Strict hierarchy among Bell Theorems

As demonstrated 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” (BI) 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 outcomes that could occur according to quantum mechanics, but we do not worry about their exact probabilities. With “Pseudotelepathy” (PT), we are satisfied if the classical simulation produces only outcomes allowed by quantum mechanics, but not necessarily all of them.Bell’s original proof of BI involved a maximally entangled bipartite state such as the singlet state. Hardy proved that BTWI are possible in dimension , but his construction used a non-maximally entangled state. Here, we prove that no maximally entangled state can serve to produce BTWI. Combining this with our earlier result that 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.