Qm Interference And Local Detection

Assume the Axiom of Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), the choice hypothesis of the cited Hilbert-space and measure suppliers. Let $(D,\mathcal{A},dx)$ be a $\sigma$-finite measure space (Measures on sigma-algebras), let $r\ge 1$ be an integer, and let $L^2(D;{\mathbb{C}}^r):=\{f:D\to {\mathbb{C}}^r\text{ measurable}:{\int }_D\mathrm{\parallel }f(x){\mathrm{\parallel }}_{{\mathbb{C}}^r}^{2}dx<\mathrm{\infty }\}/\text{\hspace{0.17em}⁣}\sim ,$ the quotient by equality $dx$-almost everywhere (The space $L^p(\mu )$ as the quotient by null functions, Complex Lp classes and Euclidean test-function conventions), with $⟨f,g⟩:={\int }_D⟨f(x),g(x){⟩}_{{\mathbb{C}}^r}dx,\mathrm{\parallel }f{\mathrm{\parallel }}^2:=⟨f,f⟩={\int }_D\mathrm{\parallel }f(x){\mathrm{\parallel }}_{{\mathbb{C}}^r}^{2}dx,$ linear in the first argument ($L^2$ with the integral pairing is a Hilbert space). For $a,b\in L^2(D;{\mathbb{C}}^r)$ and measurable $B\subseteq D$ write $⟨a,b{⟩}_B:={\int }_B⟨a(x),b(x){⟩}_{{\mathbb{C}}^r}dx,N:=\mathrm{\parallel }a{\mathrm{\parallel }}^2+\mathrm{\parallel }b{\mathrm{\parallel }}^2,c:=\mathrm{Re}⟨a,b{⟩}_D\mathrm{.}$ Then:

(a) Pointwise and integrated cross term. For $dx$-almost every $x\in D$, $\mathrm{\parallel }a(x)+b(x){\mathrm{\parallel }}_{{\mathbb{C}}^r}^2=\mathrm{\parallel }a(x){\mathrm{\parallel }}_{{\mathbb{C}}^r}^2+\mathrm{\parallel }b(x){\mathrm{\parallel }}_{{\mathbb{C}}^r}^2+2\mathrm{Re}⟨a(x),b(x){⟩}_{{\mathbb{C}}^r},$ and for every measurable $B\subseteq D$, the three functions $\mathrm{\parallel }a{\mathrm{\parallel }}^2$, $\mathrm{\parallel }b{\mathrm{\parallel }}^2$, $⟨a,b{⟩}_{{\mathbb{C}}^r}$ being $dx$-integrable on $B$, ${\int }_B\mathrm{\parallel }a+b{\mathrm{\parallel }}^{2}dx={\int }_B\mathrm{\parallel }a{\mathrm{\parallel }}^{2}dx+{\int }_B\mathrm{\parallel }b{\mathrm{\parallel }}^{2}dx+2\mathrm{Re}⟨a,b{⟩}_B\mathrm{.}$

(b) Coherent bin law. If $\mathrm{\parallel }a+b\mathrm{\parallel }>0$, then $p_{\mathrm{c}\mathrm{o}\mathrm{h}}(B):=\frac{{\displaystyle {\int }_B\mathrm{\parallel }a+b{\mathrm{\parallel }}^{2}dx}}{\mathrm{\parallel }a+b{\mathrm{\parallel }}^2}(B\in \mathcal{A})$ defines a probability measure on $(D,\mathcal{A})$ (Probability measures and probability spaces), and for every $B\in \mathcal{A}$ the expansion of (a) holds with $\mathrm{\parallel }a+b{\mathrm{\parallel }}^2=N+2c$ in the denominator.

(c) Incoherent bin law. If $\mathrm{\parallel }a\mathrm{\parallel }>0$, $\mathrm{\parallel }b\mathrm{\parallel }>0$ and $w\in [0,1]$, then $p_{\mathrm{m}\mathrm{i}\mathrm{x}}(B):=w\frac{{\displaystyle {\int }_B\mathrm{\parallel }a{\mathrm{\parallel }}^{2}dx}}{\mathrm{\parallel }a{\mathrm{\parallel }}^2}+(1-w)\frac{{\displaystyle {\int }_B\mathrm{\parallel }b{\mathrm{\parallel }}^{2}dx}}{\mathrm{\parallel }b{\mathrm{\parallel }}^2}(B\in \mathcal{A})$ defines a probability measure on $(D,\mathcal{A})$.

