Experimental data from Cluster have shown that entropy density can be generated across Earth's bow shock. These new observations are a starting point for a more sophisticated analysis that includes computer modeling of a collisionless shock using observed shock parameters as input. In this Letter, we present the first comparison between observations and particle-in-cell simulations of such entropy generation across a collisionless shock. The ion heating at the shock is dominated by the phase mixing of reflected and directly transmitted ions, which are separated from the incident ions. The electron heating is a nearly thermal process due to the conservation of their angular momentum. For both species, we calculate the entropy density across the shock, and obtain good consistency between observations and simulations on entropy generation across the shock. We also find that the entropy generation rate is reduced as the shock Mach number decreases.