Coherent and which-path detection laws of the declared two-path model

Adopt the state and composition postulate P1, the observable and probability postulate P2, the kinematics and massive-particle model postulate P5 and the internal-factor postulate P6, as read by The declared two-path preparation model and its ideal detection bins, and the preparation and apparatus discipline P7, and adopt the declared two-path preparation model of that definition with path components ${\psi }_1,{\psi }_2\in H_s=L^2({\mathbb{R}}^3;{\mathbb{C}}^r)$, relative phase $\phi \in \mathbb{R}$ and weight $w\in [0,1]$ (State and composition postulate P1: Hilbert spaces, preparations and composite systems, Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes, Kinematics and massive-particle model postulate P5: Newtonian frames and one-particle operators, Identical-particle and spin assignments P6: permutation sectors and finite-spin representations, Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions). 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). Let $G(B)=M_{{\mathbf{1}}_B}$ be the joint position PVM constructed in clause (a) of The declared two-path preparation model and its ideal detection bins, and read its outcome law by P2 as ${\mu }_{\rho }^G(B)=\mathrm{tr}(\rho G(B))$ (Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes). Put $N:=\mathrm{\parallel }{\psi }_1{\mathrm{\parallel }}^2+\mathrm{\parallel }{\psi }_2{\mathrm{\parallel }}^2,Z(B):={\int }_B⟨{\psi }_1(x),{\psi }_2(x){⟩}_{{\mathbb{C}}^r}dx,Z:=Z({\mathbb{R}}^3),c(B):=\mathrm{Re}Z(B),c:=c({\mathbb{R}}^3)\mathrm{.}$ Then:

(a) Coherent law. If the declared model is admissible at phase $\phi$, then for every Borel set $B\subseteq {\mathbb{R}}^3$ ${\mu }_{{\rho }_{\mathrm{c}\mathrm{o}\mathrm{h}}(\phi )}^G(B)=\frac{{\displaystyle {\int }_B\mathrm{\parallel }{\psi }_1(x)+e^{i\phi }{\psi }_2(x){\mathrm{\parallel }}_{{\mathbb{C}}^r}^{2}dx}}{\mathrm{\parallel }{\psi }_1+e^{i\phi }{\psi }_2{\mathrm{\parallel }}^2},$ and the numerator expands as ${\int }_B(\mathrm{\parallel }{\psi }_1{\mathrm{\parallel }}^2+\mathrm{\parallel }{\psi }_2{\mathrm{\parallel }}^2)dx+2\mathrm{Re}(e^{-i\phi }{\int }_B⟨{\psi }_1(x),{\psi }_2(x){⟩}_{{\mathbb{C}}^r}dx),$ with denominator $N+2\mathrm{Re}(e^{-i\phi }Z)$; in particular $\mathrm{\parallel }{\psi }_1+e^{i\phi }{\psi }_2{\mathrm{\parallel }}^2>0$.

(b) Which-path law. For the declared weight $w\in [0,1]$ and every Borel set $B\subseteq {\mathbb{R}}^3$, ${\mu }_{{\rho }_{\mathrm{m}\mathrm{i}\mathrm{x}}(w)}^G(B)=w\frac{{\displaystyle {\int }_B\mathrm{\parallel }{\psi }_1{\mathrm{\parallel }}^{2}dx}}{\mathrm{\parallel }{\psi }_1{\mathrm{\parallel }}^2}+(1-w)\frac{{\displaystyle {\int }_B\mathrm{\parallel }{\psi }_2{\mathrm{\parallel }}^{2}dx}}{\mathrm{\parallel }{\psi }_2{\mathrm{\parallel }}^2}\mathrm{.}$

(c) Coincidence criterion. Let $w_{\ast }:=\mathrm{\parallel }{\psi }_1{\mathrm{\parallel }}^2\mathrm{/}N$, so that ${\mu }_{{\rho }_{\mathrm{m}\mathrm{i}\mathrm{x}}(w_{\ast })}^G(B)=N^{-1}{\int }_B(\mathrm{\parallel }{\psi }_1{\mathrm{\parallel }}^2+\mathrm{\parallel }{\psi }_2{\mathrm{\parallel }}^2)dx$. If the declared model is admissible at phase $0$, then ${\mu }_{{\rho }_{\mathrm{c}\mathrm{o}\mathrm{h}}(0)}^G(B)={\mu }_{{\rho }_{\mathrm{m}\mathrm{i}\mathrm{x}}(w_{\ast })}^G(B)$ for every Borel set $B$ if and only if $N\mathrm{Re}⟨{\psi }_1(x),{\psi }_2(x){⟩}_{{\mathbb{C}}^r}=c(\mathrm{\parallel }{\psi }_1(x){\mathrm{\parallel }}_{{\mathbb{C}}^r}^2+\mathrm{\parallel }{\psi }_2(x){\mathrm{\parallel }}_{{\mathbb{C}}^r}^2)\text{for almost every }x\mathrm{.}$ In particular, if $⟨{\psi }_1(x),{\psi }_2(x){⟩}_{{\mathbb{C}}^r}=0$ for almost every $x$, then the coherent law of phase $0$ and the which-path law of weight $w_{\ast }$ agree on every bin; and if $c=0$, then, under phase-$0$ admissibility, they agree if and only if $\mathrm{Re}⟨{\psi }_1(x),{\psi }_2(x){⟩}_{{\mathbb{C}}^r}=0$ for almost every $x$.

(d) Scope. Clauses (a)-(c) are conditional on P1, P2, P5, P6 and the declared model, and identify the laws of exactly two declared preparations. They assert no path ontology, use no entangled marker state and no reduced state, assert no detector response, no loss, no sampling model, no dynamics and no empirical result.

• The cross term is physical relative phase. A change of the relative phase $\phi$ rotates the complex overlap $Z(B)$, giving the cross term $2\mathrm{Re}(e^{-i\phi }Z(B))$, without changing the two path densities; a common phase of ${\psi }_1$ and ${\psi }_2$ is immaterial.
• What the theorem does not claim. It does not assert that the coherent preparation is "more correct" than the which-path preparation, that a particle took a particular path, or that any apparatus realizes either declaration; and it makes no claim about visibility loss by entanglement or path marking, which belongs to the entanglement model of the category.