Marker overlap contracts the cross term: the detection law and the visibility bound of an overlap-normalized marked model

Adopt the declared marked two-component model of Declared marked two-component model: path components, marker states, phase family, detection weights and fringe visibility with unit path components ${\psi }_1,{\psi }_2\in H_s$, unit marker states $e_1,e_2\in K$, phase family $\rho (\phi )$, marker-carried which-path preparation ${\rho }_{\mathrm{m}\mathrm{i}\mathrm{x}}$, and a declared detection bin $B$. Assume the model is overlap-normalized, $⟨{\psi }_1,{\psi }_2⟩=0$, and 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), as required by the cited position and local-readout suppliers. Put $\gamma (B):=⟨e_1,e_2⟩⟨G(B){\psi }_1,{\psi }_2⟩,p_j(B):=⟨G(B){\psi }_j,{\psi }_j⟩(j=1,2),p_{\mathrm{m}\mathrm{i}\mathrm{x}}(B)=\frac{1}{2}(p_1(B)+p_2(B))\mathrm{.}$ Then:

(a) Unit-norm phase family. $\mathrm{\parallel }\mathrm{\Psi }(\phi )\mathrm{\parallel }=1$ for every $\phi$, so $\rho (\phi )=T_{\mathrm{\Psi }(\phi ),\mathrm{\Psi }(\phi )}$.

(b) Detection law. For every $\phi$, $p(\phi )(B)=p_{\mathrm{m}\mathrm{i}\mathrm{x}}(B)+\mathrm{Re}(e^{-i\phi }\gamma (B))\mathrm{.}$

(c) Contraction of the cross term. For every $\phi$, $\mid p(\phi )(B)-p_{\mathrm{m}\mathrm{i}\mathrm{x}}(B)\mid \le \mathrm{\mid }\gamma (B)\mathrm{\mid }\le \mathrm{\mid }⟨e_1,e_2⟩\mathrm{\mid }\sqrt{p_1(B)p_2(B)}\le \mathrm{\mid }⟨e_1,e_2⟩\mathrm{\mid }{p}_{\mathrm{m}\mathrm{i}\mathrm{x}}(B)\mathrm{.}$ In particular, if $e_1\perp e_2$, then $p(\phi )(B)=p_{\mathrm{m}\mathrm{i}\mathrm{x}}(B)$ for every $\phi$: the marked detection law is exactly the which-path law.

(d) Visibility. Writing $\gamma (B)=\mathrm{\mid }\gamma (B)\mathrm{\mid }{e}^{i\theta }$ with $\theta \in [0,2\pi )$, one has ${\sup }_{\phi }p(\phi )(B)=p_{\mathrm{m}\mathrm{i}\mathrm{x}}(B)+\mathrm{\mid }\gamma (B)\mathrm{\mid },{\inf }_{\phi }p(\phi )(B)=p_{\mathrm{m}\mathrm{i}\mathrm{x}}(B)-\mathrm{\mid }\gamma (B)\mathrm{\mid },$ so that, whenever $p_{\mathrm{m}\mathrm{i}\mathrm{x}}(B)>0$, the fringe visibility of clause (e) of the definition is $V(B)=\frac{\mathrm{\mid }\gamma (B)\mathrm{\mid }}{p_{\mathrm{m}\mathrm{i}\mathrm{x}}(B)}=\mathrm{\mid }⟨e_1,e_2⟩\mathrm{\mid }\frac{\mathrm{\mid }⟨G(B){\psi }_1,{\psi }_2⟩\mathrm{\mid }}{p_{\mathrm{m}\mathrm{i}\mathrm{x}}(B)}\le \mathrm{\mid }⟨e_1,e_2⟩\mathrm{\mid }\mathrm{.}$

(e) Extremes. If the model is trivially marked, $\mathrm{\mid }⟨e_1,e_2⟩\mathrm{\mid }=1$, then for every bin with $p_{\mathrm{m}\mathrm{i}\mathrm{x}}(B)>0$ one has $V(B)=\frac{\mathrm{\mid }⟨G(B){\psi }_1,{\psi }_2⟩\mathrm{\mid }}{p_{\mathrm{m}\mathrm{i}\mathrm{x}}(B)}\le 1$; if $e_1\perp e_2$, then $V(B)=0$ for every bin with $p_{\mathrm{m}\mathrm{i}\mathrm{x}}(B)>0$.

(f) Scope. Clauses (a)-(e) are conditional computations in the declared model. They assert no measured pattern, no trial counts, no statistical analysis, no decoherence mechanism, no irreversibility and no which-path ontology, and they contain no empirical claim.

• What the bound does and does not say. The factor $\mathrm{\mid }⟨e_1,e_2⟩\mathrm{\mid }$ is an upper bound on the visibility of the declared phase family; the actual visibility also contains the overlap factor $\mathrm{\mid }⟨G(B){\psi }_1,{\psi }_2⟩\mathrm{\mid }\mathrm{/}{p}_{\mathrm{m}\mathrm{i}\mathrm{x}}(B)$ of the path components at the bin, so a large marker overlap alone does not guarantee a visible fringe and a small marker overlap alone does not by itself describe a measured pattern.
• No mechanism claim. The marker is part of the declared preparation; the computation exhibits the correlation between the marker state and the path component as the origin of the reduced contrast in the model. It asserts no dynamical decoherence process, no irreversibility and no preference among interpretations of the alternatives.