(d) Recorded weights, sub-probability and conditioning. Let $\mu$ be a probability measure on $(D,\mathcal{A})$ and let $W:D\to [0,1]$ be measurable. Then $B\mapsto \nu (B):={\int }_{B}Wd\mu$ is a finite measure on $(D,\mathcal{A})$ with $0\le \nu (D)\le 1$; if $\nu (D)>0$, then $\bar{\nu }(B):=\nu (B)\mathrm{/}\nu (D)$ defines a probability measure on $(D,\mathcal{A})$. More generally, for finitely many measurable weights $W_1,\dots ,W_m:D\to [0,1]$ with ${\sum }_{a=1}^m{W}_a\le 1$ pointwise, the numbers ${\nu }_a:={\int }_D{W}_{a}d\mu$ satisfy ${\nu }_a\ge 0$ and ${\sum }_{a=1}^m{\nu }_a\le 1$; if ${\sum }_{a=1}^m{W}_a=1$ pointwise, then ${\sum }_{a=1}^m{\nu }_a=1$.

(e) Coincidence criterion. Let $\mathrm{\parallel }a\mathrm{\parallel }>0$, $\mathrm{\parallel }b\mathrm{\parallel }>0$ and $\mathrm{\parallel }a+b\mathrm{\parallel }>0$, and put $w_{\ast }:=\mathrm{\parallel }a{\mathrm{\parallel }}^2\mathrm{/}N$, so that $p_{\mathrm{m}\mathrm{i}\mathrm{x}}$ with weight $w_{\ast }$ is $p_{\ast }(B)=N^{-1}{\int }_B(\mathrm{\parallel }a{\mathrm{\parallel }}^2+\mathrm{\parallel }b{\mathrm{\parallel }}^2)dx$. Then $p_{\mathrm{c}\mathrm{o}\mathrm{h}}(B)=p_{\ast }(B)$ for every $B\in \mathcal{A}$ if and only if $N\mathrm{Re}⟨a(x),b(x){⟩}_{{\mathbb{C}}^r}=c(\mathrm{\parallel }a(x){\mathrm{\parallel }}_{{\mathbb{C}}^r}^2+\mathrm{\parallel }b(x){\mathrm{\parallel }}_{{\mathbb{C}}^r}^2)\text{for }dx\text{-almost every }x\in D\mathrm{.}$ In particular, if $⟨a(x),b(x){⟩}_{{\mathbb{C}}^r}=0$ for almost every $x$, then the two laws coincide; and if $c=0$, then they coincide if and only if $\mathrm{Re}⟨a(x),b(x){⟩}_{{\mathbb{C}}^r}=0$ for almost every $x$.

The assertions concern the specified classes and weights only; no probability law of an apparatus, no sample of trials and no empirical claim is asserted.

• Convention. The pairing is linear in the first argument, as fixed by the cited $L^2$ items, so the cross term of a sum $a+b$ is $2\mathrm{Re}⟨a,b⟩$ and not $2\mathrm{Re}⟨b,a⟩$; the two agree because the real part is symmetric.
• What is not asserted. The lemma is an identity and inequality package for classes and weights; it produces no detector law, no state update, no sample of trials and no measured frequency. The coincidence criterion concerns the two declared laws, not the question of which preparation a laboratory realizes.

(a) Framework data and state convention. Adopt the state and composition postulate P1, the observable and probability postulate P2, the kinematics and one-particle model postulate P5, the identical-particle and spin postulate P6, and the preparation and apparatus discipline P7 (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). Fix a spin label $s\in \{0,\frac{1}{2},1,\frac{3}{2},\dots \}$, put $r:=2s+1$, and fix a mass parameter $m>0$. P5 assigns the one-particle state space $H_s:=L^2({\mathbb{R}}^3;{\mathbb{C}}^r)$ with almost-everywhere equivalence classes and pairing $⟨f,g⟩:={\int }_{{\mathbb{R}}^3}{\sum }_{l=1}^r{f}_l(x)\overline{g_l(x)}dx,$ linear in the first argument (Kinematics and massive-particle model postulate P5: Newtonian frames and one-particle operators, The space $L^p(\mu )$ as the quotient by null functions, Real and complex inner-product spaces and their induced length). P1 identifies a pure preparation with the ray $[\psi ]:=\{z\psi :z\in \mathbb{C}\setminus \{0\}\}$ of a unit vector $\psi$ and a general preparation with a density operator, meaning a bounded self-adjoint positive trace-class operator $\rho$ with $\mathrm{tr}\rho =1$ (State and composition postulate P1: Hilbert spaces, preparations and composite systems, Trace class operator, Self-adjoint, positive, unitary and normal operators). Write $D(H_s):=\{\rho :\rho \text{ is a bounded self-adjoint positive trace-class operator on }{H}_s\text{ and }\mathrm{tr}\rho =1\}$ for the set of general preparations.

