Manuscript En06-B@1 · En06-B.tex · proposition · ee886ae8

Proposition: Bell law and CHSH value

A nontrivial finite quantum instrument › Bell correlations and unconditioned no-signalling · lines 1414–1455 · proof included or linked

En06-B@1#prop:Belllaw, En06-B@1#eq:Belllaw, En06-B@1#eq:CHSH

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For source input $∣ 00 ⟩_{Q_{L} Q_{R}}$ , any fixed memory state, and fixed local setting inputs $s, t$ ,

p (a, b ∣ s, t) = \frac{1}{4} (1 + ab cos (θ_{s} - φ_{t})) .

Both one-wing outcome marginals are uniform. The correlators

E_{s t} = a, b \sum ab p (a, b ∣ s, t)

satisfy

E_{00} + E_{01} + E_{10} - E_{11} = 22 .

Proof. The state $Φ^{+}$ has

⟨ Z \otimes Z ⟩ = 1, ⟨ X \otimes X ⟩ = 1, ⟨ Z \otimes X ⟩ = ⟨ X \otimes Z ⟩ = 0,

and vanishing one-qubit Pauli expectations. Expanding the two projectors in (eq:projectors) gives (eq:Belllaw). Summing over either outcome gives $1/2$ .

The setting table gives

E_{00} = E_{01} = E_{10} = \frac{1}{2}, E_{11} = - \frac{1}{2},

proving (eq:CHSH). The CHSH comparison is the standard one introduced in CHSH; the value here has been computed directly.

Displayed formulas in this passage

En06-B@1#eq:Belllaw · line 1418
$p (a, b ∣ s, t) = \frac{1}{4} (1 + ab cos (θ_{s} - φ_{t})) .$
unlabeled · line 1424
$E_{s t} = a, b \sum ab p (a, b ∣ s, t)$
En06-B@1#eq:CHSH · line 1428
$E_{00} + E_{01} + E_{10} - E_{11} = 22 .$
unlabeled · line 1436
$⟨ Z \otimes Z ⟩ = 1, ⟨ X \otimes X ⟩ = 1, ⟨ Z \otimes X ⟩ = ⟨ X \otimes Z ⟩ = 0,$
unlabeled · line 1447
$E_{00} = E_{01} = E_{10} = \frac{1}{2}, E_{11} = - \frac{1}{2},$

Proposition: Bell law and CHSH value

Displayed formulas in this passage

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