Composite systems of the framework: product bases, local readouts, local effects and local instruments

Adopt the state and composition postulate (P1), the observable and probability postulate (P2) and the ideal conditional-measurement postulate (P3) 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 choice hypotheses of the cited mathematical suppliers (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The Axiom of Choice). The following vocabulary is used by this page for a declared pair of distinguishable subsystems.

(a) Declared composite and its state space. A declared composite model is the assignment of two declared systems $A$ and $B$ of the massive-matter sector, their assigned nonzero separable complex Hilbert spaces $H$ and $K$ (State and composition postulate P1: Hilbert spaces, preparations and composite systems clauses (i) and (iv), Hilbert space), and the composite assigned space $H\widehat{\otimes }K,$ the completed Hilbert tensor product of $H$ and $K$ with inclusion $\kappa$ (Hilbert tensor construction: inner product, completion and product bases, clause (iv) of P1). The state space of the declared composite is $D(H\widehat{\otimes }K)$, the set of density operators on $H\widehat{\otimes }K$ (The state space of the framework: preparations, pure rays and density operators). A preparation of the composite is an element of this set; the model's assignment of a state to a preparation is the assignment of clause (iii) of P1, not a consequence of this definition.

(b) Product basis convention. If $(e_a{)}_{a\in \mathbf{A}}$ is a finite or countable orthonormal basis of $H$ and $(f_b{)}_{b\in \mathbf{B}}$ one of $K$, then $(\kappa (e_a\otimes f_b){)}_{(a,b)\in \mathbf{A}\times \mathbf{B}}$ is an orthonormal basis of $H\widehat{\otimes }K$ (Hilbert tensor construction: inner product, completion and product bases clause (d)); every $\mathrm{\Psi }\in H\widehat{\otimes }K$ has the expansion $\mathrm{\Psi }={\sum }_{a,b}{c}_{ab}\kappa (e_a\otimes f_b),c_{ab}\in \mathbb{C},$ converging in norm. A product vector is $\kappa (u\otimes v)$ for $u\in H$, $v\in K$, and a product preparation is a state of the form $T_{\mathrm{\Psi },\mathrm{\Psi }}$ with $\mathrm{\Psi }=\kappa (u\otimes v)$, $\mathrm{\parallel }u\mathrm{\parallel }=\mathrm{\parallel }v\mathrm{\parallel }=1$ (Hilbert tensor construction: inner product, completion and product bases). Pure composite preparations are projectors $T_{\mathrm{\Psi },\mathrm{\Psi }}$ for unit vectors $\mathrm{\Psi }$ given by norm-convergent superpositions of product vectors as above. General composite preparations are density operators on the completed product, including statistical mixtures of pure composite preparations; these need not factor. Superposition acts on vectors, while mixing acts on density operators (P1 clauses (ii)-(iv)).

(c) Local readout of an assigned sharp observable. Let $(A,u_A)$ be an assigned sharp observable of subsystem $A$ with spectral projection valued measure $E_A$ on the Borel $\sigma$-algebra of $\mathbb{R}$ (Assigned sharp observables and the outcome law of the framework clauses (a), (b), the spectral measure being regular by Spectral theorem for unbounded self-adjoint operators (PVM form)). Define $F_A(B):=E_A(B)\otimes I_K(B\subseteq \mathbb{R}\text{ Borel})\mathrm{.}$ By Local PVM extension: tensoring a projection-valued measure with the identity clauses (a) and (d), $F_A$ is a regular projection valued measure on the Borel $\sigma$-algebra of $\mathbb{R}$ acting on $H\widehat{\otimes }K$ (Projection valued measure). Its integral $A^{\mathrm{l}\mathrm{o}\mathrm{c}}:=\int \lambda d{F}_A$ with domain $\{\mathrm{\Psi }:\int {\lambda }^{2}d{F}_{A,\mathrm{\Psi }}(\lambda )<\mathrm{\infty }\}$ is self-adjoint with spectral projection valued measure $F_A$ by the converse direction of Spectral theorem for unbounded self-adjoint operators (PVM form). The pair $(A^{\mathrm{l}\mathrm{o}\mathrm{c}},u_A)$ is therefore an assigned sharp observable of the composite (Assigned sharp observables and the outcome law of the framework clause (a)); it is the local readout of $(A,u_A)$ on the declared composite, and the unit $u_A$ is read as before (Dimensions, SI unit conventions and quantity types of the framework). Its outcome law in a preparation $\rho \in D(H\widehat{\otimes }K)$ is ${\mu }_{\rho }^{A^{\mathrm{l}\mathrm{o}\mathrm{c}}}(B)=\mathrm{tr}(\rho (E_A(B)\otimes I_K))\mathrm{.}$ The analogous construction with the roles of $A$ and $B$ exchanged gives the local readout of an assigned sharp observable of $B$.

