Evaluating Pauli Errors on Cluster States by Weighted Distances

The problem of evaluating the differences between quantum states before and after being affected by errors encoded in unitary transformations is addressed. Standard distance functions, e.g., the Bures length, are not fully adequate for such a task. Weighted distances are instead the appropriate information measures to quantify the distinguishability of multipartite states. Here, the previously introduced weighted Bures length and the newly defined weighted Hilbert-Schmidt distance are employed to quantify how much single-qubit Pauli errors alter cluster states. For both types of weighted distances, it is found that different errors of the same dimension change cluster states in a different way, i.e., their detectability is in general different. Indeed, they transform an ideal cluster state into a state whose weighted distances from the input depends on the specific chosen Pauli rotation, as well as the position of the affected qubit in the graph related to the cluster state. As these features are undetected by using standard distances, the study proves the usefulness of weighted distances to monitor key but elusive properties of many-body quantum systems