The declared recorded detector and sampling model of the two-path arrangement

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.