Qm Composites Entanglement And Reduced States

Assume the Axiom of Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Let $H,K$ be nonzero complex Hilbert spaces (Hilbert space) and let $H\widehat{\otimes }K$ be their completed tensor product with inclusion $\kappa$ (Hilbert tensor construction: inner product, completion and product bases). Let $(X,\mathrm{\Sigma })$ be a measurable space (Measurable spaces and measurable sets) and let $E$ be a projection valued measure on $(X,\mathrm{\Sigma })$ acting on $H$ (Projection valued measure). Define, for $B\in \mathrm{\Sigma }$, $F(B):=E(B)\otimes I_K\in \mathcal{B}(H\widehat{\otimes }K),$ the bounded tensor operator of Bounded tensor maps: extension, norm estimate, adjoint and commutation clause (a). Then:

(a) $F$ is a projection valued measure on $(X,\mathrm{\Sigma })$ acting on $H\widehat{\otimes }K$;

(b) for every bounded $\mathrm{\Sigma }$-measurable function $f:X\to \mathbb{C}$, writing ${\mathrm{\Phi }}_F(f)=\int fdF$ and ${\mathrm{\Phi }}_E(f)=\int fdE$ for the integrals of Bounded borel pvm integral, one has ${\mathrm{\Phi }}_F(f)={\mathrm{\Phi }}_E(f)\otimes I_K;$

(c) if $(S_n)$ is a sequence of bounded operators on $H$ with ${\sup }_n\mathrm{\parallel }{S}_n\mathrm{\parallel }<+\mathrm{\infty }$ and $S_{n}x\to Sx$ for every $x\in H$, then $(S_n\otimes I_K)z\to (S\otimes I_K)z$ for every $z\in H\widehat{\otimes }K$;

(d) if $X$ is a locally compact Hausdorff space (Locally compact topological space: every point has a compact neighbourhood; and what this says in a metric space), $\mathrm{\Sigma }$ its Borel $\sigma$-algebra (The Borel sigma-algebra of a topological space) and $E$ regular in the sense of Projection valued measure, then $F$ is regular.

Clause (d) is what makes $F$ a regular projection valued measure on $\mathbb{R}$ in the application to assigned sharp observables, so that the converse direction of the spectral theorem supplies a self-adjoint operator having $F$ as its spectral measure. No assertion is made about unbounded operators in this lemma itself.

• Use. The lemma is the mathematical core of the local readout of a subsystem quantity: on a composite $H\widehat{\otimes }K$, the outcome structure of a quantity of the first factor alone is $E(\cdot )\otimes I_K$, and by clause (d) together with the converse direction of the spectral theorem it is the spectral measure of a self-adjoint operator on the composite. Nothing here is a physical statement about measurements; the physical reading is fixed on the composites page.
• Countable Choice. The choice hypothesis enters through the tensor construction and is carried as stated; clause (c) additionally uses the density of the algebraic tensor product, which is part of that construction.

Assume Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Let $H,K$ be arbitrary nonzero complex Hilbert spaces, with inner products linear in the first argument, and put $L=H\widehat{\otimes }K$ with the isometric inclusion $\kappa$ of Hilbert tensor construction: inner product, completion and product bases. Write $T_{x,y}z=⟨z,y⟩x$ as in Rank-one operator calculus. Set $\mathcal{P}=\{T_{\kappa (a\otimes b),\kappa (a\otimes b)}:\mathrm{\parallel }a\mathrm{\parallel }=\mathrm{\parallel }b\mathrm{\parallel }=1\},\mathcal{S}={\overline{\mathrm{conv}\mathcal{P}}}^{\mathrm{\parallel }\cdot {\mathrm{\parallel }}_1},$ with trace class as in Trace class operator, where the convex hull consists of finite convex combinations, and the closure is taken in the trace-class operators on $L$. For each unit $\mathrm{\Psi }\in L$, let $P_{\mathrm{\Psi }}=T_{\mathrm{\Psi },\mathrm{\Psi }}$. Then $P_{\mathrm{\Psi }}\in \mathcal{S}⟺\mathrm{\Psi }=\kappa (a\otimes b)\text{ for some unit }a\in H,b\in K\mathrm{.}$