For a unit vector $u$, define $P_{u}f:=⟨f,u⟩u$. By Cauchy--Schwarz (Cauchy–Schwarz: $\mathrm{\mid }⟨x,y⟩\mathrm{\mid }\le \mathrm{\parallel }x\mathrm{\parallel }\mathrm{\parallel }y\mathrm{\parallel }$, with equality exactly for dependent pairs), $\mathrm{\parallel }{P}_{u}f\mathrm{\parallel }=\mathrm{\mid }⟨f,u⟩\mathrm{\mid }\le \mathrm{\parallel }f\mathrm{\parallel }$, so $P_u$ is bounded. Then $P_u^2=P_u$ and $P_u$ is self-adjoint because for $f,g\in H_s$, $⟨P_{u}f,g⟩=⟨f,u⟩⟨u,g⟩=⟨f,P_{u}g⟩$. Moreover $⟨P_{u}f,f⟩=\mathrm{\mid }⟨f,u⟩{\mathrm{\mid }}^2\ge 0$ by the inner-product identities and $⟨u,u⟩=1$. Its range is $\mathbb{C}u$, so it has finite rank and is trace class by Trace class operator. The one-term nuclear representation $P_{u}f=⟨f,u⟩u$ gives $\mathrm{tr}{P}_u=⟨u,u⟩=1$ by Trace is absolutely convergent and basis independent. Thus $P_u$ is a density operator. For $\psi \mathrm{\ne }0$ define $\widehat{\rho }(\psi ):=P_{\psi \mathrm{/}\mathrm{\parallel }\psi \mathrm{\parallel }},\widehat{\rho }(\psi )f=\frac{⟨f,\psi ⟩\psi }{\mathrm{\parallel }\psi {\mathrm{\parallel }}^2}\mathrm{.}$ For every nonzero scalar $z$, conjugate-linearity in the second argument and $\mathrm{\parallel }z\psi {\mathrm{\parallel }}^2=\mathrm{\mid }z{\mathrm{\mid }}^2\mathrm{\parallel }\psi {\mathrm{\parallel }}^2$ give $\widehat{\rho }(z\psi )f=\frac{⟨f,z\psi ⟩z\psi }{\mathrm{\parallel }z\psi {\mathrm{\parallel }}^2}=\frac{\mathrm{\mid }z{\mathrm{\mid }}^2⟨f,\psi ⟩\psi }{\mathrm{\mid }z{\mathrm{\mid }}^2\mathrm{\parallel }\psi {\mathrm{\parallel }}^2}=\widehat{\rho }(\psi )f,$ so the pure density operator depends only on the ray.

