Quantum Measurement of Spins and Magnets, and the Classical Limit of PR-Boxes

Abstract

Using weak quantum measurements one can determine the direction in which a large ensemble of spins, as in a classical magnet, points. Assume Alice and Bob share a large ensemble of N pairs of spin-$\frac{1}{2}$. If Alice measures all her spins, all along the same direction, she prepares at a distance an ensemble of spins for Bob which, because of statistical fluctuations, have a magnetic moment of the order $\sqrt{N}$. By making N large enough, this magnetic moment can be made arbitrarily large, indicating an apparent possibility to signal. However, we show that an arbitrarily large magnetic moment is not necessarily classical in the sense that it might be fundamentally impossible to determine in which direction it points. We also consider stronger than quantum correlations and show that Tsirelson’s bound follows from the physical assumption that in the macroscopic limit all measurements are compatible and that this should not lead to signaling.