The notion of approximate information states (AIS) was introduced in (Subramanian 2020) as a methodology for learning task-relevant state representations for control in partially observable systems. They proposed particular learning objectives which attempt to reconstruct the cost and next state and provide a bound on the suboptimality of the closed-loop performance, but it is unclear whether these bounds are tight or actually lead to good performance in practice. Here we study this methodology by examining the special case of discrete approximate information states (DAIS). In this setting, we can solve for the globally optimal policy using value iteration for the DAIS model, allowing us to disambiguate the performance of the AIS objective from the policy search. Going further, for small problems with finite information states, we reformulate the DAIS learning problem as a novel mixed-integer program (MIP) and solve it to its global optimum; in the infinite information states case, we introduce clustering-based and end-to-end gradient-based optimization methods for minimizing the DAIS construction loss. We study DAIS in three partially observable environments and find that the AIS objective offers relatively loose bounds for guaranteeing monotonic performance improvement and is sufficient but not necessary for implementing optimal controllers. DAIS may even prove useful in practice by itself or as part of mixed discrete- and continuous-state representations, due to its ability to represent logical state, to its potential interpretabilty, and to the availability of these stronger algorithms.