  The geometry of Hamiltonian's eigenstates is encoded in the quantum geometric
tensor (QGT). It contains both the Berry curvature, central to the description
of topological matter and the quantum metric. So far the full QGT has been
measured only in Hermitian systems, where the role of the quantum metric is
mostly shown to determine corrections to physical effects. On the contrary, in
non-Hermitian systems, and in particular near exceptional points, the quantum
metric is expected to diverge and to often play a dominant role, for example on
the enhanced sensing and on wave packet dynamics. In this work, we report the
first experimental measurement of the quantum metric in a non-Hermitian system.
The specific platform under study is an organic microcavity with
exciton-polariton eigenstates, which demonstrate exceptional points. We measure
the quantum metric's divergence and we determine the scaling exponent
$n=-1.01\pm0.08$, which is in agreement with theoretical predictions for the
second-order exceptional points.