(d) Local effects. A local effect of subsystem $A$ is a bounded operator of the form $F\otimes I_K$ with $0\le F\le I_H$ in the quadratic-form order; the map $F\mapsto F\otimes I_K$ is well defined by Bounded tensor maps: extension, norm estimate, adjoint and commutation clause (a), and $F\otimes I_K$ is an effect on $H\widehat{\otimes }K$: it is bounded, self-adjoint $((F\otimes I{)}^{\ast }=F^{\ast }\otimes I=F\otimes I)$, and it factors as $F\otimes I_K=(F^{1\mathrm{/}2}\otimes I_K{)}^{\ast }(F^{1\mathrm{/}2}\otimes I_K),I_H\otimes I_K-F\otimes I_K=((I-F{)}^{1\mathrm{/}2}\otimes I_K{)}^{\ast }((I-F{)}^{1\mathrm{/}2}\otimes I_K),$ with the positive square roots of the continuous functional calculus (Continuous functional calculus for bounded self adjoint operators) and the product and adjoint rules of Bounded tensor maps: extension, norm estimate, adjoint and commutation clauses (b), (c); hence $0\le F\otimes I_K\le I_H\otimes I_K$ in the quadratic-form order (Non-negative and positive operators, A bounded linear operator between normed spaces). Local effects of subsystem $B$ are defined in the same way.

(e) Local instruments. Let $(M_{a,r}{)}_{\begin{array}{c}a\in A\\ r\in R_a\end{array}}$ be a declared finite or countable instrument of subsystem $A$ with Kraus family as in Declared ideal measurements and instruments of the framework clause (d), normalized in the strong sense ${\sum }_{a,r}{M}_{a,r}^{\ast }{M}_{a,r}=I_H$. The local extension of the instrument to the declared composite is the family $(M_{a,r}\otimes I_K{)}_{\begin{array}{c}a\in A\\ r\in R_a\end{array}}$ together with the outcome effects ${\sum }_r(M_{a,r}\otimes I_K{)}^{\ast }(M_{a,r}\otimes I_K)=({\sum }_r{M}_{a,r}^{\ast }{M}_{a,r})\otimes I_K$ and the conditional updates, defined only for outcomes with $p_a>0$, $\rho ⟼\frac{1}{p_a}{\sum }_r(M_{a,r}\otimes I_K)\rho (M_{a,r}\otimes I_K{)}^{\ast },p_a=\mathrm{tr}(\rho F_a^{\mathrm{l}\mathrm{o}\mathrm{c}})\mathrm{.}$ It is normalized: every finite partial sum of ${\sum }_{a,r}(M_{a,r}\otimes I_K{)}^{\ast }(M_{a,r}\otimes I_K)$ equals $\left({\sum }_{a\le N,r\le R}{M}_{a,r}^{\ast }{M}_{a,r}\right)\otimes I_K$ by linearity of tensoring with $I_K$, the finite partial sums ${\sum }_{a\le N,r\le R}{M}_{a,r}^{\ast }{M}_{a,r}$ are uniformly bounded by $I_H$ and converge strongly to $I_H$, so clause (c) of Local PVM extension: tensoring a projection-valued measure with the identity gives strong convergence of the extended partial sums to $I_H\otimes I_K$; hence the local extension is a declared finite or countable instrument of the composite in the sense of clause (d) of Declared ideal measurements and instruments of the framework. The same construction applies with the roles of $A$ and $B$ exchanged.

