  The concept of entropy is fundamental to thermalization, yet appears at odds
with basic principles in quantum mechanics. Statistical mechanics relies on the
maximization of entropy for a system at thermal equilibrium. However, an
isolated many-body system initialized in a pure state will remain pure during
Schr\"{o}dinger evolution, and in this sense has static, zero entropy. The
underlying role of quantum mechanics in many-body physics is then seemingly
antithetical to the success of statistical mechanics in a large variety of
systems. Here we experimentally study the emergence of statistical mechanics in
a quantum state, and observe the fundamental role of quantum entanglement in
facilitating this emergence. We perform microscopy on an evolving quantum
system, and we see thermalization occur on a local scale, while we measure that
the full quantum state remains pure. We directly measure entanglement entropy
and observe how it assumes the role of the thermal entropy in thermalization.
Although the full state remains measurably pure, entanglement creates local
entropy that validates the use of statistical physics for local observables. In
combination with number-resolved, single-site imaging, we demonstrate how our
measurements of a pure quantum state agree with the Eigenstate Thermalization
Hypothesis and thermal ensembles in the presence of a near-volume law in the
entanglement entropy.
