Modern quantum devices require precise implementation of desired quantum
channels. The quality of this implementation can be characterized using
the notion of operation fidelity, which measures the overlap between
initial states and their images with respect to the considered channel.
I present the results of my research, conducted together with prof.
Karol Życzkowski and dr Grzegorz Rajchel-Mieldzioć, where we analyze the
statistical properties of operation fidelity of low-dimensional
channels, in particular its distributions and extremal values.
First, I briefly revise the notions of fidelity between quantum states
and numerical range and shadow of a linear operator. Then, I show
several examples of quantum channels whose operation fidelity
distributions can be calculated exactly (namely, mixed unitary qubit
channels and unitary qutrit channels). Lastly, I present a method of
analyzing quantum channels utilizing the concepts of numerical range and
shadow of their Kraus operators, using Schur channels as an example.