The "Sum of singular values" subsection of section 6.2.4 "Singular value optimization" of the Mosek Modeling Cookbook shows two Linear SDP formulations (which are duals of each other) of this problem. These formulations allow the problem to be solved numerically with an SDP solver.
Alternatively, CVX can be used, which has a norm_nuc function which does the SDP formulation under the hood, and calls an SDP solver, such as Mosek, SeDuMi, or SDPT3, to solve it.
cvx_begin
variable z complex
minimize(norm_nuc(psi - z*eye(n))
cvx_end