Faithfulness and Sensitivity for Ancilla-Assisted Process Tomography

A system-ancilla bipartite state capable of containing the complete information of an unknown quantum channel acting on the system is called faithful. In this work, we extend the applicability and generality of faithfulness significantly by introducing its local variant and examining their relationship when applied to various classes of quantum channels. In doing so, we discovered that, in the original proof by D'Ariano and Presti, only sufficiency was shown, not the full equivalence between faithfulness of state and invertibility of the corresponding Jamiolkowski map. We complete the proof by showing necessity and examine how far this characterization of faithfulness can be generalized by applying it to various classes of quantum channels. We also explore a more general notion we call sensitivity, the property of quantum state being altered by any nontrivial action of quantum channel. We study their relationship by characterizing both properties for important classes of quantum channels such as unital channels, random unitary operations, and unitary operations. Unexpected (non)equivalence results among them shed light on the structure of quantum channels by showing that we need only two classes of quantum states for characterizing quantum states faithful or sensitive to various subclasses of quantum channels.

Automated summary

In this study, we expand the applicability and generality of faithfulness by introducing its local variant and examining their relationship in various classes of quantum channels. We complete the proof of the equivalence between faithfulness of state and invertibility of the corresponding Jamiolkowski map by showing the necessity. We also explore a more general notion called sensitivity, and study its relationship with faithfulness by characterizing both properties in important classes of quantum channels. Unexpected (non)equivalence results shed light on the structure of quantum channels.