Qm Composites Entanglement And Reduced States — Examples

Adopt the framework's composition, state and measurement readings for the internal factors of two declared spin-half particles (Tensor spin composition, Composite systems of the framework: product bases, local readouts, local effects and local instruments, Reduced preparations of a declared composite: the subsystem states of the framework, Declared ideal measurements and instruments of the framework), 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), as required by the cited spin-outcome spectral construction. Let $V={\mathbb{C}}^2$ be a declared spin-half internal factor with orthonormal basis $(e_{+},e_{-})$ and spin matrices $J_i=\frac{1}{2}{\sigma }_i$ of Finite spin matrices, so that $S_i:=\mathrm{\hslash }{J}_i$, with unit $\mathrm{J}\mathrm{s}$, and for each unit vector $n\in {\mathbb{R}}^3$ let $S_n:={\sum }_i{n}_i{S}_i$. On the declared composite of the two internal factors $V\widehat{\otimes }V$, define the total spin component for each $i\in \{x,y,z\}$ by $S_i^{\mathrm{t}\mathrm{o}\mathrm{t}}:=S_i\otimes I_V+I_V\otimes S_i\mathrm{.}$ For a unit direction $n$ the local spin observable is $S_n\otimes I_V$ or $I_V\otimes S_n$, read as a bounded assigned sharp observable of its factor with unit $\mathrm{J}\mathrm{s}$ (Orbital, spin and total angular momentum of the massive spin-s model: assigned sharp observables, units and readings clause (c), Assigned sharp observables and the outcome law of the framework). Put $\mathrm{\Psi }:=\frac{1}{\sqrt{2}}(\kappa (e_{+}\otimes e_{-})-\kappa (e_{-}\otimes e_{+})),\rho :=T_{\mathrm{\Psi },\mathrm{\Psi }}\in D(V\widehat{\otimes }V),$ the singlet preparation, and let ${\rho }_A={\mathrm{Tr}}_B\rho$, ${\rho }_B={\mathrm{Tr}}_A\rho$ be its reduced preparations. Then:

(a) Singlet data. $\mathrm{\Psi }$ is a unit vector; $S_i^{\mathrm{t}\mathrm{o}\mathrm{t}}\mathrm{\Psi }=0$ for all $i$; ${\rho }_A={\rho }_B=\frac{1}{2}I$; and the Schmidt expansion of $\mathrm{\Psi }$ is $\mathrm{\Psi }=\frac{1}{\sqrt{2}}(\kappa (e_{+}\otimes e_{-})-\kappa (e_{-}\otimes e_{+}))$ with two equal coefficients, so $\mathrm{\Psi }$ is not a product vector and both reduced preparations are mixed with $\mathrm{tr}{\rho }_A^2=\frac{1}{2}$.

(b) Joint directional laws. For unit vectors $a,b\in {\mathbb{R}}^3$ with $\theta :=\arccos (a\cdot b)\in [0,\pi ]$, put $P_{n,\epsilon }=\frac{1}{2}(I+\epsilon n\cdot \sigma )$ for $\epsilon \in \{-1,+1\}$. These are the spectral projections of $S_n$. Define the four-outcome joint PVM on the power set of $\{-1,+1{\}}^2$ by $G_{a,b}(C):={\sum }_{(\epsilon ,\delta )\in C}{P}_{a,\epsilon }\otimes P_{b,\delta }(C\subseteq \{-1,+1{\}}^2)\mathrm{.}$ The declared PVM is the ideal joint model readout for the two factors; its P2 outcome probabilities $p(\epsilon ,\delta ):=\mathrm{tr}(\rho G_{a,b}(\{(\epsilon ,\delta )\}))$ are $p(+,+)=p(-,-)=\frac{1}{2}{\sin }^2\frac{\theta }{2},p(+,-)=p(-,+)=\frac{1}{2}{\cos }^2\frac{\theta }{2},$ where the first sign refers to the outcome $+\mathrm{\hslash }\mathrm{/}2$ or $-\mathrm{\hslash }\mathrm{/}2$ of $S_a\otimes I$ and the second to that of $I\otimes S_b$; in particular the four numbers are nonnegative and sum to one.

