Strict Hierarchy of Bell Theorems

As proved by John Bell, quantum mechanics exhibits correlations in spacelike separated bipartite systems that are impossible to reproduce by classical means. There are three levels of "Bell theorems", depending on which aspects of the quantum correlations can or cannot be reproduced classically. The original "Bell inequalities" (Bl) require a perfect classical simulation of all quantum probabilities. With "Bell theorems without inequalities" (BTWI), we ask the classical simulation to be able to produce precisely the outputs that could occur according to quantum mechanics, but we do not worry about their exact probabilities. With "pseudo-telepathy" (PT), we are satisfied if the classical simulation produces only outputs allowed by quantum mechanics, but not necessarily all of them. Bell's original proof of Bl involved a maximally entangled 2times2 bipartite state such as the singlet state. Hardy proved that BTWI are possible in dimension 2times2, but his construction used a non-maximally entangled state. Here, we prove that no 2times2 maximally entangled state can serve to produce BTWI. Combining this with the fact that 2times2 entangled states cannot be used at all for the purpose of PT, it follows a strict hierarchy on the quantum resources that are required to exhibit the various levels of Bell theorems.