Quantifying the difference between many-body quantum states

The quantum state overlap, i.e. the ``bracket'', is the fundamental measure of the difference between two quantum states. Yet, it can seriously mislead when we study many-body systems. The problem is inherited by widely employed parent quantities, such as the state fidelity and related distance functions. We introduce a new class of information-theoretic measures, the weighted distances, to overcome such limitations. They quantify the difference between quantum states of many particles accounting for the size of the system dimension. The result enables to evaluate limits and precision of complex quantum processes, e.g. the performance of quantum computers.