Understanding quantum noise is essential for accurately creating desired quantum states and for examining a given state's evolution in any protocol. Using spontaneous parametric downconversion, we can create a wide variety of single- and two-qubit polarization states, including nearly perfect Bell states, mixed states (i.e., "noisy" states) and maximally entangled mixed states (MEMS). To characterize these states we use several different measures, including fidelity, "tangle" and linear entropy. In the course of our experiments, we have discovered and numerically investigated an extreme imbalance in the sensitivity of these different two-qubit state measures. We have also experimentally realized a "Procrustean" filtering technique to remove noise from MEMS. For moderate amounts of filtering, the experimental procedure works as desired to increase the tangle and decrease the linear entropy. However, for large amounts of filtering, the process becomes dominated by perturbations in the starting density matrix. The final outcome is a pure (i.e., zero entropy) product state (i.e., zero entanglement).