Position PVM and Born-density calculation. P5 assigns the componentwise position operators $Q_{j}f=x_{j}f$ on $D(Q_j):=\{f\in H_s:x_{j}f\in H_s\},j=1,2,3.$ Each $Q_j$ is self-adjoint on this domain. To see density, for $f\in H_s$ put $f_n={\mathbf{1}}_{\{\mathrm{\mid }{x}_j\mathrm{\mid }\le n\}}f$; then $f_n\in D(Q_j)$ and $f_n\to f$ in $H_s$ by dominated convergence. If $f,g\in D(Q_j)$, then $⟨Q_{j}f,g⟩=⟨f,Q_{j}g⟩$ because $x_j$ is real-valued and both products are integrable by Cauchy--Schwarz. If $g\in D(Q_j^{\ast })$ with $Q_j^{\ast }g=h$, let $E_n=\{x:\mathrm{\mid }{x}_j\mathrm{\mid }\le n\}$. For every $q\in H_s$ supported in $E_n$, one has $q\in D(Q_j)$ and $⟨q,(x_{j}g-h){\mathbf{1}}_{E_n}⟩=0$ by the defining adjoint identity. The vector $(x_{j}g-h){\mathbf{1}}_{E_n}$ lies in $H_s$ because $x_j$ is bounded on $E_n$; taking it for $q$ shows $x_{j}g=h$ almost everywhere on $E_n$. Since $E_n\uparrow {\mathbb{R}}^3$, $x_{j}g=h\in H_s$ almost everywhere, so $g\in D(Q_j)$ and $Q_j^{\ast }g=Q_{j}g$. Symmetry gives the reverse domain inclusion; hence $Q_j=Q_j^{\ast }$ by Adjoint of a densely defined operator, Symmetric, self-adjoint and essentially self-adjoint operators.

For a Borel set $S\subseteq {\mathbb{R}}^3$, define the indicator multiplication operator $M_{{\mathbf{1}}_S}:H_s\to H_s$ by $(M_{{\mathbf{1}}_S}f)(x):={\mathbf{1}}_S(x)f(x)$ almost everywhere. This is independent of the representative of the $L^2$ class and is bounded, since $\mathrm{\parallel }{M}_{{\mathbf{1}}_S}f{\mathrm{\parallel }}^2={\int }_S\mathrm{\parallel }f(x){\mathrm{\parallel }}_{{\mathbb{C}}^r}^{2}dx\le \mathrm{\parallel }f{\mathrm{\parallel }}^2\mathrm{.}$ It is linear and a bounded operator with bound one (A bounded linear operator between normed spaces). It is an orthogonal projection: ${\mathbf{1}}_S^2={\mathbf{1}}_S$ and the real indicator multiplier is self-adjoint for the declared pairing.

For Borel $C\subseteq \mathbb{R}$, define $E_j(C):=M_{{\mathbf{1}}_{\{x:x_j\in C\}}}\mathrm{.}$ Each value is a bounded orthogonal projection since $\mathrm{\parallel }{\mathbf{1}}_{\{x:x_j\in C\}}f\mathrm{\parallel }\le \mathrm{\parallel }f\mathrm{\parallel }$, and $E_j(\mathrm{\varnothing })=0$, $E_j(\mathbb{R})=I$, and $E_j(C\cap D)=E_j(C)E_j(D)$. For a disjoint sequence $(C_n)$ with union $C$, the squared norm of $E_j(C)f-{\sum }_{n\le N}{E}_j(C_n)f$ equals ${\int }_{\{x:x_j\in C\setminus {\cup }_{n\le N}{C}_n\}}\mathrm{\parallel }f(x){\mathrm{\parallel }}^{2}dx,$ which tends to zero by dominated convergence; hence $E_j$ is strongly countably additive. Its scalar measures are finite Borel measures $⟨E_j(C)f,f⟩={\int }_{\{x:x_j\in C\}}\mathrm{\parallel }f(x){\mathrm{\parallel }}^{2}dx,$ and are regular on $\mathbb{R}$ by Locally finite Borel measures on second-countable LCH spaces are regular. The push-forward integral identity ${\int }_{\mathbb{R}}g(\lambda )d⟨E_j(\lambda )f,f⟩={\int }_{{\mathbb{R}}^3}g(x_j)\mathrm{\parallel }f(x){\mathrm{\parallel }}^{2}dx$ holds first for indicators and nonnegative simple functions and then for nonnegative measurable $g$ by monotone convergence. In particular the domain of $\int \lambda d{E}_j(\lambda )$ is exactly $D(Q_j)$ by the same identity with $g(\lambda )={\lambda }^2$. For bounded Borel $g$, the bounded PVM calculus equals multiplication by $g(x_j)$: this holds for indicators by the definition of $E_j$, for finite simple functions by linearity, and for bounded Borel functions by uniform simple approximation and the norm bound of the bounded calculus (Bounded borel pvm integral). Therefore the truncations of the unbounded integral are multiplication by $x_j{\mathbf{1}}_{\{\mathrm{\mid }{x}_j\mathrm{\mid }\le n\}}$. For $f\in D(Q_j)$ these converge in $H_s$ to $x_{j}f$, since the squared tail norm is ${\int }_{\{\mathrm{\mid }{x}_j\mathrm{\mid }>n\}}{x}_j^2\mathrm{\parallel }f(x){\mathrm{\parallel }}^{2}dx\to 0$ by dominated convergence. Thus $\int \lambda d{E}_j(\lambda )=Q_j$ on exactly $D(Q_j)$, and the regular spectral theorem identifies $E_j$ as the spectral PVM of $Q_j$ (Integral of a measurable function against a projection-valued measure, Spectral theorem for unbounded self-adjoint operators (PVM form)).

