Suppose $m,n$ are positive integers.

$D$ denote the set of $n\times n$ complex semideinite positive matrices with unit trace.

$A_1,\cdots,A_m$ are all $n\times n$ Hermitians.

We are interested in the projection of $D$ onto $A_1,\cdots,A_m$ in the following way,

$$ S=\{(tr(A_1N),\cdots,tr(A_m N)):N\in D\} $$

It is clear $S$ is convex compact set. How to charactrize its boundary. Can it be characterized by polynomial inequalities of $\mathbb{R}^m$. What is the minimal degree of those polynomials? Is $m$ enough?