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Liouville theorem (LT) reveals robust incompressibility of distributionfunction in phase space, given arbitrary potentials. However, its quantumgeneralization, Wigner flow, is compressible, i.e., LT is only conditionallytrue (e.g., for perfect Harmonic potential). We develop quantum Liouvilletheorem (rigorous incompressibility) for arbitrary potentials (interacting ornot) in Hamiltonians. Haar measure, instead of symplectic measure dp^dq used inWigner's scheme, plays a central role. The argument is based on general measuretheory, independent of specific spaces or coordinates. Comparison of classicaland quantum is made: for instance, we address why Haar measure and metricpreservation do not work in the classical case. Applications of theorems instatistics, topological phase transition, ergodic theory, etc. are discussed.

Quantum Liouville's theorem based on Haar measure