Define the joint position PVM by $G(B):=M_{{\mathbf{1}}_B}(B\subseteq {\mathbb{R}}^3\text{ Borel})\mathrm{.}$ For each $f$, $\mathrm{\parallel }G(B)f\mathrm{\parallel }\le \mathrm{\parallel }f\mathrm{\parallel }$, so its values are bounded; indicator multiplication gives its projection, identity, and intersection laws. Strong countable additivity follows from dominated convergence applied to $\mathrm{\parallel }f{\mathrm{\parallel }}^2$. Its scalar measures are $B\mapsto {\int }_B\mathrm{\parallel }f(x){\mathrm{\parallel }}^{2}dx$, finite regular Borel measures on ${\mathbb{R}}^3$. On coordinate rectangles, $G(C_1\times C_2\times C_3)=E_1(C_1)E_2(C_2)E_3(C_3)\mathrm{.}$ The rectangles generate $\mathcal{B}({\mathbb{R}}^3)$; the published Borel-product and finite-measure uniqueness theorems show that these rectangle values uniquely determine the joint PVM (The Borel product of R^m and R^n is the Borel sigma-algebra of R^{m+n}, Finite measures agreeing on a generating pi-system and on the whole space are equal, Scalar and complex measures from a pvm): for any two candidate PVMs, their finite scalar measures agree on the generating rectangles and on the whole space, so they agree on every Borel set; the polarization identity in the last lemma then gives equality of the operator values. This constructs the mathematical joint position PVM; a physical joint detector reading is supplied only for the finite effect families declared below. For a density operator $\rho$ define ${\mu }_{\rho }^G(B):=\mathrm{tr}(\rho G(B))$ for each Borel $B\subseteq {\mathbb{R}}^3$. Since $G$ is a PVM, this set function is a probability measure by the general density/PVM calculation used in clause (ii) of P2. For a finite Borel partition $(B_a{)}_{a=1}^m$, the operators $G(B_a)$ are orthogonal projections and sum to $I$. For each $f\in H_s$, $⟨G(B_a)f,f⟩=\mathrm{\parallel }G(B_a)f{\mathrm{\parallel }}^2\in [0,\mathrm{\parallel }f{\mathrm{\parallel }}^2]$, so they are bounded effects with $0\le G(B_a)\le I$ in quadratic-form order. Clause (iv) of P2 therefore assigns the channel probabilities $p_a(\rho )={\mu }_{\rho }^G(B_a)$; the two-channel partition $(B,{\mathbb{R}}^3\setminus B)$ assigns ${\mu }_{\rho }^G(B)$ to bin $B$ (Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes). Thus $G$ is the mathematical joint position PVM, while the detector reading used here is the declared finite-effect model of P2(iv); this does not assert that every three sharp coordinate measurement is jointly realized by a laboratory apparatus. For a unit vector $u$, the trace formula for the rank-one operator $P_u$ gives ${\mu }_{P_u}^G(B)=\mathrm{tr}(P_{u}G(B))=\mathrm{\parallel }G(B)u{\mathrm{\parallel }}^2={\int }_B\mathrm{\parallel }u(x){\mathrm{\parallel }}_{{\mathbb{C}}^r}^{2}dx,$ and P7(iii) leaves identification of this ideal detector with a concrete apparatus as a tested physical claim with uncertainty.

Assume the Axiom of Countable Choice and the Axiom of Choice as required by the cited trace and spectral suppliers (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The Axiom of Choice).

