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I am studying (mainly using Mathematica) some constrained integration problems in which the six-dimensional convex set of $4 \times 4$ correlation matrices plays a central role. In light of this, I would like to proceed using alternative parameterizations of the matrices that might facilitate analyses. I am aware of parameterizations based on partial correlations and on Cholesky factors. So, I would appreciate any guidance in specifically formulating and implementing these and other possible parameterizations. Since I am integrating over the six-dimensional set, I also need to employ the associated jacobians.

Per the request of Alex R for more details, let me say that, just to start, I'd like an explicit parameterization of the 4 x 4 correlation matrices in terms of partial correlations. (I know that Harry Joe has papers on this--but the notation employed for correlation matrices of arbitrary dimensions is a little challenging at first glance.)

The constrained integrations I'm pursuing are over essentially a nine-dimensional set of (nonnegative definite) $4 \times 4$ "density matrices", in which the six-dimensional set of $4 \times 4$ correlation matrices can be considered to be embedded. (The diagonal entries of these "density matrices" are nonnegative and sum to 1, rather than being all equal to 1.) The ("separability probability") problem under study is one in quantum information theory, and not only involves the nine-dimensional set of $4 \times 4$ density matrices ($\rho$) , but the "partial transposes" ($\rho^{PT}$) of the density matrices, obtained by transposing in place their four $2 \times 2$ blocks/submatrices. The constrained integrations attempt to enforce the positivity of $\rho^{PT}$ in addition to that of $\rho$ itself. So, I'd like to compare the use of various parameterizations in addressing these highly computationally challenging problems.

I can give references to the considerable associated quantum information theory literature.

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    $\begingroup$ It's not really clear what you're asking for. There are infinitely many such parametrizations. What's your end goal? Maybe post the class of integrals you are trying to evaluate? $\endgroup$
    – Alex R.
    Commented Mar 28, 2017 at 17:31
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    $\begingroup$ pdfs.semanticscholar.org/2ff5/… You could also try some parametrization using matrix varietys (manifolds) $\endgroup$
    – kjetil b halvorsen ♦
    Commented Mar 28, 2017 at 18:12
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    $\begingroup$ Exactly what measure are you integrating against? The measure will suggest suitable parameterizations. $\endgroup$
    – whuber ♦
    Commented Mar 28, 2017 at 18:41
  • $\begingroup$ Paul, by that do you mean the Lebesgue measure inherited from $\mathbb{R}^{16}$ by viewing these matrices as vectors? $\endgroup$
    – whuber ♦
    Commented Mar 29, 2017 at 22:14
  • $\begingroup$ whuber--I regard the correlation matrices as forming a six-dimensional space. $\endgroup$
    – Paul B. Slater
    Commented Mar 30, 2017 at 2:25

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