Solve a constrained 4D integration problem using either Cartesian or (paired) polar coordinates

Question

I have a pair of 4D integrands (there is, in both, a fifth variable $u \geq 1$, not subject to integration), \begin{equation} -\frac{\pi ^2 \left(-\left(u^2-1\right) y_{14}^2-\left(u^2-1\right) z_{14}^2+y_{13}^2+z_{13}^2-1\right) \left(u^2 \left(y_{14}^2+z_{14}^2\right)+y_{13}^2+z_{13}^2-1\right)}{2 \left(y_{13}^2+z_{13}^2-1\right)} \end{equation} and \begin{equation} -\frac{\pi ^2 r_{13} r_{14} \left(r_{14}^2 u^2+r_{13}^2-1\right) \left(-r_{14}^2 \left(u^2-1\right)+r_{13}^2-1\right)}{2 \left(r_{13}^2-1\right)}. \end{equation} (Note four "active" variables in the first, and only two radial and no angular ones in the second.) Now I want to integrate the first of these, subject to the constraints (also reported in my earlier posting What are the new limits of integration in changing from 4-D Cartesian coordinates to two sets of polar coordinates?), \begin{equation} u>1\land -\frac{1}{u}<z_{14}<\frac{1}{u}\land -\frac{\sqrt{1-u^2 z_{14}^2}}{u}<y_{14}<\frac{\sqrt{1-u^2 z_{14}^2}}{u}\land -\sqrt{1-u^2 \left(y_{14}^2+z_{14}^2\right)}<y_{13}<\sqrt{1-u^2 \left(y_{14}^2+z_{14}^2\right)}\land -\sqrt{u^2 \left(-\left(y_{14}^2+z_{14}^2\right)\right)-y_{13}^2+1}<z_{13}<\sqrt{u^2 \left(-\left(y_{14}^2+z_{14}^2\right)\right)-y_{13}^2+1} \end{equation} and/or the second integrand, subject to the same set of constraints, after the use of the polar-coordinate transformations \begin{equation} \left\{z_{13}\to r_{13} \cos \left(t_{13}\right),z_{14}\to r_{14} \cos \left(t_{14}\right),y_{13}\to r_{13} \sin \left(t_{13}\right),y_{14}\to r_{14} \sin \left(t_{14}\right)\right\}. \end{equation} For $u=1$, the results of both integrations should be $\frac{\pi^4}{12}$. For $u=2$, the results should be $\approx 2.5637$. (Implicitly, $z_{14},y_{14},y_{13},z_{13} \in [-1,1]$ and $r_{13},r_{14} \in [0,1]$ and $t_{13},t_{14} \in [0, 2 \pi]$.)

Answer 1

We have found that the result (divided by $\frac{\pi^4}{12}$, as mentioned) of the two four-dimensional integrations that were the subject of the question is the function \begin{equation} f(u)= \frac{4 u^2-1}{3 u^4}. \end{equation} When employed in the form \begin{equation} \label{true} f(\frac{1}{\varepsilon})=\frac{1}{3} \varepsilon ^2 \left(4-\varepsilon ^2\right) \end{equation} this serves as the function \begin{equation} \tilde{\chi_2}(\varepsilon)=f(\frac{1}{\varepsilon}) \end{equation} that A. Lovai and A. Andai expressed hope in constructing in their preprint (to appear in J. Phys. A, I do believe) https://arxiv.org/pdf/1610.01410.pdf, entitled "Invariance of separability probability over reduced states in 4 × 4 bipartite systems". The use now of $\tilde{\chi_2}(\varepsilon)$ leads to the verification of the "long-standing" ($\approx$ 10 years) conjecture that the probability that a two-qubit density matrix, randomly selected with respect to Hilbert-Schmidt measure, is separable (disentangled) is \begin{equation} \frac{8}{33} \approx 0.242424. \end{equation} More specifically, the appropriate two-qubit analogue of the Lovas-Andai "two-rebit separability probability" equation (p. 12 of the indicated preprint, immediately after eq. (24)) \begin{equation} \label{sepR} \mathcal{P}_{sep}(\mathbb{R}) = \frac{\int\limits_{-1}^1\int\limits_{-1}^x \tilde{\chi_1} \left( \left.\sqrt{\frac{1-x}{1+x}}\right/ \sqrt{\frac{1-y}{1+y}} \right)(1-x^2)(1-y^2) (x-y) \mbox{d} y\mbox{d} x}{\int\limits_{-1}^1\int\limits_{-1}^x (1-x^2)(1-y^2)(x-y) \mbox{d} y \mbox{d} x}, \end{equation} is \begin{equation} \label{sepC} \mathcal{P}_{sep}(\mathbb{C}) = \frac{\int\limits_{-1}^1\int\limits_{-1}^x \tilde{\chi_2} \left( \left.\sqrt{\frac{1-x}{1+x}}\right/ \sqrt{\frac{1-y}{1+y}} \right)(1-x^2)^2(1-y^2)^2 (x-y)^2 \mbox{d} y\mbox{d} x}{\int\limits_{-1}^1\int\limits_{-1}^x (1-x^2)^2(1-y^2)^2(x-y)^2 \mbox{d} y \mbox{d} x}, \end{equation} where the real and complex domains are denoted. The denominator of the latter equation straightforwardly evaluates to \begin{equation} \frac{256}{1575}, \end{equation} while the use of the newly-constructed $\tilde{\chi_2}(\varepsilon)$ yields a numerator value of \begin{equation} \frac{2048}{51975}, \end{equation} with the ratio giving the $\frac{8}{33}$ result.

