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Worked Bell angles: the singlet against the EPR product mixture at the same directions

Physical scope

A finite conditional computation in the declared spin correlation arrangement of the framework at four specified directions: the singlet's four sign tables and CHSH value, the tables and CHSH value of the EPR-type product mixture of definite opposite spins, an explicit local-response representation of the mixture's tables, and the comparison of the two arrangements. The example asserts no measured correlation, no apparatus and no empirical fact; it makes no claim that any laboratory preparation realizes either arrangement.

Example

Adopt the declared spin correlation arrangement of Declared spin correlation arrangement of the framework: two spin-half factors, four directions, joint sign tables and local-response models over the two declared spin-half factors $V, V$ with basis $(e_{+}, e_{-})$ . Assume the Axiom of Countable Choice and the Axiom of Choice, as required by the cited singlet, spin-spectrum and arrangement suppliers (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice). Fix the orthonormal vectors $e_{1} : = (1, 0, 0)$ , $e_{2} : = (0, 1, 0)$ , $e_{3} : = (0, 0, 1)$ and the four unit directions $a_{0} : = e_{3}, a_{1} : = e_{1}, b_{0} : = \frac{e_{3} + e_{1}}{\sqrt{2}}, b_{1} : = \frac{e_{3} - e_{1}}{\sqrt{2}} .$ Consider two arrangements over these same directions:

the singlet arrangement, with the singlet preparation $ρ_{Ψ} = T_{Ψ, Ψ}$ , $Ψ = \frac{1}{\sqrt{2}} (κ (e_{+} \otimes e_{-}) - κ (e_{-} \otimes e_{+}))$ ;
the product-mixture arrangement, with the EPR-type mixture $M : = \frac{1}{2} T_{e_{+} \otimes e_{-}, e_{+} \otimes e_{-}} + \frac{1}{2} T_{e_{-} \otimes e_{+}, e_{-} \otimes e_{+}},$ a convex combination of the two product preparations of definite opposite spins along $e_{3}$ .

Then:

(a) for the singlet arrangement the four correlations are $E_{00} = E_{01} = E_{10} = - \frac{1}{\sqrt{2}}, E_{11} = + \frac{1}{\sqrt{2}}, S (ν^{ρ_{Ψ}}) = - 2 \sqrt{2};$

(b) for the product-mixture arrangement the four sign tables are $ν_{i j}^{M} (ε, δ) = \frac{1}{4} (1 - ε δ (a_{i})_{3} (b_{j})_{3}),$ so its correlations are $E_{i j} = - (a_{i})_{3} (b_{j})_{3}$ ; at the displayed directions $(a_{0})_{3} = 1$ , $(a_{1})_{3} = 0$ and $(b_{0})_{3} = (b_{1})_{3} = 1 / \sqrt{2}$ , hence $E_{00} = E_{01} = - \frac{1}{\sqrt{2}}, E_{10} = E_{11} = 0, S (ν^{M}) = - \sqrt{2};$

(c) the mixture's tables admit a local-response model (Local-response representations of a binary correlation model: one setting-independent specification space and per-station response laws): on the two-point specification space, the response laws $α_{i} (1, ε) = \frac{1}{2} (1 + ε (a_{i})_{3}), α_{i} (2, ε) = \frac{1}{2} (1 - ε (a_{i})_{3}), β_{j} (1, δ) = \frac{1}{2} (1 - δ (b_{j})_{3}), β_{j} (2, δ) = \frac{1}{2} (1 + δ (b_{j})_{3})$ with specification weights $\frac{1}{2}, \frac{1}{2}$ reproduce $ν^{M}$ ; consequently $∣ S (ν^{M}) ∣ = \sqrt{2} \leq 2$ , as the bound for representable models requires (The CHSH bound for every binary correlation model admitting a local-response representation);

(d) both arrangements have the same marginal sign laws $ν^{A} (\pm) = ν^{B} (\pm) = \frac{1}{2}$ for every direction, and the same reduced preparation ${Tr}_{B} M = \frac{1}{2} I = {Tr}_{B} ρ_{Ψ}$ on each factor, although their joint tables and CHSH functionals differ: $∣ S (ν^{ρ_{Ψ}}) ∣ = 2 \sqrt{2} > 2 \geq \sqrt{2} = ∣ S (ν^{M}) ∣$ .

All clauses are conditional computations in the declared framework; no measured correlation is asserted.