(b) Declared two-path assignment. A declared two-path preparation model of the apparatus is the assignment of two nonzero classes ${\psi }_1,{\psi }_2\in H_s$, called the path components of the model with respect to the declared pair of alternative path conditions of the apparatus, together with a declared relative phase $\phi \in \mathbb{R}$ and a declared weight $w\in [0,1]$. The assignment of ${\psi }_1,{\psi }_2$, $\phi$ and $w$ is part of the preparation model in the sense of clause (i) of P7 (Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions): it is declared, not computed from the framework, and no item of this page asserts that a particle described by the model "took" one of the two paths.

(c) Coherent preparation. The declared two-path model is admissible at phase $\phi$ when ${\psi }_1+e^{i\phi }{\psi }_2\mathrm{\ne }0$ in $H_s$. When it is admissible, the coherent preparation of relative phase $\phi$ is the pure preparation ${\rho }_{\mathrm{c}\mathrm{o}\mathrm{h}}(\phi ):=\widehat{\rho }({\psi }_1+e^{i\phi }{\psi }_2),$ the normalized rank-one density operator defined in clause (a). The relative phase $\phi$ is read in radians (dimensionless). Multiplying both path components by one common nonzero scalar leaves their associated pure density operator unchanged, while the declared relative phase remains part of the preparation assignment.

(d) Incoherent preparation. For a declared weight $w\in [0,1]$ the incoherent preparation, or which-path preparation, of the model is ${\rho }_{\mathrm{m}\mathrm{i}\mathrm{x}}(w):=w\widehat{\rho }({\psi }_1)+(1-w)\widehat{\rho }({\psi }_2)\mathrm{.}$ It is a density operator: both summands are positive trace-class operators of trace one; trace-class operators form a vector space, nonnegative real combinations preserve positivity and self-adjointness, and the trace is linear, so $\mathrm{tr}{\rho }_{\mathrm{m}\mathrm{i}\mathrm{x}}(w)=w+(1-w)=1$ (Trace class is a two sided Banach operator ideal, Trace is absolutely convergent and basis independent). The weight $w$ is a declared preparation weight, not an outcome probability.

(e) Ideal detection bins. A declared ideal detector of the model is a finite family $\mathcal{B}=(B_a{)}_{a=1}^m$ of pairwise disjoint Borel subsets of ${\mathbb{R}}^3$ with ${\bigcup }_a{B}_a={\mathbb{R}}^3$; the ideal bin probabilities of a preparation $\rho \in D(H_s)$ are the numbers $p_a(\rho ):={\mu }_{\rho }^G(B_a)=\mathrm{tr}(\rho G(B_a))(a=1,\dots ,m),$ the reading of the finite-effect clause (iv) of P2 for the declared detector and its mathematical joint position PVM (Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes, Projection valued measure). By clause (iv) of P2 these numbers are nonnegative, are at most one, and satisfy ${\sum }_a{p}_a(\rho )=1$. A single declared bin $B$ is the two-element ideal detector $(B,{\mathbb{R}}^3\setminus B)$; its probability is ${\mu }_{\rho }^G(B)$.

For the ideal support, choose a Lebesgue-measurable representative $f$ of the unit $L^2$ class and form $S_f:=\{x:\mathrm{\parallel }f(x){\mathrm{\parallel }}_{{\mathbb{C}}^r}>0\}$. The ideal support is the class of $S_f$ under equality up to a Lebesgue-null symmetric difference. This class is independent of the representative, and it is not asserted to be a Borel detector bin.

(f) Status and scope. Clauses (a)-(e) fix the model data of the pair: the framework reading, the declared path components and preparations, and the ideal bin reading of the position observable. They adopt no assumption beyond P1, P2, P5, P6, P7 and the cited mathematics; the declaration of ${\psi }_1,{\psi }_2,\phi ,w$ and of the bins is an apparatus-model input, and the claim that a concrete two-slit apparatus realizes a particular such declaration is a tested identification with uncertainty under P7(iii). The declared bins are subsets of ${\mathbb{R}}^3$; the model does not claim that a physical screen of lower dimension is literally a spatial region, and a marginal, coarse-graining or flux reduction of a state to a screen coordinate or surface is a separate detection model, not supplied here. No dynamics, no recorded-event response, no sampling model and no empirical result is asserted; the recorded-event and sampling models are declared separately on this page.