More precisely, let $\mathrm{\Psi }={\sum }_{r\in J}{s}_r\kappa (v_r\otimes u_r)$ be any finite or countably infinite positive Schmidt expansion from Infinite Schmidt decomposition and the pure product criterion, and put ${\lambda }_r=s_r^2$. The number $m={\max }_{r\in J}{\lambda }_r$ exists. Every pair of unit vectors $a,b$ satisfies $\mathrm{\mid }⟨\mathrm{\Psi },\kappa (a\otimes b)⟩{\mathrm{\mid }}^2\le m$, with equality at a Schmidt pair whose weight is $m$. Every $\sigma \in \mathcal{S}$ satisfies $\mathrm{tr}(\sigma P_{\mathrm{\Psi }})\le m$, whereas $\mathrm{tr}(P_{\mathrm{\Psi }}^2)=1$. If $\mathrm{\Psi }$ is non-product then $m<1$ and, for every $\sigma \in \mathcal{S}$, $\mathrm{\parallel }{P}_{\mathrm{\Psi }}-\sigma {\mathrm{\parallel }}_1\ge 1-m>0.$ The factors need not be separable; only the support of the single tensor vector has a countable Schmidt expansion. These are mathematical trace pairings, with no measurement or Born-rule assumption.

Assume the Axiom of Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Let $H,K$ be nonzero complex Hilbert spaces (Hilbert space) and let $H\widehat{\otimes }K$ be their completed tensor product with inclusion $\kappa$ (Hilbert tensor construction: inner product, completion and product bases). Density operators are the self-adjoint positive trace-class operators of trace one (Density operators and scalar spectral probability clause (a), Trace class operator), and for $u,u^{\mathrm{\prime }}\in H$, $v,v^{\mathrm{\prime }}\in K$ the rank-one operator $T_{u\otimes v,u^{\mathrm{\prime }}\otimes v^{\mathrm{\prime }}}$ is the one of Rank-one operator calculus.

(a) Product vectors and pure product states. A vector $\mathrm{\Psi }\in H\widehat{\otimes }K$ is a product vector when $\mathrm{\Psi }=\kappa (u\otimes v)$ for some $u\in H$, $v\in K$. A pure product state of the composite is a density operator of the form $T_{\mathrm{\Psi },\mathrm{\Psi }},\mathrm{\Psi }=\kappa (u\otimes v),\mathrm{\parallel }u\mathrm{\parallel }=\mathrm{\parallel }v\mathrm{\parallel }=1.$ A vector is non-product when it is not a product vector.

(b) Separable states. Write $\mathcal{P}$ for the set of all pure product states, a subset of the set $D(H\widehat{\otimes }K)$ of density operators. The separable states are the elements of the trace-norm closed convex hull $\mathcal{S}(H,K):=\stackrel{}{\mathrm{conv}}\mathcal{P}\subseteq D(H\widehat{\otimes }K),$ where $\mathrm{conv}$ denotes the set of finite convex combinations and the closure is taken in the trace norm $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_1$ of Trace class is a two sided Banach operator ideal. A state $\rho \in D(H\widehat{\otimes }K)$ is entangled (with respect to the declared splitting $H,K$) when $\rho \mathrm{\notin }\mathcal{S}(H,K)$.

(c) Conventions and elementary consequences of the form of the definition. The set $\mathrm{conv}\mathcal{P}$ consists exactly of the finite sums ${\sum }_{j=1}^n{w}_j{T}_{{\mathrm{\Psi }}_j,{\mathrm{\Psi }}_j}$ with $n\ge 1$, weights $w_j\ge 0$, ${\sum }_j{w}_j=1$ and unit product vectors ${\mathrm{\Psi }}_j$: such sums are convex combinations of density operators and hence density operators (The state space is convex, closed under countable mixtures, and complete in the trace norm clause (a)), without any orthogonality requirement on the product vectors. Every element of $\mathrm{conv}\mathcal{P}$ is a density operator, and $\mathcal{S}(H,K)$ is a convex subset of $D(H\widehat{\otimes }K)$: if $\rho ,\sigma$ are limits of convex combinations ${\rho }_n\to \rho$, ${\sigma }_n\to \sigma$ and $t\in [0,1]$, then $t\rho +(1-t)\sigma$ is the trace-norm limit of the convex combinations $t{\rho }_n+(1-t){\sigma }_n$. Separability of a state is a property of the declared splitting $(H,K)$; no assertion is made here about its behaviour under a different factorization of a larger space.

(d) Finite dimension. When $H$ and $K$ are finite dimensional, $\mathcal{P}$ is compact and $\mathrm{conv}\mathcal{P}$ is already closed in the trace norm (it is a compact, hence closed, convex set in the finite-dimensional space of endomorphisms of $H\widehat{\otimes }K$), so $\mathcal{S}(H,K)=\mathrm{conv}\mathcal{P}$ and no closure convention is needed. In the general case the closure in clause (b) is declared explicitly and is part of the definition.

