  The quantum state overlap is the textbook measure of the difference between
two quantum states. Yet, it is inadequate to compare the complex configurations
of many-body systems. The problem is inherited by the widely employed quantum
state fidelity and related distances. We introduce the weighted distances, a
new class of information-theoretic measures that overcome these limitations.
They quantify how hard it is to discriminate between two quantum states of many
particles, factoring in the structure of the required measurement apparatus.
Therefore, they can be used to evaluate both the theoretical and the
experimental performances of complex quantum devices. We also show that the
newly defined "weighted Bures length" between the input and output states of a
quantum process is a lower bound to the experimental cost of the
transformation. The result uncovers an exact quantum limit to our ability to
convert physical resources into computational ones.
