Macroscopically nonlocal quantum correlations

It is usually believed that coarse-graining of quantum correlations leads to classical correlations in the macroscopic limit. Such a principle is known as $macroscopic$ $locality$. In this letter we re-examine this principle and consider a macroscopic Bell experiment in which parties measure collective observables with a resolution of the order of $N^\alpha$, where $N$ is the number of particles and $\alpha$ defines the level of coarse-graining. One expects nonlocal correlations to fade away as $\alpha$ increases, so the natural question to ask is: what is the value of $\alpha$ required to restore locality, marking the quantum-to-classical transition? In a previous work [M. Navascues and H. Wunderlich, Proc. R. Soc. A. $\textbf{466}$: 881 (2010)] it was demonstrated that locality is restored at $\alpha=1/2$ in the case of $independent$ $and$ $identically$ $distributed$ (IID) entangled pairs. Here we show that this is no longer the case in the generic (non-IID) scenario, where the Hilbert space structure of quantum theory, along with the superposition principle and the Born rule, can survive in the macroscopic limit. This leads straightforwardly to a Bell violation for macroscopic observables, and thus the question of the quantum-to-classical transition remains open.