  It is a hard and important problem to find the criterion of the set of
positive-definite matrixes which can be written as reduced density operators of
a multi-partite quantum state. This problem is closely related to the study of
many-body quantum entanglement which is one of the focuses of current quantum
information theory. We give several results on the necessary compatibility
relations between a set of reduced density matrixes, including: (i)
compatibility conditions for the one-party reduced density matrixes of any
$N_A\times N_B$ dimensional bi-partite mixed quantum state, (ii) compatibility
conditions for the one-party and two-party reduced density matrixes of any
$N_A\times N_B\times N_C$ dimensional tri-partite mixed quantum state, and
(iii) compatibility conditions for the one-party reduced matrixes of any
$M$-partite pure quantum state with the dimension $N^{\otimes M}$.
