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The spin singlet of two declared spin-half factors: directional correlations, perfect anti-correlation and setting-independent marginals

Physical scope

An idealized model computation for the internal-factor composite of two declared spin-half factors in the singlet preparation: the reduced preparations, the joint PVM of the two local directional spin readouts, its outcome probabilities predicted by P2, the resulting correlation expectation and the setting-independent marginals. The declared settings and PVM are model inputs; no particular apparatus, performed laboratory measurement, recorded trial, measured correlation or empirical fact is asserted, and the item draws no conclusion about nature.

Statement

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 ( ${A C}_{ω}$ ), The Axiom of Choice), as required by the cited spin-outcome spectral construction. Let $V = C^{2}$ be a declared spin-half internal factor with orthonormal basis $(e_{+}, e_{-})$ and spin matrices $J_{i} = \frac{1}{2} σ_{i}$ of Finite spin matrices, so that $S_{i} : = ℏ J_{i}$ , with unit $J s$ , and for each unit vector $n \in R^{3}$ let $S_{n} : = \sum_{i} n_{i} S_{i}$ . On the declared composite of the two internal factors $V \hat{\otimes} V$ , define the total spin component for each $i \in {x, y, z}$ by $S_{i}^{t o t} : = S_{i} \otimes I_{V} + I_{V} \otimes S_{i} .$ 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 $J s$ (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 $Ψ : = \frac{1}{\sqrt{2}} (κ (e_{+} \otimes e_{-}) - κ (e_{-} \otimes e_{+})), ρ : = T_{Ψ, Ψ} \in D (V \hat{\otimes} V),$ the singlet preparation, and let $ρ_{A} = {Tr}_{B} ρ$ , $ρ_{B} = {Tr}_{A} ρ$ be its reduced preparations. Then:

(a) Singlet data. $Ψ$ is a unit vector; $S_{i}^{t o t} Ψ = 0$ for all $i$ ; $ρ_{A} = ρ_{B} = \frac{1}{2} I$ ; and the Schmidt expansion of $Ψ$ is $Ψ = \frac{1}{\sqrt{2}} (κ (e_{+} \otimes e_{-}) - κ (e_{-} \otimes e_{+}))$ with two equal coefficients, so $Ψ$ is not a product vector and both reduced preparations are mixed with $tr ρ_{A}^{2} = \frac{1}{2}$ .

(b) Joint directional laws. For unit vectors $a, b \in R^{3}$ with $θ : = \arccos (a \cdot b) \in [0, π]$ , put $P_{n, ε} = \frac{1}{2} (I + ε n \cdot σ)$ for $ε \in {- 1, + 1}$ . These are the spectral projections of $S_{n}$ . Define the four-outcome joint PVM on the power set of ${- 1, + 1}^{2}$ by $G_{a, b} (C) : = \sum_{(ε, δ) \in C} P_{a, ε} \otimes P_{b, δ} (C \subseteq {- 1, + 1}^{2}) .$ The declared PVM is the ideal joint model readout for the two factors; its P2 outcome probabilities $p (ε, δ) : = tr (ρ G_{a, b} ({(ε, δ)}))$ are $p (+, +) = p (-, -) = \frac{1}{2} \sin^{2} \frac{θ}{2}, p (+, -) = p (-, +) = \frac{1}{2} \cos^{2} \frac{θ}{2},$ where the first sign refers to the outcome $+ ℏ / 2$ or $- ℏ / 2$ 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 $ε δ$ of the two sign labels, $E (a, b) : = \sum_{ε, δ \in {- 1, + 1}} ε δ p (ε, δ) = - \cos θ = - a \cdot b .$ In particular $E (a, a) = - 1$ and $E (a, - a) = + 1$ : 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 $ρ_{A} = ρ_{B} = \frac{1}{2} I$ 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.

