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Consider the class $A$ of $6 \times 6$ positive definite matrices with real entries and unit trace (that is, the sum of the six diagonal entries is 1). (In quantum information-theoretic parlance, this is the class of "rebit-retrit density matrices".)

What is the twenty-dimensional Euclidean volume comprised by such matrices? This is the denominator of the ratio in which we are interested.

As to the numerator of the ratio, we want the twenty-dimensional Euclidean volume comprised by a certain subset $B$ of $A$. In addition to satisfying the conditions for membership in $A$, the "partial transposes" of such matrices must also be positive definite. To construct the partial transposes, one transposes in place the four $3 \times 3$ blocks of the rebit-retrit density matrices.

If one analogously considers the (9-dimensional) set of $4 \times 4$ "two-rebit density matrices" and the partial transposes of their $2 \times 2$ blocks, the desired ratio has been shown

Lovas-Andai1 Lovas-Andai2 Lovas-Andai3

to equal \begin{equation} \frac{29}{64} =\frac{29}{2^6} \approx 0.453125. \end{equation}

As to the presently-desired ratio in the $6 \times 6$ instance, the conjecture--based on numerical computations (RebitRetritConjecture)--of \begin{equation} \frac{860}{6561} = \frac{2^2 \cdot 5 \cdot 43}{3^8} \approx 0.131078 \end{equation} has been given.

These several ratios comprise what is termed "Hilbert-Schmidt separability probabilities".

As to the particular use of Mathematica in approaching this problem, I have considered the use of the GenericCylindricalDecomposition command (so far, with no successful outcomes). I have also considered the use of the positivity of the leading minors as a test for positive-definiteness. Perhaps the transformation of variables to such leading minors might be helpful.

Also, certain "simplified" forms of this daunting problem can be of interest--such as the 14-dimensional scenario in which the off-diagonal entries of the two diagonal (or two off-diagonal) $3 \times 3$ blocks are set to 0.

If one allows the off-diagonal entries of the matrices in question to be complex-valued, one moves to the realm of "two-qubit" and "qubit-qutrit" density matrices. In the 15-dimensional two-qubit case, the ratio (though not yet formally demonstrated to be such) is $\frac{8}{33}$. In the qubit-qutrit scenario, a conjecture of $\frac{27}{1000}$ has been stated.

According to eq. (7.7) \begin{equation} V^{(1)}_N = \frac{2^{\frac{1}{4} (N-1) N+N} \sqrt{N} \pi ^{\frac{1}{4} (N-1) N-\frac{1}{2}} \Gamma \left(\frac{N+1}{2}\right) \prod _{k=1}^N \Gamma \left(\frac{k}{2}+1\right)}{N! \Gamma \left(\frac{1}{2} N (N+1)\right)}. \end{equation} in Hilbert–Schmidt volume, with $N= 6$, the (Hilbert-Schmidt-valued) denominator of the ratio we seek should be \begin{equation} \frac{\pi ^9}{2252687044608000 \sqrt{3}} \approx \text{7.63989457197784$\grave{ }$*${}^{\wedge}$-12}. \end{equation} Alternatively, based on Lebesgue measure, the volume of the denominator should be (Theorem 1 in LebesgueVolume)

V[k_] :=  Pi^(k^2) (2 k)! Product[(2 i)!, {i, 1, 
 k - 1}]/(2^(k^2 + k) k! (2 k^2 + k - 1)!)

with $k=3$, that is, \begin{equation} \frac{\pi ^9}{1730063650258944000} \approx \text{1.7230059326999088$\grave{ }$*${}^{\wedge}$-14}. \end{equation}

The ratio of the latter (Lebesgue) volume to the former (Hilbert-Schmidt) one is \begin{equation} \frac{1}{256 \sqrt{3}}. \end{equation}

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  • $\begingroup$ Is it true that each volume individually is diverging? $\endgroup$
    – yarchik
    Jul 11 at 14:55
  • $\begingroup$ No, yarchik, the volumes are not diverging arxiv.org/abs/math-ph/0604032 $\endgroup$ Jul 11 at 15:00
  • $\begingroup$ yarchik--The "Hilbert-Schmidt metric" is defined by the line element squared--$\mbox{d}s^2_{HS}=(1/2) \mbox{Tr[}(d \rho)^2]$, where $\rho$ is a density matrix. This formula is (14.29) in the 2006 edition of "Geometry of Quantum States" by Zyczkowski and Bengtsson. I'm not sure if it is equivalently a "Haar measure". I just treat the problem in question as one in Euclidean/flat (Frobenius?) space. $\endgroup$ Jul 11 at 15:17
  • $\begingroup$ There are other possible measures of strong interest--in particular, the "Bures" (see the "Geometry of Quantum States" book--or the literature, in general). These are still more challenging to compute, it would seem. $\endgroup$ Jul 11 at 15:26

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