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

Hello pricey customer to our community We will proffer you an answer to this query pr.chance – Perform sure constrained integrations over an ordered subsection of a 3-simplex, yielding “absolute separability” possibilities ,and the respond will breathe typical by means of documented data sources, We welcome you and proffer you fresh questions and solutions, Many customer are questioning concerning the respond to this query.

## 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!

we’ll proffer you the answer to pr.chance – Perform sure constrained integrations over an ordered subsection of a 3-simplex, yielding “absolute separability” possibilities query by way of our community which brings all of the solutions from a number of dependable sources.