  This paper addresses the challenge of preparing arbitrary mixed quantum
states, an area that has not been extensively studied compared to pure states.
Two circuit design methods are presented: one via a mixture of pure states and
the other via purification. A novel strategy utilizing the Cholesky
decomposition is proposed to improve both computational efficiency during
preprocessing and circuit efficiency in the resulting circuits, offering
significant advantages, especially when the targeted density matrix is
low-ranked or sparse. By leveraging the incomplete Cholesky decomposition with
threshold dropping, we also propose an appealing strategy for generating a
high-fidelity approximation of the targeted density matrix, enabling
substantial efficiency enhancement at the cost of mild fidelity loss.
Additionally, as a closely related issue, we prove the "no-superposing
theorem": given a certain number of arbitrary unknown pure states as input, it
is impossible to devise an operation that produces an output state as the
superposition of the input states with predefined coefficients unless all but
one of the coefficients vanish.
