  Quantum fluctuations of local quantities can be a direct signature of
entanglement in an extended quantum many-body system. Hence they may serve as a
theoretical (as well as an experimental) tool to detect the spatial properties
of the entanglement entropy of a subsystem. In the ground state of quantum
many-body systems, its scaling is typically linear in the boundary of the
subsystem (area law). Here we propose a microscopic insight into the spatial
structure of entanglement and fluctuations using the concept of \emph{contour},
recently introduced to decompose the bipartite entanglement entropy of lattice
free fermions between two extended regions $A$ and $B$ into contributions from
single sites in $A$. We generalize the notion of contour to the entanglement of
any quadratic (bosonic or fermionic) lattice Hamiltonian, as well as to
particle-number fluctuations. The entanglement and fluctuations contours are
found to generally decay when moving away from the boundary between $A$ and
$B$. We show that in the case of free fermions the decay of the entanglement
contour follows closely that of the fluctuation contour. In the case of
Bose-condensed interacting bosons, treated via the Bogoliubov and spin-wave
approximations, such a link cannot be established -- fluctuation and
entanglement contours are found to be radically different, as they lead to a
logarithmically violated area law for fluctuations, and to a strict area law of
entanglement. Analyzing in depth the role of the zero-energy Goldstone mode of
spin-wave theory, and of the corresponding lowest-energy mode in the
entanglement spectrum, we unveil a subtle interplay between the special contour
and energy scaling of the latter, and universal additive logarithmic
corrections to entanglement area law discussed extensively in the recent
literature.
