What does this equation mean?

Start with the settled part, because it is smaller than most accounts suggest and it is worth stating precisely. Quantum mechanics gives the Born rule: the probability of a measurement outcome is the squared modulus of a probability amplitude, $P(a) = |\langle a \mid \psi \rangle|^2$. for a system in state |$\psi$$\rangle$ measured in a basis containing |a$\rangle$ . Every interpretation discussed here reproduces this rule for every experiment that has actually been performed. That includes Bell tests. John Bell showed in 1964 that any theory obeying two assumptions — that a measurement’s outcome depends only on a hidden variable carried by the particle and on the local setting of its own detector, not on the distant detector’s setting — predicts correlations between two separated measurements that are bounded above by a fixed value [ 1 ] . Quantum mechanics predicts correlations that exceed that bound for entangled pairs. This is a fact, not an interpretation: it was measured, most decisively in 2015 experiments that closed the two major escape routes at once — the “locality loophole,” by switching each detector’s setting after the entangled pair had already left the source, so no sub-light-speed signal could coordinate the two sides, and the “detection loophole,” by catching a high enough fraction of the pairs that a hidden mechanism selectively revealing only pairs that would violate the bound could be ruled out [ 2 ] . The 2022 Nobel Prize in Physics went to Alain Aspect, John Clauser, and Anton Zeilinger explicitly for this line of experiment, “for experiments with entangled photons, establishing the violation of Bell inequalities and pioneering quantum information science” [ 3 ] .