(e) Pure-state specialization and status. A pure state $T_{\mathrm{\Psi },\mathrm{\Psi }}$ with $\mathrm{\parallel }\mathrm{\Psi }\mathrm{\parallel }=1$ is separable in the precise trace-norm closed-convex-hull sense of clause (b) exactly when $\mathrm{\Psi }$ is a product vector of clause (a); otherwise it is entangled. This is the proved mathematical specialization of Pure-state separability is equivalent to being a product: a Schmidt overlap witness, whose locally defined $\mathcal{P},\mathcal{S}$ are exactly those above. Clauses (a)-(e) fix mathematical vocabulary and adopt no physical assumption, dynamics, measurement or probability law. Whether a laboratory preparation realizes a separable or entangled state for the declared splitting remains a tested identification with uncertainty, not a consequence about nature of this definition.

• Why the closure is explicit. In infinite dimension the convex hull of the pure product states need not be trace-norm closed, and different authors complete it differently. Clause (b) declares the trace-norm closure, which is the convention used by this category; the mathematical results of the page are stated for this declared class.

• Entanglement is relative to a splitting. The same vector of a three-factor space may be a product vector with respect to one grouping of the factors and non-product with respect to another, so a separability or entanglement statement without the declared pair $(H,K)$ is incomplete; the definition above always carries that pair.

• Finite-dimensional closed convex hull. Suppose $H$ and $K$ have complex dimensions $h,k\ge 1$, put $L=H\widehat{\otimes }K$ and $N=hk$, and let $W=\{T\in \mathcal{B}(L):T=T^{\ast }\}$, regarded as a real vector space. In orthonormal coordinates, $W$ has real dimension $d=N^2$: there are $N$ real diagonal entries and $N(N-1)\mathrm{/}2$ free complex entries above the diagonal. The product of the unit spheres of $H$ and $K$ is a closed bounded subset of ${\mathbb{R}}^{2h+2k}$, hence compact by Heine-Borel in ${\mathbb{R}}^n$: with the Euclidean metric a subset of ${\mathbb{R}}^n$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line. The map $(u,v)\mapsto \kappa (u\otimes v)$ is continuous: bilinearity and the tensor norm formula give $\mathrm{\parallel }\kappa (u\otimes v)-\kappa (u^{\mathrm{\prime }}\otimes v^{\mathrm{\prime }})\mathrm{\parallel }\le \mathrm{\parallel }u-u^{\mathrm{\prime }}\mathrm{\parallel }\mathrm{\parallel }v\mathrm{\parallel }+\mathrm{\parallel }{u}^{\mathrm{\prime }}\mathrm{\parallel }\mathrm{\parallel }v-v^{\mathrm{\prime }}\mathrm{\parallel }$. The map $q\mapsto T_{q,q}$ is trace-norm continuous by Rank-one operator calculus clause (f). Therefore $\mathcal{P}$ is compact by The image of a compact metric space under a continuous map is compact, and so is the image of any compact subset.

