Compatible sharp observables, their joint law and the ideal joint instrument

Adopt the state and composition postulate P1, the observable and probability postulate P2 in the reading of Assigned sharp observables and the outcome law of the framework, and the ideal conditional measurement postulate P3 in the reading of Declared ideal measurements and instruments of the framework (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, Ideal conditional measurement postulate P3: the projective Lueders update and discarded outcomes), 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). Let $(A,u_A)$ and $(B,u_B)$ be assigned sharp observables with spectral projection-valued measures $E_A,E_B$ (Projection valued measure, Spectral theorem for unbounded self-adjoint operators (PVM form)). Call a regular projection-valued measure $H$ on the Borel sigma-algebra of ${\mathbb{R}}^2$ a joint sharp observable of the pair when its marginals are the two spectral measures, $H(\mathrm{\Lambda }\times \mathbb{R})=E_A(\mathrm{\Lambda }),H(\mathbb{R}\times \mathrm{\Sigma })=E_B(\mathrm{\Sigma })(\mathrm{\Lambda },\mathrm{\Sigma }\text{ Borel}),$ and call the pair compatible when such an $H$ exists. Then:

(a) Criterion. The pair is compatible if and only if $A$ and $B$ strongly commute in the sense of Strong spectral commutation of self-adjoint operators clause (b), if and only if $f(A)g(B)=g(B)f(A)$ for all bounded Borel $f,g$; the joint sharp observable $H$ is then unique, and for bounded Borel $f,g$ $f(A)g(B)={\int }_{{\mathbb{R}}^2}f(a)g(b)dH(a,b)\mathrm{.}$

(b) Joint outcome law. If the pair is compatible, then for every preparation $\rho$ the assignment ${\mu }_{\rho }^H(S):=\mathrm{tr}(\rho H(S))(S\subseteq {\mathbb{R}}^2\text{ Borel})$ is a probability measure on the Borel sets of ${\mathbb{R}}^2$, the joint outcome law of the pair in $\rho$, and its marginals are the individual laws: ${\mu }_{\rho }^H(\mathrm{\Lambda }\times \mathbb{R})={\mu }_{\rho }^A(\mathrm{\Lambda }),{\mu }_{\rho }^H(\mathbb{R}\times \mathrm{\Sigma })={\mu }_{\rho }^B(\mathrm{\Sigma })\mathrm{.}$ For the pure preparation of a unit vector $\psi$ it reads ${\mu }_{\psi }^H(S)=\mathrm{\parallel }H(S)\psi {\mathrm{\parallel }}^2$.

(c) Ideal finite-resolution joint instrument. If the pair is compatible and $(P_1,\dots ,P_n)$, $(Q_1,\dots ,Q_m)$ are the projections of two finite disjoint Borel partitions of $\mathbb{R}$, $P_i:=E_A({\mathrm{\Lambda }}_i)$, $Q_j:=E_B({\mathrm{\Sigma }}_j)$ with ${\bigcup }_i{\mathrm{\Lambda }}_i={\bigcup }_j{\mathrm{\Sigma }}_j=\mathbb{R}$, then $R_{ij}:=H({\mathrm{\Lambda }}_i\times {\mathrm{\Sigma }}_j)=P_i{Q}_j$ are mutually orthogonal projections with strong sum $I$, and in the framework's declared ideal sharp measurement of the family $(R_{ij})$ the outcome probabilities are $\mathrm{tr}(\rho R_{ij})$ with marginals $\mathrm{tr}(\rho P_i)$ and $\mathrm{tr}(\rho Q_j)$. For each joint outcome $(i,j)$ with $p_{ij}:=\mathrm{tr}(\rho R_{ij})>0$, the sequential declared ideal measurements of $(P_i)$ and then $(Q_j)$, or in the reverse order, give the same conditional state ${\rho }_{ij}=\frac{R_{ij}\rho R_{ij}}{p_{ij}}=\frac{Q_j{P}_i\rho P_i{Q}_j}{p_{ij}}=\frac{P_i{Q}_j\rho Q_j{P}_i}{p_{ij}}\mathrm{.}$ In that state an immediate repetition of the joint measurement or either corresponding marginal-bin measurement returns the same outcome with probability one. If $p_{ij}=0$, no conditional state or repeatability claim is made for that outcome.

(d) Scope. Clauses (a)-(c) are conditional consequences of P1, P2, P3 and the cited mathematics. A joint sharp observable, when it exists, is a mathematical object of the framework: it supplies a joint outcome law, not an apparatus. Clause (c)'s updates and repeatability are conditional ideal-model claims under P3; no universal disturbance relation or apparatus-level disturbance claim is asserted. The claim that a concrete laboratory device realizes the joint measurement, with its resolution, losses and systematics, is a physical identification with uncertainty under the preparation and apparatus postulate P7. No dynamics or empirical result is asserted.