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mp.mathematical physics – What are the John ellipsoids for a pair of (9- and 15-dimensional) convex units of $4 instances 4$ positive-definite matrices? Answer

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mp.mathematical physics – What are the John ellipsoids for a pair of (9- and 15-dimensional) convex units of $4 instances 4$ positive-definite matrices?

What are the John ellipsoids (JohnEllipsoid) for the 9- and 15-dimensional convex units ($A,B$) of $4 instances 4$ positive-definite, trace-1 symmetric (Hermitian) matrices (in quantum-information parlance, the units of “two-rebit” and “two-qubit” “density matrices” [DensityMatrices], respectively)? (Are these our bodies “centrally-symmetric”, within the sense of 1 side of the underlying theorem JohnTheorem?)

Further, what’s the relation (intersections, …) of those ellipsoids to the essential convex subsets of $A$ and $B$ collected of these matrices that stay positive-definite beneath the (not fully constructive) operation of partial transposition—by which the 4 $2 instances 2$ blocks of the $4 instances 4$ matrices are transposed in place? (It has been established [MasterLovasAndai] that the fractions of Euclidean quantity occupied by these “PPT” [positive-partial-transpose/separable/nonentangled] convex subsets are $frac{29}{64}$ for $A$ and $frac{8}{33}$ for $B$.)

Also, what’s the additional relation of those ellipsoids to the “inspheres” (the maximal balls inscribed in $A$ and $B$ [SBZ])?
The inspheres too equivocate inside the PPT units.
Might the John ellipsoids and inspheres merely coincide?

Additionally, what energy breathe the John ellipsoids themselves for these PPT units?

There is an fascinating conception of a “steering ellipsoid”, referred to within the following citation p. 28 [SteeringEllipsoid]:

For two-qubit states, the normalized conditional states Alice can usher Bob’s system to figure an ellipsoid inside Bob’s Bloch sphere, known as the steering ellipsoid (Verstraete, 2002; Shi et al., 2011, 2012; Jevtic et al., 2014).

However, the “Bloch sphere” is three-d, so the steering ellipsoid of a two-qubit condition can’t breathe the (15-dimensional) John ellipsoid requested above.

Of passage, the query what are the John ellipsoids can breathe requested for the convex units of $m instances m$ symmetric and $n instances n$ Hermitian (positive-definite, hint 1) density matrices ($m,n geq 2$). For $m,n=2$, the solutions emerge to breathe trifling, specifically the convex units themselves. For $m,n =3$, it appears presumably nontrivial. Only, nevertheless, for amalgam values of $m,n$, do we’ve got subsidiary questions relating to the convex subsets of PPT-states.

The Wikipedia article given by the primary hyperlink above describes the
“maximum volume inscribed ellipsoid as the inner Löwner–John ellipsoid”.

[DensityMatrices]: Slater – A concise system for generalized two-qubit Hilbert–Schmidt separability possibilities

[JohnTheorem]: Howard – The John ellipsoid theorem

[MasterLovasAndai]: Slater – Master Lovas–Andai and equal formulation verifying the $frac8{33}$ two-qubit Hilbert–Schmidt separability chance and companion rational-valued conjectures

[SBZ]: Szarek, Bengtsson, and Życzkowski – On the construction of the corpse of states with constructive partial transpose

[SteeringEllipsoid]: Uola, Costa, Nguyen, and Gühne – Quantum steering

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