  The area law-like scaling of local quantum entropies is the central
characteristic of the entanglement inherent in quantum fields, many-body
systems, and spacetime. Whilst the area law is primarily associated with the
entanglement structure of the underlying quantum state, we here show that it
equally manifests in classical entropies over measurement distributions when
vacuum contributions dictated by the uncertainty principle are subtracted.
Using the examples of the Gaussian ground and thermal states, but also the
non-Gaussian particle state of a relativistic scalar field, we present
analytical and numerical area laws for the entropies of various distributions
and unveil how quantities of widespread interest such as the central charge and
the (local) temperature are encoded in classical observables. With our
approach, quantum entropies are no longer necessary to probe quantum phenomena,
thereby rendering area laws and other quantum features directly accessible to
theoretical models of high complexity as well as state-of-the-art experiments.
