  Non-linear properties of quantum states, such as entropy or entanglement,
quantify important physical resources and are frequently used in quantum
information science. They are usually calculated from a full description of a
quantum state, even though they depend only on a small number parameters that
specify the state. Here we extract a non-local and a non-linear quantity,
namely the Renyi entropy, from local measurements on two pairs of polarization
entangled photons. We also introduce a "phase marking" technique which allows
to select uncorrupted outcomes even with non-deterministic sources of entangled
photons. We use our experimental data to demonstrate the violation of entropic
inequalities. They are examples of a non-linear entanglement witnesses and
their power exceeds all linear tests for quantum entanglement based on all
possible Bell-CHSH inequalities.
