Assigned sharp observables and the outcome law of the framework

(a) Assigned sharp observables. Adopt the observable and probability postulate Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes (P2) for a declared system whose state space is the model's The state space of the framework: preparations, pure rays and density operators. An assigned sharp observable of the system is a pair $(A,u_A)$ consisting of a densely defined self-adjoint operator $A:D(A)\subset H\to H$ on the assigned Hilbert space $H$ (Adjoint of a densely defined operator, Symmetric, self-adjoint and essentially self-adjoint operators, Hilbert space) and a declared unit $u_A$ in which the real spectral coordinate of $A$ is read, in the sense of Dimensions, SI unit conventions and quantity types of the framework. The domain $D(A)$ is part of the assignment: symmetry on a smaller domain, or a formal differential expression, does not by itself supply an assigned observable (Symmetric, self-adjoint and essentially self-adjoint operators). Whether a concrete laboratory quantity of a given apparatus is described by a particular pair $(A,u_A)$ is a tested identification with uncertainty under the preparation and apparatus postulate P7 and is not produced by this definition.

(b) Spectral measure and outcome law. 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), the choice hypotheses of the cited spectral suppliers. The self-adjoint operator $A$ has a unique spectral projection-valued measure $E_A$ on the Borel $\sigma$-algebra of $\mathbb{R}$, with $D(A)=\{x:\int {\lambda }^{2}d{E}_{A,x}<\mathrm{\infty }\}$ and $Ax=\int \lambda d{E}_A(\lambda )x$, where $E_{A,x}(B)=⟨E_A(B)x,x⟩$ (Spectral theorem for unbounded self-adjoint operators (PVM form), Projection valued measure, The Borel sigma-algebra of a topological space). For a preparation $\rho$, that is a state $\rho \in D(H)$ of The state space: density operators, pure and mixed states, and purity, the outcome law of the pair $(A,u_A)$ in the preparation $\rho$ is the assignment ${\mu }_{\rho }^A(B):=\mathrm{tr}(\rho E_A(B))(B\subseteq \mathbb{R}\text{ Borel}),$ the reading of clause (ii) of P2; for the pure preparation $[\psi ]$ of a unit vector $\psi$ it is ${\mu }_{\psi }^A(B)=\mathrm{\parallel }{E}_A(B)\psi {\mathrm{\parallel }}^2$, which depends only on the ray by State vectors, unit vectors, rays, normalization and phases of a Hilbert space. The law is a probability measure on the Borel sets by Density operators and scalar spectral probability clause (b), and it is affine in the preparation: for a statistical mixture ${\sum }_n{w}_n{\rho }_n$ one has ${\mu }_{{\sum }_n{w}_n{\rho }_n}^A={\sum }_n{w}_n{\mu }_{{\rho }_n}^A$ by Affinity and trace-norm continuity of the scalar spectral law clause (b).

(c) Outcome sets and exact values. A declared finite-resolution outcome of the quantity is a Borel set $B$; its probability in the preparation $\rho$ is ${\mu }_{\rho }^A(B)$, and a "value in $B$" is read only through this number. The reading of an exact value $\lambda$ is the singleton $B=\{\lambda \}$, with probability ${\mu }_{\rho }^A(\{\lambda \})$; by Atoms of the spectral measure and eigenvalues of a self-adjoint operator clauses (a) and (b) this number is zero for every preparation unless $\lambda$ is an eigenvalue of $A$, and when $\lambda$ is an eigenvalue it is the total weight of the eigenspace. In particular the model provides no normalizable eigenvector at a non-eigenvalue of $A$. If $A$ has no eigenvalues, every preparation has an atomless outcome law: each singleton has probability zero, while $\mathbb{R}$ remains the outcome space and the law is concentrated on $\sigma (A)$ as in clause (d). Zero singleton probability does not exclude real point values as realizations of this law; it supplies no normalized exact-value eigenstate. A finite-resolution record is read through its declared Borel bin, in the sense of clause (iii) of P2.

(d) Spectral support. The outcomes of the observable lie in its spectrum: with $\sigma (A)$ the spectrum of Resolvent and spectrum of an unbounded operator, one has ${\mu }_{\rho }^A(B)={\mu }_{\rho }^A(B\cap \sigma (A))$ and ${\mu }_{\rho }^A(\sigma (A))=1$ for every preparation by Support of the spectral measure, the spectrum and the values of the scalar law clause (b). This is the precise reading of the statement that the result of a measurement lies in the spectrum of the observable.

(e) Moments. The mean and the variance of the outcome law in a preparation are the numbers $\int \lambda d{\mu }_{\rho }^A(\lambda )$ and $\int (\lambda -m{)}^{2}d{\mu }_{\rho }^A(\lambda )$ with $m$ the mean, read in the declared unit $u_A$ and its square respectively; they exist only under the integrability hypotheses of Mean, second moment and variance of the scalar spectral law, which also supplies the formulas in terms of $A$ and the state. For a pure preparation of a unit vector $\psi \in D(A)$ the mean and the second moment are $⟨A\psi ,\psi ⟩$ and $\mathrm{\parallel }A\psi {\mathrm{\parallel }}^2$; membership of $\psi$ in $D(A)$ or in $D(A^2)$ is a genuine hypothesis and is not asserted for an arbitrary preparation.

(f) Unsharp finite outcomes are separate. Clause (iv) of P2 permits a finite family of bounded effects $F_a$ with $0\le F_a\le I$ and ${\sum }_a{F}_a=I$ and outcome probabilities $\mathrm{tr}(\rho F_a)$ as an operational description of a finite-resolution apparatus. That extension is tied to an apparatus instrument and is developed, together with state updates, on the measurement-and-instruments page; this definition fixes only the sharp outcome law of clause (b) and makes no claim that every laboratory procedure realizes it.

(g) Status. Clauses (a)-(f) record the readings of P1 and P2 in the vocabulary of this page and fix the terminology used by the later pages of the category. They adopt no physical assumption beyond P1 and P2 and the choice hypotheses of the spectral theorem, assert no dynamics, no joint measurement of several observables, no detector realization and no empirical result, and make no claim that any physical system is exactly described by an assigned pair $(A,u_A)$. The additive additivity, support, atom and moment statements cited are mathematical consequences proved in the cited items; their physical reading is exactly the reading of P2 fixed here.

• Mixtures versus superpositions. The affinity of clause (b) concerns statistical mixtures with classical weights, not superpositions: a coherent superposition is a normalized vector combination of unit representatives, and its outcome law need not be the weighted average of their laws. The interference terms can vanish for a particular observable, so equality of these laws is possible. The distinction is carried by the state-space items of the preceding page.
• No discrete basis is assumed. Nothing in this definition asserts that $A$ has eigenvalues or a complete system of eigenvectors. The outcome law exists for every assigned self-adjoint $A$ through its spectral PVM; discrete sums are the special case of a purely atomic law.