Facts & Assumptions

Given: The declared spin-half internal factor $V = C^{2}$ with basis $(e_{+}, e_{-})$ and the spin matrices $S_{i} = ℏ J_{i}$ ; the declared composite $V \hat{\otimes} V$ with local observables $S_{n} \otimes I$ , $I \otimes S_{m}$ ; the singlet vector $Ψ$ and preparation $ρ$ ; unit directions $a, b$ ; the Axiom of Countable Choice and the Axiom of Choice.

[F1]

Finite spin-half matrices. $J_{i} = \frac{1}{2} σ_{i}$ is self-adjoint on $V$ with $J_{z} e_{\pm} = \pm \frac{1}{2} e_{\pm}$ , $J_{+} e_{-} = e_{+}$ , $J_{+} e_{+} = 0$ , and $σ_{i} σ_{j} = δ_{i j} I + i \sum_{k} ε_{i j k} σ_{k}$ , so $S_{n} = \frac{ℏ}{2} n \cdot σ$ with $(n \cdot σ)^{2} = I$ (Finite spin matrices clauses (a), (b), (e)).

[F2]

Directional outcome laws on the declared internal factor. Put $A_{n} = n \cdot σ$ on $V = C^{2}$ . The matrices of [F1] give $A_{n}^{*} = A_{n}$ , $A_{n}^{2} = I$ and $tr A_{n} = 0$ for unit $n$ (Finite spin matrices clause (e)). Hence $P_{n, \pm} = (I \pm A_{n}) / 2$ are self-adjoint idempotents, $P_{n, +} P_{n, -} = 0$ and $P_{n, +} + P_{n, -} = I$ . Each has trace one, so neither is zero or $I$ and each has rank one on the two-dimensional space. The identity $S_{n} = (ℏ / 2) (P_{n, +} - P_{n, -})$ therefore identifies them as the spectral projections, with eigenvalues $\pm ℏ / 2$ . For a unit $ψ \in V$ , P2 assigns $μ_{ψ}^{S_{n}} ({\pm ℏ / 2}) = tr (T_{ψ, ψ} P_{n, \pm}) = ⟨ P_{n, \pm} ψ, ψ ⟩ = ∥ P_{n, \pm} ψ ∥^{2} = (1 \pm ⟨ A_{n} ψ, ψ ⟩) / 2$ (Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes clause (ii), Rank-one operator calculus clauses (b), (d)). This finite-factor deduction is distinct from Spin spectra and outcome laws of the massive spin-s model: directional components, the Casimir law and the spin-half probabilities clauses (a), (c), whose outcome law is on $H_{1 / 2} = L^{2} (R^{3}; C^{2})$ with fibrewise projections.

[F3]

Tensor composition. For the spin-half factor, the mathematical tensor-spin supplier defines the dimensionless total components $J_{i}^{t o t} = J_{i} \otimes I_{V} + I_{V} \otimes J_{i}$ ; the completed tensor product preserves the elementary tensor pairing, and the bounded tensor calculus has the adjoint and product rules (Tensor spin composition clause (a), Hilbert tensor construction: inner product, completion and product bases clauses (a), (b), Bounded tensor maps: extension, norm estimate, adjoint and commutation clauses (a)-(c)). Thus the displayed $S_{i}^{t o t}$ is $ℏ J_{i}^{t o t}$ . No annihilation of the displayed singlet is asserted by this supplier; that action is computed below.

[F4]

Partial trace and reductions. ${Tr}_{B} T_{u \otimes v, u^{'} \otimes v^{'}} = ⟨ v, v^{'} ⟩ T_{u, u^{'}}$ , and for unit $Ψ = \frac{1}{\sqrt{2}} (κ (e_{+} \otimes e_{-}) - κ (e_{-} \otimes e_{+}))$ the reduced preparations are $ρ_{A} = \frac{1}{2} T_{e_{+}, e_{+}} + \frac{1}{2} T_{e_{-}, e_{-}}$ and $ρ_{B} = \frac{1}{2} T_{e_{-}, e_{-}} + \frac{1}{2} T_{e_{+}, e_{+}}$ ; the purity criterion gives $tr ρ_{A}^{2} = \frac{1}{2}$ and shows $Ψ$ is not a product vector (Separable partial trace and its basic properties clause (c), Schmidt reduction of a pure composite preparation, purity of the subsystems, and the product criterion clauses (a), (b)).

