Physical Review Letters

Bell’s theorem [1] states that one cannot in general reproduce the results of quantum theory with a classical, deterministic local model. Hardy’s argument of “nonlocality without inequalities” [2] is considered “the best version of Bell’s theorem” [3]. Curiously enough, while in the original proof of Bell’s theorem using inequalities the maximum discrepancy with local models occurs for maximally entangled states [4] (also known as “Bell states” [5]), Hardy’s proof is not valid for maximally entangled states. It works only for entangled but nonmaximally entangled states (also known as “Hardy states” [6]). This curious feature has led to several attempts and suggestions to develop a Hardy-like argument for maximally entangled states of two qubits. However, so far none of these proposals has worked [7]. In addition, Hardy’s argument works only for 9% of the runs of a certain experiment. On the other hand, the proof of Bell’s theorem by Greenberger, Horne, and Zeilinger (GHZ) [8–11] works for 100% of the runs but requires three observers (instead of two, as in Hardy’s proof). In this paper, I introduce a Hardy-like proof which works for 100% of the runs of an experiment. It requires only two observers and is based on maximally entangled states of two qubits. This proof is more complicated than the original by Hardy, since it involves two copies of the maximally entangled state (instead of just one), but it preserves the logical structure of the original argument. The proof is inspired in Hardy’s argument [2] (see also [3,6,12,13]) and in a proof of the Kochen-Specker theorem [14] by Peres [15] and Mermin [16] (see also [17–20]).