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Abstract

Theoretical understanding of the scaling of entropies and the mutual
information in quantum many-body systems has led to significant advances in the
research of correlated states of matter, quantum field theory, and gravity.
Although there have been several experimental measurements of R\'{e}nyi and von
Neumann entropies in systems with discrete degrees of freedom, the measurement
of entropies between extensive intervals of continuous systems has remained
elusive. In this work, we measure the von Neumann entropy of spatially extended
subsystems in an ultra-cold atom simulator of one-dimensional quantum field
theories. We experimentally verify one of the fundamental properties of
equilibrium states of gapped quantum many-body systems, the area law of quantum
mutual information. We also study the dependence of mutual information on
temperature and the separation between the subsystems. Our work demonstrates
the capability of ultra-cold atom simulators to measure entanglement in quantum
field theories.