[F5]

Joint model outcome law. Once the family $G_{a, b}$ is verified to be a declared ideal sharp measurement in step 1.2, P2 assigns probabilities $tr (ρ G_{a, b} ({(ε, δ)}))$ ; for rank-one local projections the pairing is $∣ ⟨ Ψ, κ (x \otimes y) ⟩ ∣^{2}$ (Declared ideal measurements and instruments of the framework clauses (a), (b), Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes clause (ii), Rank-one operator calculus clause (e)).

[F6]

No-signalling. A declared instrument of one factor leaves the reduced preparation of the other factor unchanged when its record is discarded (Local instruments and no-signalling: operations on one factor do not change the other factor's local statistics clause (b)).

[F7]

Choice hypothesis. The Axiom of Choice is inherited by the adopted composite interface for local spectral readouts (The Axiom of Choice). The finite-matrix derivation in [F2] uses no further choice or full-space spin outcome premise.

Proof

technique · direct; read the singlet's reduced preparations off the partial trace, project the singlet onto the product of directional spin eigenstates to obtain the joint laws, and evaluate the correlation sum and the marginals

Given: The data of the statement and [F1]-[F6].

1.1givenF1F3F4algebra

[Singlet data and reduced preparations] The basis actions of the Pauli matrices are $σ_{x} e_{+} = e_{-}$ , $σ_{x} e_{-} = e_{+}$ , $σ_{y} e_{+} = i e_{-}$ , $σ_{y} e_{-} = - i e_{+}$ , $σ_{z} e_{+} = e_{+}$ , and $σ_{z} e_{-} = - e_{-}$ . On each of the two terms $e_{+} \otimes e_{-}$ and $e_{-} \otimes e_{+}$ , the $z$ -component $σ_{z} \otimes I + I \otimes σ_{z}$ vanishes; applying the displayed $x$ - and $y$ -actions shows respectively that $(σ_{x} \otimes I + I \otimes σ_{x}) (e_{+} \otimes e_{-} - e_{-} \otimes e_{+}) = 0$ and $(σ_{y} \otimes I + I \otimes σ_{y}) (e_{+} \otimes e_{-} - e_{-} \otimes e_{+}) = 0$ . Since $S_{i}^{t o t} = \frac{ℏ}{2} (σ_{i} \otimes I + I \otimes σ_{i})$ , this proves $S_{i}^{t o t} Ψ = 0$ for all three indices. Orthonormality makes $Ψ$ unit; the partial-trace formula [F4] gives $ρ_{A} = \frac{1}{2} T_{e_{+}, e_{+}} + \frac{1}{2} T_{e_{-}, e_{-}} = \frac{1}{2} I$ and $ρ_{B} = \frac{1}{2} I$ , with $tr ρ_{A}^{2} = \frac{1}{2}$ . The two positive Schmidt coefficients show $Ψ$ is not product. This is (a).

1.2F2F3F7algebra

[The declared joint PVM] For each unit $a$ , the two operators $P_{a, ε}$ are orthogonal projections with sum $I_{V}$ by [F2]; likewise for $b$ . By [F3], each $G_{a, b} ({(ε, δ)}) = P_{a, ε} \otimes P_{b, δ}$ is an orthogonal projection. The tensor product rule shows distinct outcome pairs give orthogonal projections, and $\sum_{ε, δ} G_{a, b} ({(ε, δ)}) = (\sum_{ε} P_{a, ε}) \otimes (\sum_{δ} P_{b, δ}) = I_{V} \otimes I_{V}$ . Thus $G_{a, b}$ is a finite PVM, hence a declared ideal sharp joint measurement under P3.

