# How does one generate a random $N \times N$ density matrix with quaternionic entries — with respect to Hilbert-Schmidt measure?

In the article

the formula (eq. (1)) $$\rho_{HS} =\frac{A A^{\dagger}}{\mbox{Tr} A A^{\dagger}}$$ is given, for generating a random $N \times N$ density matrix ($\rho_{HS}$) with complex off-diagonal entries with respect to Hilbert-Schmidt (HS) measure. ("The HS measure is defined by the HS metric $\mbox{d}s^2_{HS} =\frac{1}{2} \mbox{Tr}[(\mbox{d}\rho)^2]$, which is distinguished by the fact that it induces the flat, Euclidean geometry into the set of mixed states." A density matrix is a positive semidefinite, hermitian matrix of trace 1.)

Here $A$ is a square complex random matrix of size N pertaining to the Ginibre ensemble (with real and imaginary parts of each element being independent normal random variables).

Now, if one similarly wants a random $N \times N$ density matrix with real off-diagonal entries, the matrix $A$ (with each element now being an independent normal random variable) is chosen to be of dimensions $N \times (N+1)$ (bot. p. 9 of indicated article).

So, I want to find the appropriate analogous procedure to employ when the off-diagonal entries of the density matrix are quaternions.

I've been investigating this question with $N=4$, where I anticipate the associated "separability probability" will be $\frac{26}{323}$. (In the real case, this Hilbert-Schmidt probability has recently been proven by Lovas and Andai [https://arxiv.org/pdf/1610.01410.pdf] to be $\frac{29}{64}$ and, in the complex case to be $\frac{8}{33}$ in https://arxiv.org/pdf/1701.01973.pdf . For the $\frac{26}{323}$ assertion, see http://iopscience.iop.org.proxy.library.ucsb.edu:2048/article/10.1088/1751-8113/46/44/445302/pdf .)

I don't seem to be able to succeed numerically getting near this $\frac{26}{323}$ value by choosing $A$ to have similarly random quaternionic entries and be of dimension $4 \times M$, where $M$ is some integer (3, 4, 5...). To test separability, one needs to evaluate whether the partial transpose of the density matrix is positive definite. This test can be accomplished in the $4 \times 4$ case by testing the positivity of the determinant of the partial transpose. In the (non-commutative) quaternionic case, I've been using what is termed the "Moore determinant" [Bull. Amer. Math. Soc 28 (1922): 161-162] for this purpose.

Perhaps it may be necessary to convert the $4 \times 4$ matrices A with quaternionic entries to $8 \times 8$ complex form (using the well-known quaternion-to-Pauli matrices conversion rules) and proceed using the formula above. Then, possibly convert back to $4 \times 4$ quaternionic form.

In these last regards, I've posted a Mathematica question

https://mathematica.stackexchange.com/q/148841/

concerning how to efficiently go back and forth between these $4 \times 4$ and $8 \times 8$ structures.

• Could you supply a source for the claim that the positivity of the determinant of the partial transpose is a test for separability? Also, is it known that the Peres-Horodecki condition applies to quaternionic 4x4 matrices? – Joel Klassen Jun 23 '17 at 13:34
• Thanks, Joel Klassen. As to your first question, see eq. (2) in the paper "Universal observable detecting all two-qubit entanglement and determinant-based separability tests", journals.aps.org/pra/pdf/10.1103/PhysRevA.77.030301. As to your second highly relevant question, I presently plead ignorance, but will further investigate/cogitate. Perhaps I should incorporate these points into the original question. – Paul B. Slater Jun 23 '17 at 15:38
• As to the second question of Joel Klassen, see sec. 4 of the paper of Fei and Joynt arxiv.org/pdf/1409.1993.pdf , also ac.els cdn.com.proxy.library.ucsb.edu:2048/S0034487716300611/1-s2.0-S0034487716300611-main.pdf?_tid=52a523e8-58e2-11e7-8918-00000aab0f02&acdnat=1498311653_54b2d1c34e78785661eaa556271bccfd – Paul B. Slater Jun 24 '17 at 13:55
• Thanks Paul B. Slater! This all sounds quite interesting, I wish I could be of more help to you. – Joel Klassen Jun 25 '17 at 15:34
• Well, as a research project of current interest, I could suggest using the (computationally-intensive) Fei-Joynt procedure to generate random two-quaterbit density matrices, and using them to test the conjecture that the corresponding “Lovas-Andai function” $\tilde{\chi_4}(\varepsilon)$ is $\varepsilon^4 (2-\varepsilon^4)$—a function which, in the Lovas-Andai framework, yields the presumed \frac{26}{323}\$ Hilbert-Schmidt separability probability. (See the two cited preprints in the statement of the question for more details.) – Paul B. Slater Jun 26 '17 at 13:54