You can switch back and forth between the two views, obviously, and sometimes you do, but I think the most natural reason is because the operators you get are trace 1 positive semidefinite matrices, and there’s a lot of theory on PSD matrices waiting for you. Also, the natural maps on density matrices, the quantum channels or trace preserving completely positive maps have a pretty nice representation in terms of conjugation when you think of density matrices as matrices: \rho \mapsto \sum_i K_i \rho K_i^* for some operators K_i that satisfy \sum_i K_i^*K_i = I
Obviously all of these translates to the (0,2) tensor view, but a lot of theory was already built for thinking of these as linear maps on matrix spaces (or c* algebras or whatever fancier generalizations mathematicians had already been looking at)
You can switch back and forth between the two views, obviously, and sometimes you do, but I think the most natural reason is because the operators you get are trace 1 positive semidefinite matrices, and there’s a lot of theory on PSD matrices waiting for you. Also, the natural maps on density matrices, the quantum channels or trace preserving completely positive maps have a pretty nice representation in terms of conjugation when you think of density matrices as matrices: \rho \mapsto \sum_i K_i \rho K_i^* for some operators K_i that satisfy \sum_i K_i^*K_i = I
Obviously all of these translates to the (0,2) tensor view, but a lot of theory was already built for thinking of these as linear maps on matrix spaces (or c* algebras or whatever fancier generalizations mathematicians had already been looking at)