Given an operator $\rho$ on a Hilbert space $H$, is there a notion of nearest self-adjoint (positive) approximation of $\rho$ for a suitable norm? More specifically, does there exist an algebraic formula for it?
For example, the assignment $\rho \mapsto |\rho|:=\sqrt{\rho^\dagger\rho}$ is idempotent and yields a self-adjoint positive operator. It can therefore be seen as giving a solution to this problem. However, it is not clear that it gives an optimal solution with respect to any norm (left to specify).
I am mostly interested in the finite dimensional case but not exclusively.