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## pr.chance – Perform sure constrained integrations over an ordered subsection of a 3-simplex, yielding “absolute separability” possibilities

In precept, right here is how the 2008 assertion that the two-qubit Hilbert-Schmidt absolute separability chance is

commence{equation}

frac{29902415923}{497664}+frac{-3217542976+5120883075 pi -16386825840 tan

^{-1}left(sqrt{2}privilege)}{32768 sqrt{2}}approx 0.00365826.

aim{equation}

can breathe reappraised–which was the unprejudiced of the issue posed.

Issue the five-command Mathematica sequence for a two-dimensional integration (for simplicity of presentation, we take $lambda_1=l1$, $lambda_2=l2$, $lambda_3=l3$, $lambda_4=l4$),

l4 = 1 – l1 – l2 –

l3; g={l1,l2,l3,l4};HS = Product[(g[[i]]-g[[j]])^2,{i,1,3},{j,i+1,4}];t =

GenericCylindricalDecomposition[

l1 >= l2 && l2 >= l3 && l3 >= l4 && l4 >= 0 &&

l1 – l3 <= 2 Sqrt[l2 l4], {l3, l2, l1}][[1]]; s =

9081072000 Integrate[HS Boole[t], {l2, 0, 1}, {l1, 0, 1}];

This yields a sizable (LeafCount=5354) output, now requiring three one-dimensional integrations of capabilities of l3 over the intervals $[frac{1}{8} left(2-sqrt{2}right),frac{1}{6}]$, $[frac{1}{6},frac{1}{4}]$ and $[frac{1}{4},frac{1}{3}]$. Quite a lot of phrases moving sq. roots of sq. roots within the Mathematica output resist preliminary integration, however can breathe simplified to phrases moving solely a sole sq. root (https://math.stackexchange.com/questions/3375075/transform-certain-integrands-so-that-they-involve-a-single-square-root-rather-th), after which can breathe built-in out.

However, the resultant output is to commence quiet very sizable, moving immediate to 200 phrases, many with arcsines of varied arguments, and can clearly require a really substantial additional quantity of simplifications and transformations. (As indicated within the assertion of the query, within the cited 2008 paper, “[C]opious employ was manufactured from trigonometric identities moving the tetrahedral dihedral angle $phi=cos ^{-1}left(frac{1}{3}privilege)$”, assisted by V. Jovovic.) Further specifics will breathe supplied when and if workable. But is there a extra direct, much less “tedious” route–and, how if workable, might the clearly quiet extra inspiring Bures and Wigner-Yanase counterparts breathe addressed? Probably not doable!

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