  Motivated by the very recent observation of the $B^+_c\to J/\psi+\pi^+
+\pi^0$ decay using proton-proton collision data by the LHCb collaboration, we
study the four-body angular distributions and the quantum entanglement effects
in the $B^+_c\to J/\psi+\pi^+ +\pi^0$ associated with $J/\psi\to \mu^++\mu^-$.
The helicity angular distributions are given in the QCD effective theory and
the von Neumann entropy is obtained in $B^\pm_c\to J/\psi(\to
\mu^+\mu^-)+\rho^\pm(\to \pi^\pm \pi^0)$ decay process.
