  In this work, we perform an in-depth study of recently introduced
average-case quantum distances. The average-case distances approximate the
average Total-Variation (TV) distance between measurement outputs of two
quantum processes, in which quantum objects of interest (states, measurements,
or channels) are intertwined with random circuits. Contrary to conventional
distances, such as trace distance or diamond norm, they quantify
$\textit{average-case}$ statistical distinguishability via random circuits.
  We prove that once a family of random circuits forms an $\delta$-approximate
$4$-design, with $\delta=o(d^{-8})$, then the average-case distances can be
approximated by simple explicit functions that can be expressed via degree two
polynomials in objects of interest. We prove that those functions, which we
call quantum average-case distances, have a plethora of desirable properties,
such as subadditivity, joint convexity, and (restricted) data-processing
inequalities. Notably, all of the distances utilize the Hilbert-Schmidt norm
which provides an operational interpretation it did not possess before. We also
derive upper bounds on the maximal ratio between worst-case and average-case
distances. For each dimension $d$ this ratio is at most $d^{\frac{1}{2}},\ d, \
d^{\frac{3}{2}}$ for states, measurements, and channels, respectively. To
support the practical usefulness of our findings, we study multiple examples in
which average-case quantum distances can be calculated analytically.

Quantum Physics