  Every finite convex combination of more than $d+1$ members $P_1,\dots ,P_m\in \mathcal{P}$ with positive weights $w_i$ can be reduced. The vectors $(P_i,1)$ in $W\times \mathbb{R}$, a real vector space of dimension $d+1$, are linearly dependent; hence there are real $c_i$, not all zero, with ${\sum }_i{c}_i{P}_i=0$ and ${\sum }_i{c}_i=0$. The $c_i$ have both signs. Set $t={\min }_{c_i>0}{w}_i\mathrm{/}{c}_i$ and $w_i^{\mathrm{\prime }}=w_i-t{c}_i$. Then every $w_i^{\mathrm{\prime }}\ge 0$, ${\sum }_i{w}_i^{\mathrm{\prime }}=1$, ${\sum }_i{w}_i^{\mathrm{\prime }}{P}_i={\sum }_i{w}_i{P}_i$, and at least one $w_i^{\mathrm{\prime }}$ is zero. Removing zero weights and repeating leaves at most $d+1$ terms. Since $\mathcal{P}$ is nonempty, shorter combinations can be padded with zero weights. Thus $\mathrm{conv}\mathcal{P}$ is the image of ${\mathcal{P}}^{d+1}\times {\mathrm{\Delta }}_d$ under the continuous map $((P_i),(w_i))\mapsto {\sum }_i{w}_i{P}_i$, where ${\mathrm{\Delta }}_d=\{(w_1,\dots ,w_{d+1}):w_i\ge 0,{\sum }_i{w}_i=1\}$. The trace-norm and Euclidean coordinate topologies on $W$ agree. In an orthonormal basis $(g_i{)}_{i=1}^N$ of $L$, write $T={\sum }_{i,j}{t}_{ij}{T}_{g_i,g_j}$ and use the Euclidean coordinate norm $\mathrm{\parallel }T{\mathrm{\parallel }}_2:=({\sum }_{i,j}\mathrm{\mid }{t}_{ij}{\mathrm{\mid }}^2{)}^{1\mathrm{/}2}$. Each coordinate satisfies $\mathrm{\mid }{t}_{ij}\mathrm{\mid }=\mathrm{\mid }⟨T{g}_j,g_i⟩\mathrm{\mid }\le \mathrm{\parallel }T\mathrm{\parallel }\le \mathrm{\parallel }T{\mathrm{\parallel }}_1$; conversely, $\mathrm{\parallel }T{\mathrm{\parallel }}_1\le {\sum }_{i,j}\mathrm{\mid }{t}_{ij}\mathrm{\mid }\mathrm{\parallel }{T}_{g_i,g_j}{\mathrm{\parallel }}_1={\sum }_{i,j}\mathrm{\mid }{t}_{ij}\mathrm{\mid }\le N\mathrm{\parallel }T{\mathrm{\parallel }}_2$, since the matrix units have trace norm one by Rank-one operator calculus and there are $N^2$ coordinates. Thus $\mathrm{\parallel }T{\mathrm{\parallel }}_2\le N\mathrm{\parallel }T{\mathrm{\parallel }}_1$ and $\mathrm{\parallel }T{\mathrm{\parallel }}_1\le N\mathrm{\parallel }T{\mathrm{\parallel }}_2$, proving equality of the topologies. Hence in finite coordinates $\mathcal{P}$ is closed and bounded because it is compact; the simplex ${\mathrm{\Delta }}_d$ is also closed and bounded. Therefore ${\mathcal{P}}^{d+1}\times {\mathrm{\Delta }}_d$ is closed and bounded in a Euclidean space, hence compact by Heine-Borel in ${\mathbb{R}}^n$: with the Euclidean metric a subset of ${\mathbb{R}}^n$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line. Its image is compact and therefore closed in the trace-norm space. Finally, $\mathrm{conv}\mathcal{P}\subseteq D(L)$ by The state space is convex, closed under countable mixtures, and complete in the trace norm clause (a), and its trace-norm closure remains in $D(L)$ by that lemma's clause (d). This proves the finite-dimensional assertion of clause (d); clause (b)'s trace-norm closure remains the declared convention in every dimension.

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.

Adopt the declared composite model of Composite systems of the framework: product bases, local readouts, local effects and local instruments with subsystems $A,B$, assigned spaces $H,K$ and composite space $H\widehat{\otimes }K$, the state postulate P1 (State and composition postulate P1: Hilbert spaces, preparations and composite systems) and the state space $D(H\widehat{\otimes }K)$ of the framework (The state space of the framework: preparations, pure rays and density operators). Assume the Axiom of Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), the hypothesis of the cited partial-trace and tensor construction results. The following vocabulary is used by this page.

(a) Reduced preparations. For a preparation $\rho \in D(H\widehat{\otimes }K)$ of the declared composite, the reduced preparation of subsystem $A$ is ${\rho }_A:={\mathrm{Tr}}_K\rho \in D(H),$ the image of $\rho$ under the partial trace ${\mathrm{Tr}}_K$ of Separable partial trace and its basic properties clause (a), characterized by $\mathrm{tr}({\rho }_{A}X)=\mathrm{tr}(\rho (X\otimes I_K))\text{for every bounded }X\in \mathcal{B}(H);$ the reduced preparation of subsystem $B$ is ${\rho }_B:={\mathrm{Tr}}_H\rho \in D(K)$, characterized by $\mathrm{tr}({\rho }_{B}Y)=\mathrm{tr}(\rho (I_H\otimes Y))$ for every bounded $Y\in \mathcal{B}(K)$. By clause (b) of Separable partial trace and its basic properties the partial trace maps density operators to density operators and is trace preserving, so ${\rho }_A$ and ${\rho }_B$ are states of the respective subsystems in the sense of the state space of the framework.