# calculation and analysis: approximate / estimate the relationship of two multidimensional restricted integrals

I hypothesize that a certain "probability of qubit-qutrit separability" should assume the value $$frac {5} {3} left (112 pi ^ 2-1105 right) approx 0.659488$$. In its initial formulation, this is a problem of 15 dimensions. I think it can be reduced to the calculation of the relationship of a restricted integration of nine dimensions to one of eight dimensions.

These two integrals do not currently seem to be computable exactly.
(I think it's too big, besides including a cosine function) for the GenericCylindricalDecomposition command application. Therefore, I would like at least a numerical evaluation of their relationship, to test the plausibility of the hypothesis.

There are two sets of restrictions, the first related to the positivity of a $$6 times 6$$ "density matrix", and the second to the positivity of the "partial transposition" of the density matrix.

Denote the first set of restrictions by

A = r[1, 4]^ 2 + r[3, 4]^ 2 < 1 && (-1 + r[1, 6]^2) (-1 + r[3, 4]^2) + r[1, 4]^2 (-1 + r[3, 6]^2) >
r[3, 6] (2 Cos[t[t[t[t[1, 4] – t[1, 6] – t[3, 4] + t[3, 6]]r[1, 4] r[1,
6] r[3, 4] + r[3, 6])

and the second set by

B = 1 – r[1, 4]^ 2 –
U r[1, 6]^ 2 && -U (-1 + r[1, 4]^ 2 + U r[1, 6]^ 2) + (-1 +
U r[1, 6]^ 2) r[3, 4]^ 2 +
U r[3, 6] (-2 Cos[t[t[t[t[1, 4] – t[1, 6] – t[3, 4] + t[3, 6]]r[1, 4] r[
1, 6] r[3, 4] + (-1 + r[1, 4]^ 2) r[3, 6]).

(The notation could, of course, be simplified: as it is now, the indices refer to the inputs of the density matrix, with the $$r$$being radial variables and the $$t$$being angular)

The integral denominator of eight dimensions is

NIntegrar[
Boole[A] v, {r[1, 4], 0, 1}, {r[1, 6], 0, 1}, {r[3, 4]0
1}, {r[3, 6], 0, 1}, {t[1, 4], 0, 2 Pi}, {t[1, 6]0
2 Pi}, {t[3, 4], 0, 2 Pi}, {t[3, 6], 0, 2 Pi}],

where

v = r[1, 4] r[1, 6] r[3, 4] r[3, 6].

The integral of the nine-dimensional numerator (even more challenging) is

``````NIntegrar[Boole[FullSimplify[A&&B] v w, {U, 0,1}, {r[1, 4], 0, 1}, {r[1, 6], 0, 1}, {r[3, 4], 0,1}, {r[3, 6], 0, 1}, {t[1, 4], 0, 2 Pi}, {t[1, 6], 0, 2 Pi}, {t[3, 4], 0, 2 Pi}, {t[3, 6], 0, 2 Pi}],
``````

where

w = (56 U ^ 2 ((-1 + U) (3 + U (178 + U (478 + U (178 + 3 U))))
60 U (1 + U) (1 + U (5 + U)) Log[U])) / (- 1 + U) ^ 9,

the integral of which about $$U in [0,1]$$ is 1