  We analyze in this work the regularity properties of the density operator
solu- tion to the quantum Liouville equation. As was recently done for the
Strichartz inequalities, we extend to systems of orthonormal functions the
local smoothing estimates satisfied by the solutions to the Schr\"odinger
equation. We show in par- ticular that the local density associated to the
solution to the free, linear, quantum Liouville equation admits locally
fractional derivatives of given order provided the data belong to some Schatten
spaces.
