I will discuss some mathematical tools based on the notion of statistical distance, or statistical distinguishability. These tools can be used to reveal deep facts about quantum theory, the nature of quantum measurement, and also to quantify the power of quantum technologies. I will present some published work regarding quantum metrology, and the possible advantages of a technique called ‘weak-value amplification’, as well as some ongoing calculations concerning the accuracy of quantum simulators. Can uniquely quantum effects accelerate or improve information processing tasks?