Greenberger-Horne-Zeilinger-like proof of Bell's theorem involving observers who do not share a reference frame

Vaidman described how a team of three players, each of them isolated in a remote booth, could use a three-qubit Greenberger-Horne-Zeilinger state to always win a game which would be impossible to always win without quantum resources. However, Vaidman's method requires all three players to share a common reference frame; it does not work if the adversary is allowed to disorientate one player. Here we show how to always win the game, even if the players do not share any reference frame. The introduced method uses a 12-qubit state which is invariant under any transformation R-a x R-b x R-c (where R-a = U-a x U-a x U-a x U-a, where U-j is a unitary operation on a single qubit) and requires only single-qubit measurements. A number of further applications of this 12-qubit state are described.