Exploring Quantum Average-Case Distances: Proofs, Properties, and Examples

In this work, we present an in-depth study of average-case quantum distances introduced in Maciejewski et al. (2022). The average-case distances approximate, up to the relative error, 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 quantum circuits. Contrary to conventional distances, such as trace distance or diamond norm, they quantify
average-case
statistical distinguishability via random quantum 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 simple degree two polynomials in objects of interest. For systems of moderate dimension, they can be easily explicitly computed – no optimization is needed as opposed to diamond norm distance between channels or operational distance between measurements. We prove that those functions, which we call quantum average-case distances, have a plethora of desirable properties, such as subadditivity w.r.t. tensor products, joint convexity, and (restricted) data-processing inequalities. Notably, all distances utilize the Hilbert-Schmidt (HS) norm, which provides this norm with a new operational interpretation. We also provide upper bounds on the maximal ratio between worst-case and average-case distances, and for each of them, we provide an example that saturates the bound. Specifically, we show that 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.