Faithfulness and sensitivity for ancilla-assisted process tomography

TL;DR

This paper completes the proof of the equivalence between faithfulness of bipartite states and invertibility of the Jamiolkowski map for quantum channels, introduces the concept of sensitivity, and explores their relationships across various quantum channel classes.

Contribution

It finalizes the proof of faithfulness equivalence for all quantum channels and introduces the notion of sensitivity, analyzing their relationships for key classes of quantum channels.

Findings

Faithfulness is equivalent to invertibility of the Jamiolkowski map for all quantum channels.

Only bipartite states with no local classical observable can sense unital channels.

Different classes of quantum channels require only two types of states for characterization.

Abstract

A system-ancilla bipartite state capable of containing the complete information of an unknown quantum channel acting on the system is called faithful. The equivalence between faithfulness of state and invertibility of the corresponding Jamiolkowski map proved by D'Ariano and Presti has been a useful characterization for ancilla-assisted process tomography albeit the proof was incomplete as they assumed trace nonincreasing quantum operations, not quantum channels. We complete the proof of the equivalence and introduce the generalization of faithfulness 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. For example, it reveals the relation between quantum process tomography and quantum correlation as it turns out that only bipartite states that has no local classical observable at all can be used to sense the effect of unital channels.