  We introduce a version of the chained Bell inequality for an arbitrary number
of measurement outcomes, and use it to give a simple proof that the maximally
entangled state of two d dimensional quantum systems has no local component.
That is, if we write its quantum correlations as a mixture of local
correlations and general (not necessarily quantum) correlations, the
coefficient of the local correlations must be zero. This suggests an
experimental programme to obtain as good an upper bound as possible on the
fraction of local states, and provides a lower bound on the amount of classical
communication needed to simulate a maximally entangled state in dxd dimensions.
We also prove that the quantum correlations violating the inequality are
monogamous among non-signalling correlations, and hence can be used for quantum
key distribution secure against post-quantum (but non-signalling)
eavesdroppers.
