I saw a reference book,there is a statement:the pure states of $M_n(\Bbb C)$ are the rank 1 projections of $M_n(\Bbb C)$.
By definition of states,they should be the positive linear functional of norm 1.So the pure states should be positive linear functional of $M_n(\Bbb C)$.How to interpret the statement.
It is easy to check that any linear functional on $M_n(\mathbb C)$ is of the form $X\longmapsto \operatorname{Tr}(AX)$ for some $A\in M_n(\mathbb C)$. Under this correspondence positive functional correspond with positive matrices, and so states correspond with positive matrices of trace 1. Among these, one can check that the pure states are precisely those given by the rank-one projections.