2011

Bell inequalities for the simplest exclusivity graph

Sadiq, Muhamad, Badziag, Piotr, Bourennane, Mohamed et al.

Understand

Which is the simplest logical structure for which there is quantum nonlocality? We show that there are only three bipartite Bell inequalities with quantum violation associated with the simplest graph of relationships of exclusivity with a quantum-classical gap.

These are the most elementary logical Bell inequalities.
We show that the quantum violation of some well-known Bell inequalities is related to them.
We test the three Bell inequalities with pairs of polarization-entangled photons and report violations in good agreement with the quantum predictions.

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The maximum quantum value can be obtained preparing the state ⟨ ψ | = ( 0 , 0 , 1 ) \langle\psi|=(0,0,1) and with the tests i = 2 | v i ⟩ ⟨ v i | − 𝟙 i=2|v_{i}\rangle\langle v_{i}|-\openone with ⟨ v 0 | = N 0 ( 1 , 0 , r ) \langle v_{0}|=N_{0}\left(1,0,r\right) , ⟨ v 1 , 4 | = N 1 ( c , ± s , r ) \langle v_{1,4}|=N_{1}\left(c,\pm s,r\right) , ⟨ v 2 , 3 | = N 2 ( C , ∓ S , r ) \langle v_{2,3}|=N_{2}\left(C,\mp S,r\right) , where r = cos ⁡ ( π 5 ) r=\sqrt{\cos{\left(\frac{\pi}{5}\right)}} , c = cos ⁡ ( 4 π 5 ) c=\cos{\left(\frac{4\pi}{5}\right)} , s = sin ⁡ ( 4 π 5 ) s=\sin{\left(\frac{4\pi}{5}\right)} , C = cos ⁡ ( 2 π 5 ) C=\cos{\left(\frac{2\pi}{5}\right)} , S = sin ⁡ ( 2 π 5 ) S=\sin{\left(\frac{2\pi}{5}\right)} , and N i N_{i} are suitable normalization factors
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