Declared spin correlation arrangement of the framework: two spin-half factors, four directions, joint sign tables and local-response models

Adopt the state and composition postulate P1, the observable and probability postulate P2, the ideal conditional-measurement postulate P3, the kinematics and massive-particle model postulate P5 as inherited through the cited spin-$s$ model and spin-spectrum result (only their internal-factor content is used here), the internal-factor assignment of the identical-particle and spin postulate P6 and the preparation and apparatus discipline P7 (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, Kinematics and massive-particle model postulate P5: Newtonian frames and one-particle operators, Identical-particle and spin assignments P6: permutation sectors and finite-spin representations, Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions), together with the composite, observable and measurement readings of Composite systems of the framework: product bases, local readouts, local effects and local instruments, Assigned sharp observables and the outcome law of the framework and Declared ideal measurements and instruments of the framework, the angular momentum readings of Orbital, spin and total angular momentum of the massive spin-s model: assigned sharp observables, units and readings and the units of Dimensions, SI unit conventions and quantity types of the framework. Assume the Axiom of Countable Choice and the Axiom of Choice, the hypotheses of the cited mathematical suppliers. The following vocabulary is used by this page.

(a) Declared spin-half factors. A declared spin-half factor is the internal factor $V={\mathbb{C}}^2$ of the spin-$s$ model The massive spin-s particle model of the framework: state space, operators, units and frame readings with spin label $s=\frac{1}{2}$, with its orthonormal basis $(e_{+},e_{-})$, the self-adjoint matrices $J_i=\frac{1}{2}{\sigma }_i$ of Finite spin matrices clause (e) and the spin observables $S_i:=\mathrm{\hslash }{J}_i$, explicitly assigned under P2 on $V$ as sharp observables with unit $\mathrm{J}\mathrm{s}$, using the internal-factor assignment of P6 clause (ii). For a unit direction $n\in {\mathbb{R}}^3$ put $S_n:={\sum }_i{n}_i{S}_i$. Put $A_n:=n\cdot \sigma$. The Pauli identities of Finite spin matrices clause (e) give $A_n^{\ast }=A_n$ and $A_n^2=I$ for unit $n$. Each Pauli matrix has trace zero, so $A_n$ has trace zero; its eigenvalues are $+1,-1$, each once. Thus $S_n=(\mathrm{\hslash }\mathrm{/}2)A_n$ has eigenvalues $\pm \mathrm{\hslash }\mathrm{/}2$ with projections $P_{n,\epsilon }:=\frac{1}{2}(I+\epsilon n\cdot \sigma ),\epsilon \in \mathrm{\Sigma }=\{-1,+1\},$ and the label $\epsilon$ attached to the projection $P_{n,\epsilon }$ is the sign label of the outcome; the outcome coordinate is $\epsilon \mathrm{\hslash }\mathrm{/}2$, read in the unit $\mathrm{J}\mathrm{s}$, while the sign label itself is dimensionless.

(b) Declared arrangement. A declared spin correlation arrangement consists of: the declared composite $V\widehat{\otimes }V$ of two declared spin-half factors, with inclusion $\kappa$, in the sense of clause (a) of Composite systems of the framework: product bases, local readouts, local effects and local instruments (a declared pair of distinguishable subsystems whose assigned spaces are the two internal factors); a declared preparation $\rho \in D(V\widehat{\otimes }V)$, the set of density operators on the composite (The state space of the framework: preparations, pure rays and density operators, The state space: density operators, pure and mixed states, and purity); and two unit setting directions $a_0,a_1\in {\mathbb{R}}^3$ for the first station and $b_0,b_1\in {\mathbb{R}}^3$ for the second station. The setting labels are $I=J=\{0,1\}$.

(c) Declared local sign measurements. For the first station and setting $i\in I$, the declared local sign measurement is the declared ideal sharp measurement whose outcome structure is the pair $(P_{a_i,+}\otimes I,P_{a_i,-}\otimes I)$ with the sign labels $\pm 1$: these are the spectral projections of the local readout of the assigned observable $(S_{a_i},\mathrm{J}\mathrm{s})$ on the declared composite, the local readout being constructed through the spectral measure by clause (c) of Composite systems of the framework: product bases, local readouts, local effects and local instruments and Local PVM extension: tensoring a projection-valued measure with the identity. The second station's declared local sign measurement for setting $j$ is $(I\otimes P_{b_j,+},I\otimes P_{b_j,-})$, defined symmetrically.

