  We study anomalous couplings among neutral gauge bosons in $ZZ$ production at
the LHC for $\sqrt{s}=13$ TeV in $4$-lepton final state. We use the cross
section and polarization asymmetries of the $Z$ boson to estimate simultaneous
limits on anomalous coupling using Markov-Chain--Monte-Carlo (MCMC) method for
luminosities $35.9$ fb$^{-1}$, $150$ fb$^{-1}$, $300$ fb$^{-1}$ and $1000$
fb$^{-1}$. The $CP$-even polarization asymmetry $A_{x^2-y^2}$ is sensitive
mainly to the $CP$-odd couplings $f_4^{Z/\gamma}$ (quadratically) providing a
probe to identify $CP$-odd nature of interaction at the LHC. We find that the
polarization asymmetries significantly improve the estimation of anomalous
couplings should a deviation from the Standard Model (SM) be observed.
