Let $A$ be an $n \times n$ (Ginibre) matrix of complex-valued entries, the real and imaginary parts of which are, thus, standard normal variates, and $U$ be an independent such matrix, the rows and columns of which have been subsequently orthogonalized (so that $U$ is unitary).

Then, the formula, (eq. (24) in https://arxiv.org/abs/0909.5094) with $y=1-x$,
\begin{equation} \label{JointBuresHSformula} \rho_x= \frac{(y \mathbb{I} +x U) AA^{\dagger}(y \mathbb{I}+x U^{\dagger})}{\mbox{Tr} (y \mathbb{I}+x U) A A^{\dagger} (y \mathbb{I} +x U^{\dagger})}, \end{equation} Is of considerable quantum information-theoretic interest.

Let $n=4$. Then, for $x=0$, $\rho_x$ is a random $4 \times 4$ ("two-qubit") density matrix with respect to Hilbert-Schmidt measure, and for $x=\frac{1}{2}$, with respect to Bures ("minimal monotone") measure.

Now, one can compute the partial transpose ($\rho_x^{PT}$) of such a matrix, by transposing in place its four $2 \times 2$ blocks. If the determinant of the partial transpose is positive, then $\rho$ is designated as separable (disentangled). (In this particular case, the partial transpose can have at most one negative eigenvalue, with $\rho_x$ having none.)

A considerable body of evidence ( https://arxiv.org/abs/1701.01973) supports the proposition that the Hilbert-Schmidt ($x=0$) “separability probability” is $\frac{8}{33}$—while for another (non-Bures) monotone measure, it is $1-\frac{256}{27 \pi^2}$.

Since the Bures measure itself is of fundamental interest, it is an appealing conjecture to explore the possibility that the Bures two-qubit separability probability also assumes some strikingly elegant form.

However, the volume formula for the Bures measure is of a more complicated form that so far has resisted parallel analyses. So, it appears that we must presently have recourse to numerical methods.

Using the RandomVariate[NormalDistribution[]] command of Mathematica, in conjunction with the formula above with $x= \frac{1}{2}$, 4,372,000,000 realizations were generated, of which 320,546,752 proved to be separable, giving us an estimate of the Bures separability probability of $\frac{5008543}{683125000} \approx 0.0733181$ (p. 19 of https://arxiv.org/abs/1809.09040).

Now, alternatively the interesting contribution of Martin Roberts in https://math.stackexchange.com/questions/2231391/how-can-one-generate-an-open-ended-sequence-of-low-discrepancy-points-in-3d/2845473#2845473 led us to consider (Can/should one generate Ginibre ensembles of random matrices using low-discrepancy normal variates--and if so. how?), in the hope of stronger convergence, the use of his newly-developed open-ended (golden-ratio-related) algorithm for quasirandom sequences over [0,1], which we have been exploring (sec. XI of https://arxiv.org/abs/1809.09040). The $n$-th term in the quasirandom (Korobov) sequence, where $\phi_D$ is the smallest, positive real-value of x such that \begin{equation} x^{D+1}=x+1, \end{equation} ($D=1$, yielding the golden ratio, and $D=2$, the ``plastic constant'') takes the form $(\alpha _0+n \vec{\alpha}) \bmod 1, n = 1, 2, 3, \ldots$,

where we have the $D$-dimensional vector, \begin{equation} \vec{\alpha} =(\frac{1}{\phi_D},\frac{1}{\phi_D^2},\frac{1}{\phi_D^3},\ldots,\frac{1}{\phi_D^D}). \end{equation} The additive constant $\alpha_0$ is typically taken to be 0. ``However, there are some arguments, relating to symmetry, that suggest that $\alpha_0=\frac{1}{2}$ is a better choice,''

I’d been pursuing this approach by generating numerical vectors of length 64 (and using the inverse cumulative function to get the normal variate counterparts—see the extended discussion concerning the use of the Mathematica command InverseCDF in https://mathematica.stackexchange.com/questions/181099/can-i-use-compile-to-speed-up-inversecdf). The length 64 is based on the observation that the matrices $A$ and $U$ in the formula above both require 32 normal variates for their generation.

However, in the penultimate comment in https://math.stackexchange.com/questions/2231391/how-can-one-generate-an-open-ended-sequence-of-low-discrepancy-points-in-3d/2845473#2845473 , Roberts advises: “Definitely do not use the first 32 components of a 64D sequence for Matrix A, and the last 32 components for Matrix B, as the components are highly correlated. I do not think that scrambling improves anything. Thus, the most reasonable (but still not ideal) solution is to use two independent 32D sequences…”

So, my question is: Can I still effectively work with 64D sequences if I randomly permute them (I take it that “scrambling” $\neq$ “randomly permuting”), using say the RandomSample command of Mathematica—and then only “use the first 32 components of [the permuted] 64D sequence for Matrix A, and the last 32 components for Matrix B”? (I have initially explored the two [nonpermuted] 32D sequence option suggested by Roberts—with so far inconclusive results. Working with simply a single 64D sequence has a certain elegance/simplicity, however, it would seem.)

Let me conclude by indicating that there is an analogous problem for $4 \times 4$ (“two-rebit”) density matrices $\rho_x$ with real (as opposed to complex-valued) entries, that is also under investigation. (The Hilbert-Schmidt separability probability in that case has been formally proven by Lovas and Andai [https://arxiv.org/abs/1610.01410] to be $\frac{29}{64}$.) Then, rather than 64D sequences, the use of 36D sequences is indicated, with the Ginibre matrix $A$ now being $4 \times 5$ (sec. 4 of https://arxiv.org/abs/0909.5094 ).


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