• Classes, not functions. The path components are almost-everywhere classes; pointwise values such as ${\psi }_j(x)$ are read only through the measurable representatives, densities and pairings defined above, and the interference identities of the page hold up to null sets.
• "Path" is a declared decomposition. The phrase "path component" names an element of the declared decomposition of the preparation model. The pair proves laws of the declared coherent and incoherent preparations; it does not assert which-path ontology, does not assign a trajectory to the particle, and does not assume that the two components are orthogonal or have disjoint ideal supports. Whether a preparation model with an orthogonal or disjoint-support decomposition describes a given apparatus is a physical identification under P7.
• Units. The outcome sets are sets of position with unit metre, the probability numbers are dimensionless, and the phase is read in radians (Dimensions, SI unit conventions and quantity types of the framework).

Adopt the declared two-path preparation model of The declared two-path preparation model and its ideal detection bins, the unsharp finite-outcome clause (iv) of the observable and probability postulate P2, and the preparation and apparatus discipline P7 (Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes, Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions). Assume the Axiom of Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). The following vocabulary is used by this page.

(a) Declared recorded detector. A declared recorded detector of the arrangement is a finite set $A$ of recorded outcome labels with $0\mathrm{\notin }A$, together with measurable recorded weights $w_a:{\mathbb{R}}^3\to [0,1]$ for $a\in A$ such that ${\sum }_{a\in A}{w}_a(x)\le 1$ for almost every $x\in {\mathbb{R}}^3$. The effect of the label $a$ is the multiplication operator $F_a:=M_{w_a},(M_w\psi )(x):=w(x)\psi (x)\text{a.e.},$ on $H_s=L^2({\mathbb{R}}^3;{\mathbb{C}}^r)$, and the no-record effect is $F_0:=I-{\sum }_{a\in A}{F}_a\mathrm{.}$ The multiplier is a well-defined linear map on almost-everywhere classes, since changing a representative on a null set changes its product only on that null set. Each $F_a$ is bounded: for $\psi \in H_s$, $\mathrm{\parallel }{M}_{w_a}\psi {\mathrm{\parallel }}^2={\int }_{{\mathbb{R}}^3}{w}_a(x{)}^2\mathrm{\parallel }\psi (x){\mathrm{\parallel }}_{{\mathbb{C}}^r}^{2}dx\le \mathrm{\parallel }\psi {\mathrm{\parallel }}^2,$ so the boundedness inequality holds with bound $C=1$ by monotonicity of the nonnegative integral (Monotonicity and nonnegative homogeneity of the nonnegative integral). Thus $F_a$ is a bounded linear operator (A bounded linear operator between normed spaces). Because $w_a$ is real, the componentwise pairing gives $⟨M_{w_a}\psi ,\varphi ⟩=⟨\psi ,M_{w_a}\varphi ⟩$ for all $\psi ,\varphi \in H_s$; hence $F_a$ is self-adjoint by The Hilbert-space adjoint of a bounded operator. At almost every $x$, the finite-dimensional operator $w_a(x)I_{{\mathbb{C}}^r}$ is nonnegative in the sense of Non-negative and positive operators, and therefore $0\le ⟨F_a\psi ,\psi ⟩={\int }_{{\mathbb{R}}^3}{w}_a(x)\mathrm{\parallel }\psi (x){\mathrm{\parallel }}_{{\mathbb{C}}^r}^{2}dx\le \mathrm{\parallel }\psi {\mathrm{\parallel }}^2\mathrm{.}$ Here $0\le F_a\le I$ means these quadratic-form inequalities for every $\psi \in H_s$.

Define $w_0(x):=\max \{0,1-{\sum }_{a\in A}{w}_a(x)\}$. This is measurable, $0\le w_0\le 1$, and it equals $1-{\sum }_a{w}_a$ almost everywhere by the admissibility condition. Thus $F_0=M_{w_0}=I-{\sum }_a{F}_a$ on $H_s$, and the same argument gives $0\le F_0\le I$ and self-adjointness. Consequently ${\sum }_{a\in A}{F}_a+F_0=I$, so the completed family is a finite family of bounded effects of clause (iv) of P2 (Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes).

