Users Daniel Huber and Dominic have provided an interesting answer each to the question JohnEllipsoids of constructing circumscribing and inscribing ellipsoids for a certain three-dimensional convex set of quantum-information-theoretic interest (that is, the "ordered spectra of absolutely separable two-qubit states").
We have attempted here to adopt their specific Mathematica codes to two other sets of strongly related interest, and would appreciate feedback on how the methodology we adopt could possibly be improved for these two cases, as well as how the theoretically-important issues of minimality and maximality of fitted ellipoids could be more directly addressed than has so far been. Also, the expansion of their graphics to show three sets of relevant interest--rather than the current two displayed--should be of interest.
To begin, we note that the defining constraint for the convex set under study in JohnEllipsoids was
1 > x && x >= y && y >= z && z >= 1 - x - y - z >= 0 && x - z < 2 Sqrt[y (1 - x - y - z)] Further, it was also noted that there are two other sets of associated interest Adhikari, also circumscribing and inscribed within the convex body under examination. These are given by the constraints
1 > x && x >= y && y >= z && z >= 1 - x - y - z >= 0 && x^2 + y^2 + (1 - x - y - z)^2 + z^2 < 3/8 and
1 > x && x >= y && y >= z && z >= 1 - x - y - z >= 0 && x^2 + y^2 + (1 - x - y - z)^2 + z^2 < 1/3 . (Certainly, at least, the latter set is convex--corresponding to separable density matrices--so the question of the John ellipsoids for it seem valid ones to ask.)
We have attempted (clearly not fully successfully) to adopt the answer of Huber to the construction of a circumscribing ellipsoid for the set defined by the latter constraint and the answer of Dominic for the former case. (The volume [separability probability] of the former set is $\frac{\left(14-3 \sqrt{6}\right) \pi }{3456 \sqrt{3}} \approx 0.0034909$ and of the latter set, $\frac{\pi }{864 \sqrt{3}} \approx 0.0020993$, while that of the original main set of interest is $\frac{1}{576} \left(8-6 \sqrt{2}-9 \sqrt{2} \pi +24 \sqrt{2} \cos^{-1}\left(\frac{1}{3}\right)\right) \approx 0.00227243$.)
In the circumscribed case, Huber employed the four extremal points
pts={{1/3, 1/3, 1/3}, {1/4, 1/4, 1/4}, {1/2, 1/6, 1/6}, {1/8 (2 + Sqrt[2]), 1/8 (2 + Sqrt[2]), 1/2 (1 + 1/4 (-2 - Sqrt[2]))}} It appears possible to pursue the modified problem with three of the same points, only replacing
{1/8 (2 + Sqrt[2]), 1/8 (2 + Sqrt[2]), 1/2 (1 + 1/4 (-2 - Sqrt[2]))} by
{1/12 (3 + Sqrt[3]), 1/12 (3 + Sqrt[3]), 1/12 (3 - Sqrt[3])} (The second and fourth points remain the furthest apart, as Huber requires, with Norm[pts[2] - pts[4]] equalling $\frac{1}{2 \sqrt{3}}$.)
My modification of the Huber code and its output is ModifiedHuber
However, this led to a set of semiaxes
{1/3 I Sqrt[2/949 (81 + 56 Sqrt[3])], Sqrt[ 2/429 (-17 + 12 Sqrt[3])], 1/(4 Sqrt[3])} which when inputted to the volume formula
4/3 a1 a2 a3 \[Pi] gives a clearly unacceptable (imaginary!) volume of
2/351 I Sqrt[1/803 (639 + 20 Sqrt[3])] \[Pi] \approx 0. + 0.0163957 I .
Now, here is our adoption of the inscribed ellipsoid code of Dominic (only replacing the original defining constraint by the one with the 3/8 bound).
The output certainly seems less problematical than our modification of the circumscribed ellipsoid code (no imaginary volume,...).
Incorporation into the modified Huber and Dominic two-set plots of a third--the original set (involving the x - z < 2 Sqrt[y (1 - x - y - z) inequality)--should be of interest.
Computation of the volumes of new ellipsoids should be of interest. (For thoroughness, although not of immediate relevance to the problems at hand, let us note that the volumes [separability probabilities] of the three sets of interest have also been computed when Hilbert-Schmidt measure is attached to them ThreeVolumes. Listed increasingly, these volumes are $\frac{35 \pi }{23328 \sqrt{3}} \approx 0.00272132$, $\frac{29902415923}{497664}-\frac{50274109}{512 \sqrt{2}}-\frac{3072529845 \pi }{32768 \sqrt{2}}+\frac{1024176615 \cos ^{-1}\left(\frac{1}{3}\right)}{4096 \sqrt{2}} \approx 0.00365826$ and $\frac{35 \sqrt{\frac{1}{3} \left(2692167889921345-919847607929856 \sqrt{6}\right)} \pi}{27518828544} \approx 0.0483353$. These calculations are of relevance when the sets are viewed in the 15-dimensional context of the two-qubit density matrices, rather than strictly a three-dimensional one.)
The issue posed of minimality and maximality of volume has, however, not yet been addressed. Since the inscribed approach of Dominic involves numerical fitting it seems unlikely to be strictly maximal, but the minimality of the circumscribed one of Huber seems less clear. Dominic has noted the availability of a python code "Inner and outer Löwner-John Ellipsoids" Mosekpythoncode for these tasks. However, it appears to require the convex set in question to be a polytope, which is not the case at hand, although the code might be adoptable to a non-polytope input [see comments in Adamaszek1 Adamaszek2].
