  We discuss the possibility of estimating experimentally the von Neumann
entanglement entropy $S_{A}^{vN}$ of a symmetric bi-partite quantum system $AB$
by using the basic measurement counts for a it $single$ copy of a prepared
state. Using exact diagonalization and analog simulations performed with the
publicly available QuEra facilities for chains and ladders of Rydberg atoms, we
calculate the Shannon entropy $S_{AB}^X$ associated with the experimental
measurements of adiabatically prepared ground states and the reduced entropy
$S_A^X$ obtained by tracing the experimental probabilities over the $B$ half of
the system. We show several examples for which, in good approximation,
$S_{A}^{vN}\propto (2S_A^X-S_{AB}^X)$ with a constant of proportionality
slightly larger than one. We argue that one should have the inequality
$S_{A}^{vN}\geq(2S_A^X-S_{AB}^X)$ holding in more general circumstances.
$2S_A^X-S_{AB}^X$ can be calculated easily for many qubit platforms and appears
to be generically robust under measurement errors. Similar results are found
for the second order R\'enyi entanglement entropy.
