Suppose $m,n$ are positive integers.

$D$ denotes the set of $n\times n$ complex semidefinite positive matrices with unit trace.

$A_1,\cdots,A_m$ are $n\times n$ 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 characterize its boundary.
Can it be characterized by polynomial inequalities of $\mathbb{R}^m$. What is the minimal degree of those polynomials? Is $m$ enough? How many polynomials are needed?

In particular, for the case $m=4$, what do we know? Can we compute the boundary efficiently?