Declared marked two-component model: path components, marker states, phase family, detection weights and fringe visibility

Adopt the framework's composition and state readings for a declared composite of two distinguishable subsystems (Composite systems of the framework: product bases, local readouts, local effects and local instruments, Reduced preparations of a declared composite: the subsystem states of the framework), the kinematics and massive-particle model of P5 for a particle subsystem of spin label $s\in \{0,\frac{1}{2},1,\frac{3}{2},\dots \}$ with mass parameter $m>0$ and space $H_s=L^2({\mathbb{R}}^3;{\mathbb{C}}^r)$, $r=2s+1$ (The massive spin-s particle model of the framework: state space, operators, units and frame readings, Kinematics and massive-particle model postulate P5: Newtonian frames and one-particle operators), and the preparation and apparatus discipline P7 (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), as required by the cited position-outcome law. In the model below the particle subsystem is $A$ and the marker subsystem is $B$. The following vocabulary is used by this page.

(a) Position readout of the particle factor. The model's componentwise position family has joint spectral projection valued measure $G$ on the Borel sets of ${\mathbb{R}}^3$, with $G(B)=M_{{\mathbf{1}}_B},{\mu }_{\psi }^G(B)=\mathrm{\parallel }G(B)\psi {\mathrm{\parallel }}^2={\int }_B\mathrm{\parallel }\psi (x){\mathrm{\parallel }}_{{\mathbb{C}}^r}^{2}dx$ for $\psi \in H_s$ of unit norm and Borel $B\subseteq {\mathbb{R}}^3$, read in the metre unit of Dimensions, SI unit conventions and quantity types of the framework (Position and momentum outcome laws of the massive spin-s model clause (b), Assigned sharp observables and the outcome law of the framework clause (b)). A declared detection bin is a Borel set $B\subseteq {\mathbb{R}}^3$; the local position-bin effect of the declared composite is the bounded effect $G(B)\otimes I_K$ of Composite systems of the framework: product bases, local readouts, local effects and local instruments clause (d), and the detection weight of a preparation ${\rho }^{\mathrm{\prime }}$ of the composite is $p_{{\rho }^{\mathrm{\prime }}}(B):=\mathrm{tr}({\rho }^{\mathrm{\prime }}(G(B)\otimes I_K))\in [0,1],$ the reading of clause (iv) of P2 for that local effect.

(b) Declared marked two-component model. A declared marked two-component model of the arrangement consists of: two unit vectors ${\psi }_1,{\psi }_2\in H_s$, the path components relative to the declared pair of alternative path conditions; and two unit vectors $e_1,e_2\in K$, the marker states correlated with the alternatives. The assignment of ${\psi }_1,{\psi }_2,e_1,e_2$ and of the alternatives is declared apparatus data in the sense of P7; it is not computed from the framework, and no element of this definition asserts that a particle described by the model "took" one of the two alternatives. A declared marked model is overlap-normalized when $⟨{\psi }_1,{\psi }_2⟩=0$, and trivially marked when $\mathrm{\mid }⟨e_1,e_2⟩\mathrm{\mid }=1$.

(c) Phase family and marked preparations. For $\phi \in \mathbb{R}$ put $\mathrm{\Psi }(\phi ):=\frac{1}{\sqrt{2}}(\kappa ({\psi }_1\otimes e_1)+e^{i\phi }\kappa ({\psi }_2\otimes e_2))\in H_s\widehat{\otimes }K,$ where $\kappa$ is the embedding of Hilbert tensor construction: inner product, completion and product bases. Define the admissible phase set $I:=\{\phi \in \mathbb{R}:\mathrm{\Psi }(\phi )\mathrm{\ne }0\}$. It is nonempty: $\mathrm{\Psi }(0)+\mathrm{\Psi }(\pi )=\sqrt{2}\kappa ({\psi }_1\otimes e_1)\mathrm{\ne }0$, so those two vectors cannot both vanish, using $e^{i\pi }=-1$ from $\exp (x+iy)=e^x(\cos y+i\sin y)$, $\mathrm{\mid }\exp (x+iy)\mathrm{\mid }=e^x$, and $e^{i\pi }+1=0$. The marked preparation at an admissible phase $\phi \in I$ is the state assigned to the nonzero vector $\mathrm{\Psi }(\phi )$, $\rho (\phi ):=\widehat{\rho }(\mathrm{\Psi }(\phi ))=\frac{T_{\mathrm{\Psi }(\phi ),\mathrm{\Psi }(\phi )}}{\mathrm{\parallel }\mathrm{\Psi }(\phi ){\mathrm{\parallel }}^2}\in D(H_s\widehat{\otimes }K)$ (The state space of the framework: preparations, pure rays and density operators clause (c)); the phase $\phi$ is read in radians (dimensionless). The marker-carried which-path preparation is the statistical mixture ${\rho }_{\mathrm{m}\mathrm{i}\mathrm{x}}:=\frac{1}{2}{T}_{\kappa ({\psi }_1\otimes e_1),\kappa ({\psi }_1\otimes e_1)}+\frac{1}{2}{T}_{\kappa ({\psi }_2\otimes e_2),\kappa ({\psi }_2\otimes e_2)},$ a preparation of the declared composite by clause (iii) of P1.

(d) Detection weights of the family. For a declared detection bin $B$ and admissible phase $\phi \in I$ put $p(\phi )(B):=p_{\rho (\phi )}(B)=\mathrm{tr}(\rho (\phi )(G(B)\otimes I_K)),p_{\mathrm{m}\mathrm{i}\mathrm{x}}(B):=p_{{\rho }_{\mathrm{m}\mathrm{i}\mathrm{x}}}(B)=\frac{1}{2}(⟨G(B){\psi }_1,{\psi }_1⟩+⟨G(B){\psi }_2,{\psi }_2⟩)\mathrm{.}$ The numbers $p(\phi )(B)$ and $p_{\mathrm{m}\mathrm{i}\mathrm{x}}(B)$ lie in $[0,1]$; they are probabilities of the declared model, not measured frequencies.

(e) Fringe visibility. The fringe visibility of the declared phase family at the bin $B$ is $V(B):=\frac{{\sup }_{\phi \in I}p(\phi )(B)-{\inf }_{\phi \in I}p(\phi )(B)}{{\sup }_{\phi \in I}p(\phi )(B)+{\inf }_{\phi \in I}p(\phi )(B)}\text{whenever }{\sup }_{\phi \in I}p(\phi )(B)+{\inf }_{\phi \in I}p(\phi )(B)>0,$ and is left undefined otherwise; all extrema, including the denominator, are taken over the nonempty admissible phase set $I$ of clause (c). The number $V(B)$ is a property of the declared model and the declared bin, read as the modulation contrast of the family of probabilities; it is not a measured visibility and carries no statement about an apparatus.

(f) Status. Clauses (a)-(e) fix the model vocabulary. The path components, marker states and the identification of a concrete apparatus with the declared model are declared and tested inputs in the sense of clauses (i)-(iii) of P7; no dynamics, no interaction, no environment, no detector response, no counts and no empirical result is asserted. In particular the model does not claim that the marker is measured or read out, that the alternatives are physically realized, or that the loss of contrast described by $V$ is caused by any particular process; the quantitative statement about $V$ is the content of a separate theorem of this page.