1.3givenF1F2F3algebra

Fix unit $a$ and choose an orthonormal eigenbasis $(u_{+}, u_{-})$ of $a \cdot σ$ , with eigenvalues $+ 1, - 1$ . Write its two columns in the fixed basis $(e_{+}, e_{-})$ as the unitary matrix $W$ and set $Δ = \det W$ . Expanding the two elementary tensors gives $κ (u_{+} \otimes u_{-}) - κ (u_{-} \otimes u_{+}) = Δ [κ (e_{+} \otimes e_{-}) - κ (e_{-} \otimes e_{+})]$ . Orthonormality gives $∣ Δ ∣^{2} = ∥ u_{+} ∥^{2} ∥ u_{-} ∥^{2} - ∣ ⟨ u_{+}, u_{-} ⟩ ∣^{2} = 1$ , by direct two-coordinate expansion. Thus with $η = Δ^{- 1}$ , the fixed vector is $Ψ = η [κ (u_{+} \otimes u_{-}) - κ (u_{-} \otimes u_{+})] / \sqrt{2}$ , where $∣ η ∣ = 1$ . For unit $b$ use $P_{b, \pm} = (I \pm b \cdot σ) / 2$ from [F2]. This phase changes no projection norm or probability.

2.1givenF1F2F3F5F7step 1.2step 1.3algebra

[Joint outcome probabilities] By step 1.3, $(P_{a, +} \otimes P_{b, +}) Ψ = η κ (u_{+} \otimes P_{b, +} u_{-}) / \sqrt{2}$ . Since $∣ η ∣ = 1$ , [F5] and [F2] give $p (+, +) = ∥ P_{b, +} u_{-} ∥^{2} / 2 = [1 + ⟨ (b \cdot σ) u_{-}, u_{-} ⟩] / 4$ . The Pauli identities [F1] give $(a \cdot σ) (b \cdot σ) + (b \cdot σ) (a \cdot σ) = 2 (a \cdot b) I$ . Pairing this identity against $u_{-}$ and using self-adjointness and $(a \cdot σ) u_{-} = - u_{-}$ gives $⟨ (b \cdot σ) u_{-}, u_{-} ⟩ = - a \cdot b$ ; the same calculation for $u_{+}$ gives $+ a \cdot b$ . Applying the other three projection pairs therefore gives $p (+, +) = p (-, -) = (1 - a \cdot b) / 4 = \sin^{2} (θ / 2) / 2$ and $p (+, -) = p (-, +) = (1 + a \cdot b) / 4 = \cos^{2} (θ / 2) / 2$ , where $\cos θ = a \cdot b$ . These nonnegative numbers sum to one, proving (b).

3.1givenF2F6step 2.1algebra

[Correlation and marginals] Summing with the product signs, $E (a, b) = p (+, +) + p (-, -) - p (+, -) - p (-, +) = \frac{1}{2} \sin^{2} \frac{θ}{2} + \frac{1}{2} \sin^{2} \frac{θ}{2} - \frac{1}{2} \cos^{2} \frac{θ}{2} - \frac{1}{2} \cos^{2} \frac{θ}{2} = \sin^{2} \frac{θ}{2} - \cos^{2} \frac{θ}{2} = - \cos θ$ , which is (c) since $θ$ is the angle between $a$ and $b$ ; the marginals are $p_{A} (+) = p (+, +) + p (+, -) = \frac{1}{2}$ and $p_{A} (-) = \frac{1}{2}$ , and likewise $p_{B} (\pm) = \frac{1}{2}$ , independent of the other direction. The preparations $ρ_{A} = ρ_{B} = \frac{1}{2} I$ are unchanged by a discarded local instrument on the other factor by [F6]; this is (d).

4.1step 1.1step 1.2step 2.1step 3.1∎

[Assembly and scope] Step 1.1 is (a), step 2.1 is (b), step 3.1 is (c), (d); only the declared internal-factor composite and the cited mathematics were used. No spatial, temporal, relativistic, concrete-apparatus or empirical input enters, which is (e).

Remarks

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 $a ⊥ b$ the correlation vanishes, $E = 0$ .
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.

The spin singlet of two declared spin-half factors: directional correlations, perfect anti-correlation and setting-independent marginals

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources