Fig. 1

a Schematic showing the physical space as a convex subset of the set of unconstrained matrices, The minimum of the cost function is often outside of the physical space. b Illustration of the PGD process for a qubit. A step in the gradient direction can yield a non physically-allowed density matrix, but the projection brings the estimate back into the desired search space, (i.e., the Bloch sphere in the case of a qubit). c This process lowers the cost function, and is repeated until progress is sufficiently small and the final density matrix estimate is as close as desired to the maximum likelihood state ρ _ML