Edit: As discussed in comments by @Yemon Choi and @Nathaniel Johnston , there is a simpler solution if $\psi$ is Hermitian. So the remainder of my post addresses the non-Hermitian case, although it is valid as well in the Hermitian case.
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 as a convex optimization problem 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