(c) Correlation function. With the product $\epsilon \delta$ of the two sign labels, $\mathbb{E}(a,b):={\sum }_{\epsilon ,\delta \in \{-1,+1\}}\epsilon \delta p(\epsilon ,\delta )=-\cos \theta =-a\cdot b\mathrm{.}$ In particular $\mathbb{E}(a,a)=-1$ and $\mathbb{E}(a,-a)=+1$: the outcomes are perfectly anti-correlated for equal directions.

(d) Setting-independent marginals. The marginal laws are $p_A(\pm )=\frac{1}{2}$ and $p_B(\pm )=\frac{1}{2}$, independent of $b$ and of $a$ respectively; the reduced preparations ${\rho }_A={\rho }_B=\frac{1}{2}I$ are unchanged under the other factor's declared ideal local measurement model followed by discarding its record, in agreement with Local instruments and no-signalling: operations on one factor do not change the other factor's local statistics clause (b).

(e) Scope. This is a thought experiment of the declared framework: the composite is an internal-factor composite, and the PVM and probabilities in clause (b) are ideal model objects and predictions under P2. The particles are not assigned spatial positions or worldlines here; no spatial separation, temporal order, relativistic consideration, concrete apparatus, performed measurement, recorded data or empirical correlation enters. The computation draws no conclusion about nature and makes no claim about the completeness of the framework or about any hidden-variable model.

• Anti-correlation and the singlet. For $b=\pm a$ the correlation is $\mp 1$, the perfect anti-correlation (or correlation for the reversed direction) of the model. For $a\perp b$ the correlation vanishes, $\mathbb{E}=0$.
• Correlated but not signalling. The joint laws depend on both directions, while each marginal is the fixed law $\frac{1}{2}$ of the maximally mixed reduced preparation: the model's correlations are not a dependence of one factor's local statistics on the other factor's setting.
• Not an experiment. No counts, apparatus or uncertainty analysis appears; a comparison of such a model with recorded data would require a separate experiment item with its own preparation, efficiency and sampling qualifications, and the Bell comparison belongs to a later page.

Adopt the declared marked two-component model of Declared marked two-component model: path components, marker states, phase family, detection weights and fringe visibility with $s=0$ (scalar particle), $m>0$, $H_0=L^2({\mathbb{R}}^3)$, and the marker space $K={\mathbb{C}}^2$, 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), as required by the cited position and visibility laws. Declare the following two-cell model:

(a) fix the length $\mathrm{\ell }:=1\mathrm{m}$ and a unit vector $\eta \in L^2({\mathbb{R}}^2)$, whose representatives have dimension ${\mathrm{m}}^{-1}$ for transverse coordinates in metres. Define the longitudinal profiles $f_1(x_1):={\mathrm{\ell }}^{-1\mathrm{/}2}{\mathbf{1}}_{[0,\mathrm{\ell }]}(x_1),f_2(x_1):={\mathrm{\ell }}^{-1\mathrm{/}2}({\mathbf{1}}_{[0,\mathrm{\ell }\mathrm{/}2]}(x_1)-{\mathbf{1}}_{[\mathrm{\ell }\mathrm{/}2,\mathrm{\ell }]}(x_1)),$ of dimension ${\mathrm{m}}^{-1\mathrm{/}2}$, and the unit path components, as equivalence classes of measurable representatives, ${\psi }_j(x_1,x_2,x_3):=f_j(x_1)\eta (x_2,x_3)(j=1,2),$ of dimension ${\mathrm{m}}^{-3\mathrm{/}2}$. Changes at cell boundaries are null-set changes.

(b) declare the marker states $e_1:=(1,0)$, $e_2:=(\cos \vartheta ,\sin \vartheta )$ for a fixed $\vartheta \in [0,\pi \mathrm{/}2]$, so that $⟨e_1,e_2⟩=\cos \vartheta$;

(c) declare the detection bin $B:=[0,\mathrm{\ell }\mathrm{/}2]\times {\mathbb{R}}^2$ with position effect $G(B)=M_{{\mathbf{1}}_B}$.

