I'm currently developing a data set that consists of two $50 \times 50$ matrices, which I designate as q1 and Q1.

I strongly believe (bordering on formal proof [cf. Corollary 1 in marginalinvariance]) that the ratio of Q1 to q1 constitutes a sample from a (symmetric) bivariate copula (alternatively termed "permutons or doubly-stochastic measures" WikiCopula) $f(x,y)$ with uniform marginals over [0,1]. Now $x$ and $y$ ("Bloch radii" of quantum bit [qubit] systems--also "quadratic Casimir invariants" CasimirInvariants) appear "repulsive" in nature RepulsiveBehavior, that is the 45-degree line $x=y$ has relatively low values.

Unfortunately, the sampling process to generate the data sets yields more lower values of $x$ and $y$ $\in [0,1]$ than higher values--so some trimming of q1 and Q1 might be appropriate in analyses.

So, my question is can the ratio of Q1 to q1 be well fitted by any of the standard Gaussian or Archimedean or other copulas?

The developing Q1 and q1 data sets as well as a plot of Q1/q1--are displayed in BivariateCopulaRepulsive, which I hope is usable by interested parties. I intend to update q1 and Q1 as their entries further increase in size.