It is somewhat startling to compare the quite simple nature of $\tilde{\chi_2}(\varepsilon)$ with its "two-re[al]bit" counterpart (p. 5 of the preprint) \begin{equation} \tilde{\chi_1} (\varepsilon ) = 1-\frac{4}{\pi^2}\int\limits_\varepsilon^1 \left( s+\frac{1}{s}- \frac{1}{2}\left(s-\frac{1}{s}\right)^2\log \left(\frac{1+s}{1-s}\right) \right)\frac{1}{s} \mbox{d} s \end{equation} \begin{equation} = \frac{4}{\pi^2}\int\limits_0^\varepsilon \left( s+\frac{1}{s}- \frac{1}{2}\left(s-\frac{1}{s}\right)^2\log \left(\frac{1+s}{1-s}\right) \right)\frac{1}{s} \mbox{d} s \end{equation} \begin{equation} =\frac{2 \left(\varepsilon ^2 \left(4 \text{Li}_2(\varepsilon )-\text{Li}_2\left(\varepsilon ^2\right)\right)+\varepsilon ^4 \left(-\tanh ^{-1}(\varepsilon )\right)+\varepsilon ^3-\varepsilon +\tanh ^{-1}(\varepsilon )\right)}{\pi ^2 \varepsilon ^2}, \end{equation} where the polylogarithmic function is indicated. (Lovas and Andai employed $\tilde{\chi_1} (\varepsilon)$ to verify the also previously conjectured two-rebit separability probability of $\frac{29}{64}$.) Let us interestingly note that it was conjectured in 2007 that the "two-qubit separability function" had the form \begin{equation} \frac{6}{71} (3 -u^2) u^2, \end{equation} somewhat similar to the second equation above. (For the 2007 reference in which the $\frac{8}{33}$ conjecture was first put forth, see https://arxiv.org/abs/0704.3723, while for the more current research see https://arxiv.org/pdf/1701.01973.pdf.)

By way of further background, the 4D integrands that are the subject of this math.stack.exchange question were derived with the use of the Mathematica command GenericCylindricalDecomposition applied to an eight-dimensional set of positivity conditions, enforcing the positive-definite nature of two-qubit ($4 \times 4$) density matrices and of their partial transposes. (These density matrices had their two $2 \times 2$ diagonal blocks, themselves set diagonal in nature. An earlier parallel two-rebit analysis helped us reconstruct $\tilde{\chi_1} (\varepsilon )$, giving us confidence in this strategy. This pair of reduction strategies rendered the corresponding sets of density matrices as 11-dimensional and 7-dimensional in nature, rather than the standard full 15- and 9-dimensions, respectively.)

We were able to use the results of this cylindrical algebraic decomposition to integrate over four of the eight variables, leaving us with the first of the integrands above. (The reduction from eleven variables to eight is possible through a certain useful reparameterization of the diagonal entries--which we do not detail at this point.) Transformation to a pair of polar coordinates gave us the simpler second integrand, with the (rather complicated) integration constraints stated in the question above now being simply reducible to $r_{13}^2 +r_{14}^2 u^2 <1$, with $u>1$. (This constitutes the answer to the question posed by me earlier What are the new limits of integration in changing from 4-D Cartesian coordinates to two sets of polar coordinates?.) The integration result, $f(u)= \frac{4 u^2-1}{3 u^4}$, immediately followed.

A formidable challenge, to continue this line of research, is now to establish that the "two-quater[nionic]bit" Hilbert-Schmidt separability probability is $\frac{26}{323}$. This would move us, first, from the original 9-dimensional two-rebit and 15-dimensional two-qubit settings to a 27-dimensional one. (But these dimensions can be reduced to 7-, 11- and 19-, using the apparently acceptable strategy--that has given us $\tilde{\chi_1} (\varepsilon )$ and $\tilde{\chi_2} (\varepsilon )$--of setting the two $2 \times 2$ diagonal blocks themselves to diagonal form. In turn, this leads to cylindrical algebraic decompositions with 4, 8 and 16 variables--with the last, quaternionic one, still seemingly computationally unfeasible.)

Would the associated Lovas-Andai two-quaterbit separability function $\tilde{\chi_4} (\varepsilon)$, then, more resemble the simple-natured $\tilde{\chi_2} (\varepsilon)$ or the more complicated appearing $\tilde{\chi_1} (\varepsilon)$? We conjecture the latter, as the complex domain appears to be privileged in these respects.

In conclusion, let us state that, at this point in time, it is not yet fully clear to us whether our integration results yielding $\tilde{\chi_2} (\varepsilon )$ and, then, the $\frac{8}{33}$ ratio constitutes--in full conjunction with the foundational work of Lovas and Andai--what can be considered as a formally rigorous confirmation of the $\frac{8}{33}$ two-qubit Hilbert-Schmidt separability probability conjecture. (Any remaining concerns/clarifications would seem to be connected with the fact that the computations immediately pertain to 11- rather than 15-dimensions.)

Let me also point out that the manner of derivation of $\tilde{\chi_2} (\varepsilon )$ here is distinctly different from that employed by Lovas and Andai (preprint App. A) in obtaining the form of $\tilde{\chi_1} (\varepsilon )$, though I also been able to find this result using the cylindrical algebraic decomposition approach.

The arguments given above have been incorporated into sec. IV.B.1 of my preprint https://arxiv.org/pdf/1701.01973.pdf, entitled "Construction of the Lovas-Andai Two-Qubit Function $\tilde{\chi}_2 (\varepsilon )=\frac{1}{3} \varepsilon ^2 \left(4-\varepsilon ^2\right)$ Verifies the $\frac{8}{33}$-Hilbert Schmidt Separability Probability Conjecture"