  Starting from a four-partite photonic hyper-entangled Dicke resource, we
report the full tomographic characterization of three-, two-, and one-qubit
states obtained by projecting out part of the computational register. The
reduced states thus obtained correspond to fidelities with the expected states
larger than 87%, therefore certifying the faithfulness of the
entanglement-sharing structure within the original four-qubit resource. The
high quality of the reduced three-qubit state allows for the experimental
verification of the Koashi-Winter relation for the monogamy of correlations
within a tripartite state. We show that, by exploiting the symmetries of the
three-qubit state obtained upon projection over the four-qubit Dicke resource,
such relation can be experimentally fully characterized using only 5
measurement settings. We highlight the limitations of such approach and sketch
an experimentally-oriented way to overcome them.

Quantum Physics