High-dimensional entanglement certification

We show how to certify that a bipartite system is entangled through a simple procedure that works for arbitrary system dimension $d$ and employs only two local measurements of complementary properties. The procedure can also certify whether the system is maximally entangled. We experimentally demonstrate it for families of bipartite states of qudits with dimension up to $d=32$ composed of entangled pairs of polarised photons. We also confirm a surprising result that states with quantum discord but with no entanglement have less correlation in the measurement outcomes than classically correlated states.