(b) Recorded probabilities. For a preparation $\rho \in D(H_s)$ the recorded probability of a label $a\in A$ and the no-record probability are $\nu (a):=\mathrm{tr}\left(\rho F_a\right),\nu (0):=\mathrm{tr}\left(\rho F_0\right)=1-{\sum }_{a\in A}\nu (a),$ the outcome reading of clause (iv) of P2 for the declared effects (Assigned sharp observables and the outcome law of the framework clause (f) records the reading; the numbers are nonnegative and sum to one). The recorded-event mass is $\nu (A):={\sum }_{a\in A}\nu (a)=1-\nu (0)\in [0,1]$. A detector is full-efficiency when ${\sum }_a{w}_a=1$ almost everywhere, so that $F_0=0$ and $\nu (0)=0$ for every preparation. For a fixed preparation $\rho$, the model loses events when $\nu (0)>0$, and the no-record channel carries this lost mass. A detector that is not full-efficiency can nevertheless have $\nu (0)=0$ for a particular preparation. A declared recorded bin $B$ is the special case with one label $a$ and $w_a={\mathbf{1}}_B$; then $F_a=G(B)$ is the position projection of clause (e) of The declared two-path preparation model and its ideal detection bins, with the position set $B$ read in the metre unit of Dimensions, SI unit conventions and quantity types of the framework, and the recorded probability is the ideal bin probability.

(c) Loss conditioning. When $\nu (A)>0$, the conditioned recorded law is $\bar{\nu }(a):=\frac{\nu (a)}{\nu (A)}(a\in A)\mathrm{.}$ It is a probability measure on the finite set $A$ (Probability measures and probability spaces); it is the law of the recorded label under the declared model conditioned on the occurrence of a record. Conditioning is a statistical operation on the recorded data; it is not a state update, and no claim of P3 is made here.

(d) Sampling model. A declared $N$-trial sampling model for a fixed preparation $\rho$ is a probability space $(\mathrm{\Omega },\mathcal{F},\mathbb{P})$ carrying measurable maps $O_1,\dots ,O_N:(\mathrm{\Omega },\mathcal{F})\to (A\cup \{0\},\mathcal{P}(A\cup \{0\}))$ ($N\ge 1$), where $0$ is the no-record label of clause (b) and is distinct from every recorded label because $0\mathrm{\notin }A$. The maps are mutually independent in the sense of Pairwise and mutual independence of finite-valued random variables and identically distributed with $\mathbb{P}(O_k=a)=\nu (a)$ for every $a\in A$, $\mathbb{P}(O_k=0)=\nu (0)$ and every $k$, where $\nu$ is the recorded law of clause (b); the numbers $\nu (a)$ and $\nu (0)$ sum to one, so this is a probability law on the finite set $A\cup \{0\}$. The empirical frequency of the label $a$ in the sample is $F_N(a):=\frac{1}{N}\mathrm{\#}\{k:O_k=a\}\mathrm{.}$ A declared fringe criterion for sample size $N$ is a subset $C\subseteq (A\cup \{0\}{)}^N$; the model accepts the sample when $(O_1,\dots ,O_N)\in C$ and fails the criterion otherwise. The identification of repetitions of a concrete procedure with the independent identically distributed model is an additional statistical assumption in the sense of clause (ii) of P7; it is not a consequence of P1-P6, and correlated trials, drifting preparation, dead time and selection must be modeled or recorded as limitations if present (Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions).

(e) Status. Clauses (a)-(d) fix the recorded-detector and sampling vocabulary of the pair and read clause (iv) of P2 and clauses (i)-(iii) of P7. The recorded weights, the label set, the conditioning convention and the sampling model are declared apparatus and procedure inputs. The claim "this apparatus realizes the effect family $F_a$" and the claim "its repetitions are independent and identically distributed draws of $\nu$" are physical identifications with uncertainty under P7(iii) and P7(ii), can fail, and are not derived here. No calibration, efficiency, background or resolution value is asserted, no counts are recorded, and no statistical-inference conclusion is drawn by this definition.

• Relations between the detector families. The ideal detector of clause (e) of The declared two-path preparation model and its ideal detection bins is the full-efficiency case $w_a={\mathbf{1}}_{B_a}$ of a declared recorded detector whose labels partition the bin family; a general recorded detector with non-indicator weights represents finite resolution, response and efficiency through its weights, and its no-record channel represents lost events.
• No state update is declared. The item declares outcome probabilities only. Which post-measurement state, if any, a registration leaves behind is a separate instrument-model question and is not part of this page.

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.