(f) Scope. Clauses (a)-(e) fix the reading of clause (iv) of P1 together with clauses (i), (ii) and (iv) of P2 and clause (v) of P3 for a declared pair of distinguishable subsystems. They add no physical assumption to those postulates beyond the readings displayed. In particular: the subsystems are distinguishable, so no exchange sector, symmetrization or exclusion statement is introduced (those are separate assumptions of P6); no superselection restriction or algebraic replacement of the tensor product is adopted; no interaction, dynamics, disturbance of the other subsystem or correlation is asserted; and whether a concrete laboratory arrangement realizes a declared composite, a declared local readout or a declared local instrument is a tested identification with uncertainty under P7, not a consequence of this definition.

• Countable instrument normalization. In clause (e), write $\mathcal{I}=\{(a,r):a\in A,r\in R_a\}$ for the finite or countable Kraus index set of the declared instrument and enumerate it. For each fixed outcome $a$, enumerate its finite or countable fiber $R_a$ and let $S_{a,N}$ be the sum of $M_{a,r}^{\ast }{M}_{a,r}$ over its first $N$ indices. The declared instrument definition says these partial sums converge strongly to $F_a$; they are positive contractions because they are sub-sums of the positive normalized total family. For every finite partial sum the bounded tensor calculus gives ${\sum }_{r\le N}(M_{a,r}\otimes I_K{)}^{\ast }(M_{a,r}\otimes I_K)=S_{a,N}\otimes I_K$. Applying clause (c) of Local PVM extension: tensoring a projection-valued measure with the identity to $S_{a,N}\to F_a$ proves strong convergence of these outcome effects to $F_a\otimes I_K$. Applying the same clause to the enumerated total partial sums proves that the full extended family has strong sum $I_H\otimes I_K$. This verifies both the per-outcome identity and the normalization asserted in clause (e).

• Why the local readout is constructed through the spectral measure. The composite operator $A\otimes I_K$ for an unbounded assigned observable $A$ is not defined by this page directly; the local readout is defined through its spectral projection valued measure $E_A(\cdot )\otimes I_K$, whose regularity is supplied by Local PVM extension: tensoring a projection-valued measure with the identity clause (d). For a bounded $A$, the local readout coincides with the bounded operator $A\otimes I_K$ of Bounded tensor maps: extension, norm estimate, adjoint and commutation clause (a). Choose $R>\mathrm{\parallel }A\mathrm{\parallel }$ and the bounded Borel function $g_R(\lambda )=\max (-R,\min (\lambda ,R))$. The spectral measure $E_A$ is supported in $\sigma (A)\subset [-\mathrm{\parallel }A\mathrm{\parallel },\mathrm{\parallel }A\mathrm{\parallel }]$, and $F_A=E_A(\cdot )\otimes I_K$ has the same support. Thus $g_R(\lambda )$ agrees with $\lambda$ on both spectral supports. The bounded spectral calculus and the unbounded spectral theorem identify ${\mathrm{\Phi }}_{E_A}(g_R)=A$ and $\int \lambda d{F}_A={\mathrm{\Phi }}_{F_A}(g_R)$. Clause (b) of Local PVM extension: tensoring a projection-valued measure with the identity now gives ${\mathrm{\Phi }}_{F_A}(g_R)={\mathrm{\Phi }}_{E_A}(g_R)\otimes I_K=A\otimes I_K$.

• Choice hypotheses. The tensor construction uses Countable Choice; the spectral measure of an assigned sharp observable and its converse use the Axiom of Choice. Both are carried as stated and are inherited by the results of this page that invoke the local readout.