  Entropic uncertainty relation (EUR) formulates the restriction of the
inherent uncertainty of quantum mechanics from the information-theoretic
perspective. A tighter lower bound for uncertainty relations can provide
information-theoretic security to quantum communication protocols. Recently, a
generalized EUR (GEUR) for the measurement of multiple observables in arbitrary
many-body systems has been formulated. Here, we experimentally test this GEUR
using a four-photon entangled state with a controllable decoherence channel and
show that for the tripartite scenario, the GEUR improves the entropic bound
from Renes--Boileau's famous results. As an application, we further demonstrate
an improvement of the secure key rate in quantum key distribution from the
GEUR. Our results extend the test of EURs into multipartite regimes and may
find applications in practical quantum cryptography tasks.