(d) Declared joint sign measurement and tables. For $(i,j)\in I\times J$ the declared joint sign measurement of the setting pair is the declared ideal sharp measurement of the composite whose outcome structure is $E_{ij}:=(P_{a_i,\epsilon }\otimes P_{b_j,\delta }{)}_{(\epsilon ,\delta )\in \mathrm{\Sigma }\times \mathrm{\Sigma }},E_{ij}(\epsilon ,\delta )=P_{a_i,\epsilon }\otimes P_{b_j,\delta },$ with the pair of sign labels as its labels: by the identities $P_{n,\epsilon }^{\ast }=P_{n,\epsilon }$, $P_{n,\epsilon }^2=P_{n,\epsilon }$, $P_{n,+}{P}_{n,-}=0$ and $P_{n,+}+P_{n,-}=I$ from (a), each factor is an orthogonal projection, and the tensor calculus of Bounded tensor maps: extension, norm estimate, adjoint and commutation gives $E_{ij}(\epsilon ,\delta )E_{ij}({\epsilon }^{\mathrm{\prime }},{\delta }^{\mathrm{\prime }})={\delta }_{\epsilon {\epsilon }^{\mathrm{\prime }}}{\delta }_{\delta {\delta }^{\mathrm{\prime }}}{E}_{ij}(\epsilon ,\delta )$ and ${\sum }_{\epsilon ,\delta }{E}_{ij}(\epsilon ,\delta )=I$, so this is a finite family of mutually orthogonal projections resolving the identity, as required by clause (a) of Declared ideal measurements and instruments of the framework. Its outcome law in the declared preparation $\rho$ is ${\nu }_{ij}^{\rho }(\epsilon ,\delta ):=\mathrm{tr}(\rho E_{ij}(\epsilon ,\delta )),(\epsilon ,\delta )\in \mathrm{\Sigma }\times \mathrm{\Sigma },$ read from clause (b) of Declared ideal measurements and instruments of the framework (compare P2 clause (ii), Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes). The correlation table of the arrangement is the family ${\nu }^{\rho }=({\nu }_{ij}^{\rho }{)}_{(i,j)\in I\times J}$, which is a binary correlation model on $(I,J)$ in the sense of Binary correlation models: setting-indexed tables of two-valued outcome laws, marginals and the CHSH functional clause (a); its correlations $E_{ij}({\nu }^{\rho })$, its marginals and its CHSH functional $S({\nu }^{\rho })$ are the ones of clauses (b) and (d) of that definition, and are written $S(\rho ;a_0,a_1;b_0,b_1)$ when the arrangement data are displayed.

(e) Declared local-response models of the arrangement. A declared local-response model of the arrangement is a local-response representation, in the sense of Local-response representations of a binary correlation model: one setting-independent specification space and per-station response laws, of the correlation table ${\nu }^{\rho }$. It is a model of the same four tables: it consists of a probability space of specifications and of per-station response laws, subject to the representation identity, and it need not be a model of the framework's state space. The arrangement admits a local-response model when such a representation exists.

(f) Scope. Clauses (a)-(e) fix model vocabulary. They assign no spatial position, worldline, spatial separation or temporal order to the two factors; they assert no dynamics, no interaction, no apparatus, no detection procedure and no measured correlation; they do not assert that the composite is prepared by any particular laboratory procedure. Whether a laboratory arrangement realizes the declared preparation, the declared directions and the declared sign measurements is a tested identification with uncertainty under P7 (Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions), not a consequence of this definition; the model's assignments of states to preparations and of spectral projections to quantities are the readings of P1, P2 and P3 fixed on the earlier pages.

• Why the joint measurement is declared and not derived. The framework's measurement vocabulary declares outcome structures; it does not assert that arbitrary observables of two factors possess a joint sharp measurement. For the arrangement the product family of clause (d) is itself a finite family of mutually orthogonal projections resolving the identity, so the joint measurement of the two sign measurements is a declared ideal sharp measurement of the composite in the framework's own sense. No claim is made that an arbitrary pair of laboratory procedures realizes it.
• Sign labels versus coordinates. The sign labels $\pm 1$ are the outcome indices used by the correlation tables; the physical outcome coordinates are $\pm \mathrm{\hslash }\mathrm{/}2$ in the unit $\mathrm{J}\mathrm{s}$. The correlation and the CHSH functional of clause (d) are functionals of the sign labels and are dimensionless; no numerical value of an angular momentum coordinate is asserted anywhere.