  Entanglement is a distinguishing feature of quantum many-body systems, and
uncovering the entanglement structure for large particle numbers in quantum
simulation experiments is a fundamental challenge in quantum information
science. Here we perform experimental investigations of entanglement based on
the entanglement Hamiltonian, as an effective description of the reduced
density operator for large subsystems. We prepare ground and excited states of
a 1D XXZ Heisenberg chain on a 51-ion programmable quantum simulator and
perform sample-efficient `learning' of the entanglement Hamiltonian for
subsystems of up to 20 lattice sites. Our experiments provide compelling
evidence for a local structure of the entanglement Hamiltonian. This
observation marks the first instance of confirming the fundamental predictions
of quantum field theory by Bisognano and Wichmann, adapted to lattice models
that represent correlated quantum matter. The reduced state takes the form of a
Gibbs ensemble, with a spatially-varying temperature profile as a signature of
entanglement. Our results also show the transition from area to volume-law
scaling of Von Neumann entanglement entropies from ground to excited states. As
we venture towards achieving quantum advantage, we anticipate that our findings
and methods have wide-ranging applicability to revealing and understanding
entanglement in many-body problems with local interactions including higher
spatial dimensions.
