Empirical two-point correlation functions

Abstract

Let A ₁ , A ₂ , A ₃ A ₄ be four observables, the compatible observables among them being (A ₁ , A ₃ ), (A ₁ , A ₄ ), (A ₂ , A ₃ ), (A ₂ , A ₄ ). In order that the empirical data be reproducible by a quantum or a classical theory, the two-point correlation functions

$$\{ C_{ij} = \left\langle {A_i A_j } \right\rangle :i,j a compatible pair\} $$

must necessarily satisfy

$$|X_{13} X_{14} - X_{23} X_{24} | \leqslant \left( {1 - X_{13} ^2 } \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \left( {1 - X_{14} ^2 } \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} + \left( {1 - X_{23} ^2 } \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \left( {1 - X_{24} ^2 } \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} (*)$$

where X_ij=C_ijC ^−1/2_ii C ^−1/2_jj . In the case ofGaussian data, this inequality is alsosufficient; If (*) holds, there is a Gaussian joint distribution for A ₁ , A ₂ , A ₃ , A ₄ which reproduces the Gaussian data for compatible pairs. It follows that Bell's inequality is satisfied by all true-false propositions about the Gaussian data. A further consequence of the analysis is thatquantum Gaussian fields satisfy Bell's inequality for all true-false propositions aboutfield measurements.

The maximum violation of (*) corresponds to Rastall's example in the case of two-valued observables.