**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][1] 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


  [1]: https://docs.mosek.com/modeling-cookbook/sdo.html#semidefinite-modeling