Joint distributional approximation for multiple output probabilities

Establish whether, under the strong-design assumption, the Kolmogorov bound for individual output-probability marginals extends to joint distribution functions of multiple output probabilities with at most a linear loss in the number of entries.

Background

The main results establish quantitative Kolmogorov and total-variation guarantees for each individual output probability of a strong approximate unitary design relative to the corresponding Haar distribution. The paper leaves unresolved whether analogous guarantees hold jointly for several output probabilities, which would require controlling correlations rather than only one-dimensional marginals. The conjectured extension is measured uniformly over coordinatewise thresholds and is intended to quantify how accurately designs reproduce correlations among output probabilities.