Trace distance

In quantum mechanics, and especially quantum information and the study of open quantum systems, the trace distance T is a metric on the space of density matrices and gives a measure of the distinguishability between two states. It is the quantum generalization of the Kolmogorov distance for classical probability distributions.The trace distance is just half of the trace norm of the difference of the matrices:By using the Hölder duality for Schatten norms, the trace distance can be written in variational form as The trace distance has the following properties