Then:

(i) ${\psi }_1,{\psi }_2$ are unit vectors with $⟨{\psi }_1,{\psi }_2⟩=0$ and $\mathrm{\parallel }\mathrm{\Psi }(\phi )\mathrm{\parallel }=1$ for every phase $\phi$, so the model is overlap-normalized;

(ii) $p_1(B)=p_2(B)=\frac{1}{2}$, $⟨G(B){\psi }_1,{\psi }_2⟩=\frac{1}{2}$ and $p_{\mathrm{m}\mathrm{i}\mathrm{x}}(B)=\frac{1}{2}$, hence $p(\phi )(B)=\frac{1}{2}(1+\cos \vartheta \cos \phi ),V(B)=\cos \vartheta =\mathrm{\mid }⟨e_1,e_2⟩\mathrm{\mid }:$ the visibility equals the marker overlap exactly, saturating the bound of Marker overlap contracts the cross term: the detection law and the visibility bound of an overlap-normalized marked model clause (d);

(iii) at $\vartheta =0$ (identical markers) $p(\phi )(B)=\frac{1}{2}(1+\cos \phi )$ and $V(B)=1$; at $\vartheta =\pi \mathrm{/}2$ (orthogonal markers) $p(\phi )(B)=\frac{1}{2}$ for every phase and $V(B)=0$, so the marked detection law is exactly the which-path law.

The profiles, the bin and the marker states are declared idealizations of the model; the computation asserts no measured pattern, no counts, no detector response and no empirical result.

Adopt the declared internal-factor composite of two spin-half factors and the singlet preparation $\rho =T_{\mathrm{\Psi },\mathrm{\Psi }}$ of The spin singlet of two declared spin-half factors: directional correlations, perfect anti-correlation and setting-independent marginals, with $V={\mathbb{C}}^2$, orthonormal basis $(e_{+},e_{-})$, spin matrices $S_i=\mathrm{\hslash }{J}_i$ and $S_n={\sum }_i{n}_i{S}_i$ for unit $n\in {\mathbb{R}}^3$ (Finite spin matrices, Spin spectra and outcome laws of the massive spin-s model: directional components, the Casimir law and the spin-half probabilities), and let Reduced preparations of a declared composite: the subsystem states of the framework and Schmidt reduction of a pure composite preparation, purity of the subsystems, and the product criterion supply the reading of the reduced preparations. 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), as required by the cited spin-outcome and singlet suppliers. Declare:

(a) the improper mixture of factor $A$: the reduced preparation ${\rho }_A={\mathrm{Tr}}_B\rho$ of the singlet;

(b) the proper mixture of factor $A$: the declared preparation in which a fair classical randomization selects the pure preparation $T_{e_{+},e_{+}}$ with probability $\frac{1}{2}$ and $T_{e_{-},e_{-}}$ with probability $\frac{1}{2}$, so that ${\rho }_{\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{p}\mathrm{e}\mathrm{r}}:=\frac{1}{2}{T}_{e_{+},e_{+}}+\frac{1}{2}{T}_{e_{-},e_{-}};$

(c) the EPR-type product mixture of the composite: the declared preparation $M:=\frac{1}{2}{T}_{e_{+}\otimes e_{-},e_{+}\otimes e_{-}}+\frac{1}{2}{T}_{e_{-}\otimes e_{+},e_{-}\otimes e_{+}},$ in which each factor carries a definite spin along the $z$-axis and the two spins are opposite.

Then:

(i) ${\rho }_A={\rho }_{\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{p}\mathrm{e}\mathrm{r}}=\frac{1}{2}I$ and the marginals of $M$ are ${\mathrm{Tr}}_{B}M=\frac{1}{2}I={\mathrm{Tr}}_{A}M$: the two preparations of factor $A$ and the marginal of $M$ are the same density operator, and $\rho$ and $M$ are not the same composite preparation;

