Using MATLAB's CVX Package for Semidefinite Programming in Quantum Information
The new SDP that you obtain by replacing minimize by maximize is not necessarily closely related to the first one. Right now the value is infinity, and if you replaced 0 by $Q_a$ on the right side of the constraint, the value is always going to be 0 or infinity (if it's not 0, you can just multiply any solution with positive value by an arbitrary constant, and increase the value).

The way I went about this general issue when solving these SDPs was just implementing the partial trace, and then coding the dual and primal problem separately. The implementation of the partial trace is at https://cs.uwaterloo.ca/~amolinap/prtrace.m, hope it is useful!