Given $n$ hermitian (positive-semidefinite) operators $Q_1,\cdots,Q_n$ in finite dimensional Hilbert space (the dimension can be very large), is there a mapping $\phi$ maps $Q_i$ to $P_i$, which preserves inner product, i.e. $\langle P_i, P_j\rangle =\langle Q_i,Q_j\rangle$, and all $P_i$'s are hermitian (positive-semidefinite) operators staying in a smaller dimensional space, say $poly(n)$ ?

Further question, given $n^2$ real numbers $c_{ij}$ $1\leq i,j\leq n$, how to decide if there exist $n$ hermitian (positive-semidefinite) operators $P_1,\cdots,P_n$ satisfying $\langle P_i,P_j\rangle=c_{ij}$? If exists, what is the minimum dimension of operators?

For vectors, the questions above are trivial. I wonder if there are any known results for operators? Thanks.