Verification

Given: The declared arrangements with the singlet preparation $ρ_{Ψ}$ and the mixture $M$ ; the four directions $a_{0} = e_{3}$ , $a_{1} = e_{1}$ , $b_{0} = (e_{3} + e_{1}) / \sqrt{2}$ , $b_{1} = (e_{3} - e_{1}) / \sqrt{2}$ ; the spin-half projections $P_{a, ε} = \frac{1}{2} (I + ε a \cdot σ)$ ; the Axiom of Countable Choice and the Axiom of Choice.

[L1] The arrangement's outcome law is $ν_{i j}^{ρ} (ε, δ) = tr (ρ (P_{a_{i}, ε} \otimes P_{b_{j}, δ}))$ and its correlations and CHSH functional are read from Binary correlation models: setting-indexed tables of two-valued outcome laws, marginals and the CHSH functional clause (d) (Declared spin correlation arrangement of the framework: two spin-half factors, four directions, joint sign tables and local-response models clauses (a), (d)).

[L2] The singlet pairings and their marginals and correlation are $⟨ (P_{a, ε} \otimes P_{b, δ}) Ψ, Ψ ⟩ = \frac{1}{4} (1 - ε δ a \cdot b)$ with marginals $\frac{1}{2}$ and correlation $- a \cdot b$ (Singlet pairings: the joint laws of the two spin-half factors along two directions clauses (b), (c), (d)); at the displayed directions the singlet's CHSH value is $- 2 \sqrt{2}$ (The singlet arrangement: directional sign tables, CHSH value 2 sqrt 2 and the failure of local-response models at violating settings clause (b)).

[L3] For a product preparation $T_{u \otimes v, u \otimes v}$ the joint law of the local sign measurements factorizes: $tr (T_{u \otimes v, u \otimes v} (P_{a, ε} \otimes P_{b, δ})) = ∥ P_{a, ε} u ∥^{2} ∥ P_{b, δ} v ∥^{2}$ (Factorization of local statistics for product preparations, and its failure for non-product pure preparations clause (a), Rank-one operator calculus clauses (b), (d)); and for the spin-half basis states $⟨ (n \cdot σ) e_{+}, e_{+} ⟩ = n_{3}$ , $⟨ (n \cdot σ) e_{-}, e_{-} ⟩ = - n_{3}$ , so $∥ P_{n, ε} e_{+} ∥^{2} = \frac{1}{2} (1 + ε n_{3})$ and $∥ P_{n, ε} e_{-} ∥^{2} = \frac{1}{2} (1 - ε n_{3})$ by the Pauli matrices of Finite spin matrices clause (e): $P_{n, ε}^{*} = P_{n, ε} = P_{n, ε}^{2}$ , so $∥ P_{n, ε} e_{\pm} ∥^{2} = ⟨ P_{n, ε} e_{\pm}, e_{\pm} ⟩$ .

[L4] The partial trace of a rank-one product operator is ${Tr}_{B} T_{u \otimes v, u^{'} \otimes v^{'}} = ⟨ v, v^{'} ⟩ T_{u, u^{'}}$ , and $tr$ is linear on the trace class (Separable partial trace and its basic properties clause (c), Rank-one operator calculus clause (b)).

[L5] Every binary correlation model admitting a local-response representation satisfies $∣ S ∣ \leq 2$ (The CHSH bound for every binary correlation model admitting a local-response representation).

Proof technique: direct; evaluate the singlet numbers from the singlet pairings, compute the mixture's tables as the average of the two product tables, exhibit the two-specification representation and check its identity, and compare the marginals.

Given: The data of the statement and [L1]-[L5].

1.1givenL1L2algebra

[Directions and singlet values] The four directions are unit vectors with $a_{0} \cdot b_{0} = a_{0} \cdot b_{1} = a_{1} \cdot b_{0} = 1 / \sqrt{2}$ and $a_{1} \cdot b_{1} = - 1 / \sqrt{2}$ . By [L1] and [L2], $E_{i j} (ν^{ρ_{Ψ}}) = - a_{i} \cdot b_{j}$ , so $E_{00} = E_{01} = E_{10} = - 1 / \sqrt{2}$ , $E_{11} = + 1 / \sqrt{2}$ and $S (ν^{ρ_{Ψ}}) = - 2 \sqrt{2}$ ; this is (a).

1.2givenL3algebra

[Product-state tables] Write $u_{+} : = e_{+}$ , $u_{-} : = e_{-}$ . By [L3], for the product preparation $T_{u_{τ} \otimes u_{- τ}, u_{τ} \otimes u_{- τ}}$ (with $τ = +$ or $-$ ) the joint sign law at $(i, j)$ is $\frac{1}{2} (1 + ε τ (a_{i})_{3}) \cdot \frac{1}{2} (1 - δ τ (b_{j})_{3})$ .

