  A prerequisite for the comprehensive understanding of many-body quantum
systems is a characterization in terms of their entanglement structure. The
experimental detection of entanglement in spatially extended many-body systems
describable by quantum fields still presents a major challenge. We develop a
general scheme for certifying entanglement and demonstrate it by revealing
entanglement between distinct subsystems of a spinor Bose-Einstein condensate.
Our scheme builds on the spatially resolved simultaneous detection of the
quantum field in two conjugate observables which allows the experimental
confirmation of quantum correlations between local as well as non-local
partitions of the system. The detection of squeezing in Bogoliubov modes in a
multi-mode setting illustrates its potential to boost the capabilities of
quantum simulations to study entanglement in spatially extended many-body
systems.
