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

Further, what is the relation (intersections, …) of these ellipsoids to the important convex subsets of $A$ and $B$ composed of those matrices that remain positive-definite under the (not completely positive) operation of partial transposition—by which the four $2 \times 2$ blocks of the $4 \times 4$ matrices are transposed in place? (It has been established [MasterLovasAndai] that the fractions of Euclidean volume 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 is the further relation of these ellipsoids to the "inspheres" (the maximal balls inscribed in $A$ and $B$ [SBZ])? The inspheres also lie within the PPT sets. Might the John ellipsoids and inspheres simply coincide?

Additionally, what might be the John ellipsoids themselves for these PPT sets?

There is an interesting concept of a "steering ellipsoid", referred to in the following quotation p. 28 [SteeringEllipsoid]:

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

However, the "Bloch sphere" is 3-dimensional, so the steering ellipsoid of a two-qubit state can not be the (15-dimensional) John ellipsoid requested above.

Of course, the question what are the John ellipsoids can be asked for the convex sets of $m \times m$ symmetric and $n \times n$ Hermitian (positive-definite, trace 1) density matrices ($m,n \geq 2$). For $m,n=2$, the answers appear to be trivial, namely the convex sets themselves. For $m,n =3$, it seems possibly nontrivial. Only, however, for composite values of $m,n$, do we have subsidiary questions regarding the convex subsets of PPT-states.

The Wikipedia article given by the first hyperlink above describes the

"maximum volume inscribed ellipsoid as the inner Löwner–John ellipsoid".

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

[JohnTheorem]: Howard - The John ellipsoid theorem

[MasterLovasAndai]: Slater - Master Lovas–Andai and equivalent formulas verifying the $\frac8{33}$ two-qubit Hilbert–Schmidt separability probability and companion rational-valued conjectures

[SBZ]: Szarek, Bengtsson, and Życzkowski - On the structure of the body of states with positive partial transpose

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