I believe you can enter the problem in the image almost “as is”, using CVX’s rel_entr function.
As for solving it, use Mosek if available to you, otherwise follow the instructions at 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