2.1step 1.2L1algebra

[Mixture tables] By [L1] the outcome law of a convex mixture is the corresponding convex combination of the laws, so the two product tables of step 1.2 average to $ν_{i j}^{M} (ε, δ) = \frac{1}{4} [\frac{1}{2} (1 + ε (a_{i})_{3}) (1 - δ (b_{j})_{3}) + \frac{1}{2} (1 - ε (a_{i})_{3}) (1 + δ (b_{j})_{3})] = \frac{1}{4} (1 - ε δ (a_{i})_{3} (b_{j})_{3})$ ; this is the table formula of (b).

3.1step 2.1algebra

[Mixture correlations and CHSH value] Summing the table of step 2.1 with the product signs gives $E_{i j} (ν^{M}) = - (a_{i})_{3} (b_{j})_{3}$ ; substituting $(a_{0})_{3} = 1$ , $(a_{1})_{3} = 0$ , $(b_{0})_{3} = (b_{1})_{3} = 1 / \sqrt{2}$ gives $E_{00} = E_{01} = - 1 / \sqrt{2}$ and $E_{10} = E_{11} = 0$ , hence $S (ν^{M}) = - 1 / \sqrt{2} - 1 / \sqrt{2} - 0 + 0 = - \sqrt{2}$ ; this completes (b).

3.2step 2.1L5algebra

[Explicit representation] Choose a two-point specification space $Λ = {1, 2}$ with weights $P ({1}) = P ({2}) = \frac{1}{2}$ and define the response laws by the display of (c). Then for each $(i, j, ε, δ)$ one has $\int_{Λ} α_{i} (λ, ε) β_{j} (λ, δ) d P (λ) = \frac{1}{2} \cdot \frac{1}{2} (1 + ε (a_{i})_{3}) \cdot \frac{1}{2} (1 - δ (b_{j})_{3}) + \frac{1}{2} \cdot \frac{1}{2} (1 - ε (a_{i})_{3}) \cdot \frac{1}{2} (1 + δ (b_{j})_{3}) = \frac{1}{4} (1 - ε δ (a_{i})_{3} (b_{j})_{3}) = ν_{i j}^{M} (ε, δ)$ by step 2.1, so $(α, β, P)$ is a local-response representation of $ν^{M}$ ; the bound $∣ S (ν^{M}) ∣ \leq 2$ then follows from [L5] and is consistent with the value $- \sqrt{2}$ of (b), since $\sqrt{2} \leq 2$ . This is (c).

4.1step 1.1step 2.1step 3.1L2L4algebra

[Common marginals and comparison] Summing the mixture table of step 2.1 over the distant label gives $\frac{1}{2}$ for each local label, and the singlet's marginals are $\frac{1}{2}$ by [L2]; the reduced preparations are ${Tr}_{B} M = \frac{1}{2} T_{e_{+}, e_{+}} + \frac{1}{2} T_{e_{-}, e_{-}} = \frac{1}{2} I$ and, expanding $T_{Ψ, Ψ}$ in the product basis and applying [L4] linearly, ${Tr}_{B} ρ_{Ψ} = \frac{1}{2} (T_{e_{+}, e_{+}} + T_{e_{-}, e_{-}}) = \frac{1}{2} I$ , so both arrangements share every single-station statistic while their CHSH functionals are $2 \sqrt{2}$ and $\sqrt{2}$ in absolute value. This is (d).

5.1step 1.1step 2.1step 3.1step 3.2step 4.1∎

[Assembly and scope] Steps 1.1, 2.1, 3.1, 3.2 and 4.1 prove (a)-(d); only the declared arrangements, the singlet pairings, the product-factorization theorem, the spin-half outcome laws, the partial trace and the CHSH bound for representable models were used, and no measured correlation is asserted.

Remarks

Why the mixture does not distinguish the two routes. The mixture's tables are represented by a probabilistic local-response model with two specifications, not by a deterministic response function of a single specification; this is why clause (a) of Local-response representations of a binary correlation model: one setting-independent specification space and per-station response laws is stated for response laws rather than for response functions. The singlet's tables at the displayed directions admit no representation of either form.
The lesson of the comparison. Identical marginals and identical reduced preparations on both factors do not determine the joint tables: the contrast between $2 \sqrt{2}$ and $\sqrt{2}$ is invisible to every single-station statistic and is a property of the correlations alone.

Worked Bell angles: the singlet against the EPR product mixture at the same directions

Example

Verification

Remarks

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