)$ of clause (c) of that theorem holds, then the recorded coherent law of phase $0$ and the recorded which-path law of weight $w_{\ast }$ agree, ${\nu }_{\mathrm{c}\mathrm{o}\mathrm{h}(0)}(a)={\nu }_{\mathrm{m}\mathrm{i}\mathrm{x}(w_{\ast })}(a)$, for every label $a$ of every declared recorded detector.

(e) Scope. Clauses (a)-(d) are conditional on P2, the declared preparation and detector models and the cited mathematics; they assert no calibration, no resolution, background or efficiency value, no state update, no counts and no empirical result, and the claim that a concrete apparatus realizes the declared effects is a tested identification with uncertainty.

• Where the response enters. The recorded weights $w_a$ carry the declared finite resolution and efficiency of the model; the bin partition of the ideal detector is the special case of indicator weights. No value of any weight is asserted, and no apparatus realization is claimed.
• Conditioning is not a state update. The conditioned law is a statistical operation on the recorded sample; no claim is made about the post-detection state of the particle, and destructive or absorbing detection channels are outside the model's declared effects.