A particle-in-cell simulation is used to examine the conservative decay of 3D quasi-geostrophic turbulence. It is well-known that such decay involves an inverse energy cascade, in which small-scale potential vorticity (PV) anomalies coalesce into large-scale, approximately stationary vortices. Following Miller (PRL 65, 1990), and Robert and Sommeria (JFM 229, 1991), this study posits that the initial turbulence randomly redistributes PV, and that the relaxed flow maximizes an entropy functional. A maximum entropy state, that accounts for all pertinent quasi-geostrophic invariants, is calculated. It is shown that the 3D PV distribution of the maximum entropy state compares favorably to the final PV distribution of the simulation. However, the large-scale vortices that emerge in the simulation have higher than "expected" PV in their cores, suggesting that ergodicity is not fully realized.