Semidefinite programming with rank 1 Hermitian semidefinite matric variable

That’is why I wrote
The scaling of your problem may present difficulties. I’ll let you look into that.

You need to improve the scaling of your problem. The solver apparently is not able to deal well with such small values of b.

Also, I rrecommend you install CVXQUAD per CVXQUAD: How to use CVXQUAD's Pade Approximant instead of CVX's unreliable Successive Approximation for GP mode, log, exp, entr, rel_entr, kl_div, log_det, det_rootn, exponential cone. CVXQUAD's Quantum (Matrix) Entropy & Matrix Log related functions . However, given that CVX’s Successive Approximation method appears to have succeeded, i doubt CVXQUAD will make your results come out better in this case, but I don’t know for sure.

Let’s see whether someone clever comes along who can come up with a nice change of variables or something to improve the formulation of the that the solver is presented a numeriically nicer problem.