I have employed the GenericCylindricalDecomposition command (applied to $6 \times 6$ "density matrix" positivity constraints) to compute the following expression
0 < U < 1 && -1 < a < 1 && -Sqrt[1 - a^2] < b < Sqrt[1 - a^2] && ((-Sqrt[U - b^2 U] <
c < -Sqrt[a^2 U] && (a b c)/(-1 + b^2) -
Sqrt[(-c^2 + b^2 c^2 + U - 2 b^2 U + b^4 U + a^2 c^2 U -
a^2 U^2 + a^2 b^2 U^2)/((-1 + b^2)^2 U)] <
d < (a b c)/(-1 + b^2) +
Sqrt[(-c^2 + b^2 c^2 + U - 2 b^2 U + b^4 U + a^2 c^2 U -
a^2 U^2 + a^2 b^2 U^2)/((-1 + b^2)^2 U)]) || (-Sqrt[a^2 U] <
c < Sqrt[
a^2 U] && (a b c)/(-1 + b^2) - Sqrt[(
1 - a^2 - 2 b^2 + a^2 b^2 + b^4 - c^2 + a^2 c^2 +
b^2 c^2)/(-1 + b^2)^2] <
d < (a b c)/(-1 + b^2) + Sqrt[(
1 - a^2 - 2 b^2 + a^2 b^2 + b^4 - c^2 + a^2 c^2 +
b^2 c^2)/(-1 + b^2)^2]) || (Sqrt[a^2 U] < c < Sqrt[
U - b^2 U] && (a b c)/(-1 + b^2) -
Sqrt[(-c^2 + b^2 c^2 + U - 2 b^2 U + b^4 U + a^2 c^2 U -
a^2 U^2 + a^2 b^2 U^2)/((-1 + b^2)^2 U)] <
d < (a b c)/(-1 + b^2) +
Sqrt[(-c^2 + b^2 c^2 + U - 2 b^2 U + b^4 U + a^2 c^2 U -
a^2 U^2 + a^2 b^2 U^2)/((-1 + b^2)^2 U)]))
Subject to these constraints, I want to integrate $\frac{3}{2 \pi^2}$ over $a,b,c,d \in [-1,1]$, yielding a function of $U$ (so the Boole command should be employed, it seems). (The $\frac{3}{2 \pi^2}$ comes from enforcement of the positivity of the original $6 \times 6$ density matrix, while the further constraints come from enforcement of the positivity--required for separability--of the "partial transpose" of the density matrix.)
This function when multiplied by
(12 U ((-1 + U) (1 + U (10 + U)) - 6 U (1 + U) Log[U]))/(-1 + U)^5
and integrated over $U \in [0,1]$ should yield a (ten-dimensional) "rebit-retrit Hilbert-Schmidt separability probability" roughly (I guess) equal to 0.6 (cf. sec. IV of https://arxiv.org/abs/1809.09040 ). (An eight-dimensional X-states counterpart problem yields the probability $\frac{16}{3 \pi^2} \approx 0.54038$ https://arxiv.org/abs/1501.02289 p.3.)
If the exact integrations can not be successfully conducted, then it may nevertheless be able to intuit an exact value for the desired probability, by performing a sufficiently high-precision NUMERICAL integration, following the lead of Thies Heidecke in his comment below.
The positivity-based inequalities to which the GenericCylindricalDecomposition command were applied (responding to the request of Heidecke) took the form
1 - a^2 - b^2 > 0 && 1 - a^2 - b^2 > 0 &&
(-1 + a^2) (-1 + c^2) - 2 a b c d - d^2 + b^2 (-1 + d^2) > 0 &&
-c^2 + U - b^2 U > 0 &&
-c^2 - U (-1 + 2 a b c d + d^2 - b^2 (-1 + d^2) + a^2 (-c^2 + U)) > 0 &&
-1 < a < 1 && -1 < b < 1 && -1 < c < 1 && -1 < d < 1 && 0 < U < 1
and after application of FullSimplify
-1 < d < 1 && U < 1 && U > c^2 + b^2 U && a^2 + b^2 < 1 &&
(-1 + a^2) (-1 + c^2) + b^2 (-1 + d^2) > d (2 a b c + d) &&
c^2 + U (-1 + 2 a b c d + d^2 - b^2 (-1 + d^2) + a^2 (-c^2 + U)) < 0 .
Let us omit the constraint
-c^2 - U (-1 + 2 a b c d + d^2 - b^2 (-1 + d^2) + a^2 (-c^2 + U)) > 0
in the original (first)--corresponding to the positivity of the determinant of the partial transpose of the $6 \times 6$ density matrix--of these two sets of constraints. (The remaining partial transpose constraint -c^2 + U - b^2 U > 0 corresponds to the positivity of a $5 \times 5$ minor.) Then, based on the associated ("truncated") GenericCylindricalDecomposition,
0 < U < 1 && -1 < b < 1 && -Sqrt[U - b^2 U] < c < Sqrt[U - b^2 U] && -Sqrt[1 - b^2] < a < Sqrt[ 1 - b^2] && (a b c)/(-1 + b^2) - Sqrt[( 1 - a^2 - 2 b^2 + a^2 b^2 + b^4 - c^2 + a^2 c^2 + b^2 c^2)/(-1 + b^2)^2] < d < (a b c)/(-1 + b^2) + Sqrt[( 1 - a^2 - 2 b^2 + a^2 b^2 + b^4 - c^2 + a^2 c^2 + b^2 c^2)/(-1 + b^2)^2]
we can, in fact, proceed analytically, integrating over d, a, c, b, in that order, and obtain an associated "separability function", \begin{equation} \frac{2 \left(\sqrt{(1-U) U}+\sin ^{-1}\left(\sqrt{U}\right)\right)}{\pi }. \end{equation} When multiplied--following our basic algorithm--by
(12 U ((-1 + U) (1 + U (10 + U)) - 6 U (1 + U) Log[U]))/(-1 + U)^5
and integrated over $U \in [0,1]$, we obtain an upper bound of \begin{equation} \frac{919}{5}-264 \log (2) \approx 0.809144 \end{equation} on the separability probability, following the AdaptiveQuasiMonteCarlo approach of Heidecke, that appears to be on the order of 0.744.
NIntegrate[Boole[firstexpression]*secondexpression, {a, -1, 1}, {b, -1, 1}, {c, -1, 1}, {d, -1, 1}, {U, 0, 1} ]
either gives ~4.8 or diverges, depending on theMethod
. $\endgroup$Log
term is quite well behaving, without singularities so shouldn't be a problem. Do you still have the logic expression in the original form, before cylindrical decomposition? It might improve the integrand evaluation speed a lot. $\endgroup$