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.