  Quantum correlations between spatially separated parts of a $d$-dimensional
bipartite system ($d\geq 2$) have no classical analog. Such correlations, also
called entanglements, are not only conceptually important, but also have a
profound impact on information science. In theory the violation of Bell
inequalities based on local realistic theories for $d$-dimensional systems
provides evidence of quantum nonlocality. Experimental verification is required
to confirm whether a quantum system of extremely large dimension can possess
this feature, however it has never been performed for large dimension (e.g.,
$d\geq 1000$). Here, we report that Bell inequalities are experimentally
violated for bipartite quantum systems of extremely high dimensionality with
the usual ensembles of polarization-entangled photon pairs. Our entanglement
source violates Bell inequalities for extremely high dimensionality of
$d>4000$. The designed scenario offers a possible new method to investigate the
entanglement of multipartite systems of large dimensionality and their
application in quantum information processing.
