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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

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    $\begingroup$ Thanks--LSpice! $\endgroup$ – Paul B. Slater Aug 16 at 17:12
  • $\begingroup$ They are not centrally symmetric. $\endgroup$ – Fedor Petrov Aug 16 at 18:10
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Let us begin with two apparently relevant formulas. The first is for the volume of a $k$-dimensional ellipsoid [Thm. 2.1, EllipsoidVolume], \begin{equation} vol_k=\frac{2 \pi ^{k/2} \prod _{i=1}^k a_i}{k \Gamma \left(\frac{k}{2}\right)}, \end{equation} where the $a_i$’s are the lengths of the semi-axes.

The other is for the volume of the set of $m \times m$ symmetric, positive-definite matrices of trace 1 [(7.7), RebitVolume]. \begin{equation} Vol_m=\frac{2^{\frac{1}{4} (m-1) m+m} \sqrt{m} \pi ^{\frac{1}{4} (m-1) m-\frac{1}{2}} \Gamma \left(\frac{m+1}{2}\right) \prod _{l=1}^m \Gamma \left(\frac{l}{2}+1\right)}{m! \Gamma \left(\frac{1}{2} m (m+1)\right)}. \end{equation}

For the (“two-rebit”) case $m=4$ ($k=9$) of immediate interest, the formula yields \begin{equation} \frac{\pi ^4}{60480} \approx 0.0016106. \end{equation}

So, the question of particular interest to us is what proportion of this volume is occupied by the inner Lowner-John ellipsoid for the convex set of the indicated 9-dimensional set of $4 \times 4$ (density) matrices. Further, what is its magnitude in comparison with $\frac{29}{64}$, the fraction established by Lovas and Andai for the separability—equivalently, PPT—probability of the two-rebit states? Also, in comparison with the volume of the insphere (for which we have no immediate present computation).

So, to approach these questions, we generated pairs of randomly-generated “two-rebit density matrices” (sec, 4, RandomDensityMatrices), using Ginibre-ensemble methods. Then, we took the absolute values of their differences and divided by 2. Nine independent entries (three diagonal ones, and the six upper off-diagonal ones) of the resultant matrix, were taken as the semi-axes.

At this point in time, we have generated close to sixteen million such pairs. The pair of $4 \times 4$ density matrices for which we have found the associated maximum ellipsoid volume, $6.98613 \cdot 10^{-8}$ (only 0.0000432642 of $\frac{\pi ^4}{60480} \approx 0.0016106$), so far are \begin{equation} \left( \begin{array}{cccc} 0.424772 & -0.147161 & -0.3345 & -0.177458 \\ -0.147161 & 0.164668 & 0.146384 & 0.0925659 \\ -0.3345 & 0.146384 & 0.29387 & 0.157489 \\ -0.177458 & 0.0925659 & 0.157489 & 0.11669 \\ \end{array} \right) \end{equation} and \begin{equation} \left( \begin{array}{cccc} 0.135144 & 0.189631 & -0.03164 & 0.145386 \\ 0.189631 & 0.449171 & -0.180868 & 0.347037 \\ -0.03164 & -0.180868 & 0.126351 & -0.128246 \\ 0.145386 & 0.347037 & -0.128246 & 0.289334 \\ \end{array} \right). \end{equation} One-half of the absolute differences for these two matrices of the leading three diagonal entries and the upper six off-diagonal entries are used as the nine semi-axes in the first formula given above.

Let us also point out that there is an alternative—but equivalent up to certain normalization factors—approach to computing the volumes of $m \times m$ density matrices (AndaiVolume). Andai, however, restricted attention to the $2 \times 2$ Hermitian case, and did not give an explicit alternative to the volume formula of Zyczkowski and Sommers presented above--so, at this point in time, we are not sure of what form it would take.

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