State and prove quantum Liouville equation.
The quantum Liouville equation describes the time evolution of the density operator (\hat{\rho}) in quantum mechanics. It is the quantum analog of the classical Liouville equation and governs the dynamics of statistical ensembles of quantum systems. The density operator (\hat{\rho}) is defined for a system in a mixed or pure ___ ______ _____ ____ ________ ___.
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