Operational Quantum Average-Case Distances

We introduce distance measures between quantum states, measurements, and channels based on their statistical distinguishability in generic experiments. Specifically, we analyze the average Total Variation Distance (TVD) between output statistics of protocols in which quantum objects are intertwined with random circuits and measured in standard basis. We show that for circuits forming approximate 4-designs, the average TVDs can be approximated by simple explicit functions of the underlying objects -- the average-case distances (ACDs). We apply them to analyze the effects of noise in quantum advantage experiments and for efficient discrimination of high-dimensional states and channels without quantum memory. We argue that ACDs are better suited for assessing the quality of NISQ devices than common distance measures such as trace distance or the diamond norm.