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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 rather involved quantum-information-theoretic process RandomMatrixGeneration employed to generate the data sets yields more lower values of $x$ and $y$ $\in [0,1]$ than higher values. It is not computationally feasible to uniformly sample $x$ and $y$ over [0,1] and obtain the $4 \times 4$ density matrices that fulfill the requisite properties RequirementsForDensityMatrices.

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. I intend to update q1 and Q1 as their entries further increase in size. So, the $\{i,j\}$ cells of q1 and Q1 should be considered to correspond to $x=\frac{2 i-1}{100}$ and $y=\frac{2 j-1}{100}$.

Two calculations now inserted near the end of the linked Mathematica program show the near uniformity over [0,1] of the two marginal distributions.

Can any known forms (Gaussian, Archimedean,...) of copulas be well-fitted to these data? A Gaussian copula with uniform marginals over [0,1] would be quite appealing conceptually--due to the wide range of applicability of multivariate normal distributions.

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    $\begingroup$ Are you trying to tell us the sampling process is biased? If not, then "yields more lower values" is prima facie evidence that the marginals are not uniform. If so, then exactly in what way is it biased? We can't help you correct that bias until we have some clear characterization of it. Regarding your title, by definition there exists some copula that models this process, so the answer is trivially "yes." $\endgroup$
    – whuber ♦
    Commented Dec 20, 2021 at 14:38
  • $\begingroup$ whuber, thanks! I'm NOT sampling uniformly over x and y in [0,1]. I get the results by generating "density matrices"( in quantum parlance)--giving me the q1 results and testing whether they pass a "separability" test--in which case they are counted in the Q1 set. The z in the z-axis is the "separability probability". $\endgroup$
    – Paul B. Slater
    Commented Dec 20, 2021 at 14:47
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    $\begingroup$ Your setting is obscure. We really need more details about what you are doing. As it stands, your question is asking for the trivial answer I gave: yes, assuming your sampling is unbiased, you can fit $Q_1/q_1$ with a copula. If by "standard" you mean some severely restricted class of copulas (which doesn't seem to be the case, since you invite "other" forms), then the answer might change--but again we suffer from a lack of detail and clarity in your statement of the problem. $\endgroup$
    – whuber ♦
    Commented Dec 20, 2021 at 14:50
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    $\begingroup$ Cross-posted on Quantitative Finance SE. $\endgroup$
    – Richard Hardy
    Commented Dec 20, 2021 at 15:23
  • $\begingroup$ Could you create a more concise and complete description of what you are doing. Currently it is a broad collection of separate terms and links to other pages from which it is not easy to grasp the idea. Just in order to get the plot of Q1/q1 I have to download a 9.3 mB and 20 page large article and browse through it. And, that is just one of the many links and pieces of information in your post. If you put your question in a more approachable format, then you will reach a much larger public and number of people that might answer your question. $\endgroup$
    – Sextus Empiricus
    Commented Dec 30, 2021 at 12:06

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