  Topological phases are characterized by their entanglement properties, which
is manifest in a direct relation between entanglement spectra and edge states
discovered by Li and Haldane. We propose to leverage the power of synthetic
quantum systems for measuring entanglement via the Entanglement Hamiltonian to
probe this relationship experimentally. This is made possible by exploiting the
quasi-local structure of Entanglement Hamiltonians. The feasibility of this
proposal is illustrated for two paradigmatic examples realizable with current
technology, an integer quantum Hall state of non-interacting fermions on a 2D
lattice and a symmetry protected topological state of interacting fermions on a
1D chain. Our results pave the road towards an experimental identification of
topological order in strongly correlated quantum many-body systems.
