We investigate the quantum entanglement entropy for the four-dimensional Euclidean SU(3) gauge theory. We present the first non-perturbative calculation of the entropic $c$-function ($C(l)$) of SU(3) gauge theory in lattice Monte Carlo simulation using the replica method. For $0 leqslant l leqslant 0.7$~fm, where $l$ is the length of the subspace, the entropic $c$-function is almost constant, indicating conformally invariant dynamics. The value of the constant agrees with that perturbatively obtained from free gluons, with 20 % discrepancy. When $l$ is close to the Hadronic scale, the entropic $c$-function decreases smoothly, and it is consistent with zero within error bars at $l gtrsim 0.9$ fm.
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