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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 $y=\frac{2 j-1}{100}$.

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.

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.

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.

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.

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 $y=\frac{2 j-1}{100}$.

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.