(ii) for unit directions $a,b$, writing $a_3=a\cdot e_z$ and $b_3=b\cdot e_z$ with $e_z$ the direction of the $z$-axis of the basis $(e_{+},e_{-})$, the directional correlation of the singlet is ${\mathbb{E}}_{\rho }(a,b)=-\cos \theta =-a\cdot b$ while that of the product mixture is ${\mathbb{E}}_M(a,b)=-\frac{1}{2}{a}_3{b}_3-\frac{1}{2}{a}_3{b}_3=-a_3{b}_3;$ in particular ${\mathbb{E}}_{\rho }(z,z)={\mathbb{E}}_M(z,z)=-1$ but ${\mathbb{E}}_{\rho }(x,x)=-1$ and ${\mathbb{E}}_M(x,x)=0$ for the orthogonal direction $x$;

(iii) $M$ is a convex combination of the two pure product states $T_{e_{+}\otimes e_{-},e_{+}\otimes e_{-}}$ and $T_{e_{-}\otimes e_{+},e_{-}\otimes e_{+}}$, hence a separable preparation in the sense of clause (b) of Product vectors, separable states and entangled states of a two-factor composite, while $\rho$ is a non-product pure preparation; and every local readout law of factor $A$ is the same in $\rho$, in ${\rho }_{\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{p}\mathrm{e}\mathrm{r}}$ extended by any preparation of $B$, and in $M$, because all three have the reduced preparation $\frac{1}{2}I$ on $A$;

(iv) consequently the density operator $\frac{1}{2}I$ alone does not determine whether the preparation of factor $A$ arises as the reduction of an entangled composite preparation (the improper case) or from a classical randomization of factor preparations (the proper case): the two cases are distinguished by the joint statistics with the other factor, not by any local statistic of $A$.

The comparison is a conditional computation of the declared framework; it asserts no measured correlation, no experiment and no empirical fact.

• The reduction is not an ensemble decomposition. Nothing in the example presents ${\rho }_A=\frac{1}{2}I$ as produced by a classical randomization; the reduction is the partial trace of the composite preparation, and the randomization of the proper mixture is a separate declared preparation of factor $A$ that happens to assign the same density operator.
• Same local state, different composite. The comparison is a precise instance of the general fact that the reduced preparation determines all local statistics of its own factor while the joint statistics depend on the composite preparation.

• What the page contains. Three conditional computations in the declared framework: the singlet thought experiment of The spin singlet of two declared spin-half factors: directional correlations, perfect anti-correlation and setting-independent marginals with its reduced preparations, directional joint laws, correlation function and setting-independent marginals; the two-cell marked model of A two-cell marked model: the fringe visibility equals the marker overlap exactly, where the fringe visibility equals the marker overlap exactly and the theorem Marker overlap contracts the cross term: the detection law and the visibility bound of an overlap-normalized marked model is saturated; and the comparison of Improper and proper mixtures with the same reduced preparation: the singlet marginal versus a fair random mixture between the singlet's reduced preparation and a fair random mixture of factor preparations that assigns the same density operator. These computations retain the choice hypotheses stated by their suppliers, Countable Choice and (where a cited spectral PVM construction requires it) Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The Axiom of Choice).
• What the page does not contain. No experiment, no reported count, no calibration, no efficiency, no sampling model, no statistical test and no measured correlation appears on the page; no item is a report of an observed outcome, and none of the computations establishes that the framework or any of its postulates describes nature. The model data of the marked example and of the singlet are declared idealizations; identifying a concrete apparatus with them is a tested identification with uncertainty under P7 (Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions).
• Statistics of the displayed probabilities. The detection weights and joint laws are probabilities assigned by the declared models to single-trial outcomes or phases. A correlation function is the expectation of a product of outcome labels computed from a joint law; it is not itself a probability. A finite sample of trials would require its own sampling model and would show fluctuations; no inference, error rate or conclusion is drawn here from the displayed numbers, and a measured fringe pattern would concern the accumulated distribution of many detections, not a single event.
• Deferred comparisons. The Bell-type comparison of the singlet correlations with specified local-response models, with its angles, finite-data selection and loss caveats, belongs to the later Bell-models pair and is neither stated nor assumed here; nor does the page settle any interpretation of the alternatives, of the marker or of "which path", and it does not model any dynamical decoherence process.