Rational-Valued, Small-Prime-Based Qubit-Qutrit and Rebit-Retrit Rank-4/Rank-6 Conjectured Hilbert-Schmidt Separability Probability Ratios

TL;DR

This paper estimates Hilbert-Schmidt separability probabilities for low-rank quantum states using a Wishart-Laguerre based random matrix generation, proposing conjectures for specific ratios and probabilities in rebit-retrit systems.

Contribution

It introduces a procedure for generating random density matrices of specific ranks and conjectures exact separability probabilities and ratios for rebit-retrit systems, supported by extensive numerical simulations.

Findings

Estimated separability probability for rank-4 rebit-retrit states: approximately 0.00774.

Conjectured full-rank rebit-retrit separability probability: 860/6561 (~0.1311).

Proposed ratio of rank-4 to rank-6 probabilities: 59049/1000000 (~0.05905).

Abstract

We implement a procedure-based on the Wishart-Laguerre distribution-recently outlined by {\.Z}yczkowski and Khvedelidze, Rogojin and Abgaryan, for the generation of random (complex or real) $N \times N$ density matrices of rank $k \leq N$ with respect to Hilbert-Schmidt (HS) measure. In the complex case, one commences with a Ginibre matrix $A$ of dimensions $k \times k+ 2 (N-k)$, while for a real scenario, one employs a Ginibre matrix $B$ of dimensions $k \times k+1+ 2 (N-k)$. Then, the $k \times k$ product $A A^{\dagger}$ or $B B^T$ is diagonalized-padded with zeros to size $N \times N$-and rotated, obtaining a random density matrix. Implementing the procedure for rank-4 rebit-retrit states, for 800 million Ginibre-matrix realizations, 6,192,047 were found separable, for a sample probability of .00774006-suggestive of an exact value $\frac{387}{5000} =\frac{3^2 \cdot 43}{2^3 \cdot 5^4}=.0774$. A conjecture for the HS separability probability of rebit-retrit systems of full rank is $\frac{860}{6561} =\frac{2^2 \cdot 5 \cdot 43}{3^8} \approx 0.1310775$ (the two-rebit counterpart has been proven to be $\frac{29}{64}=\frac{29}{2^6}$). Subject to these conjectures, the ratio of the rank-4 to rank-6 probabilities would be $\frac{59049}{1000000}=\frac{3^{10}}{2^6 \cdot 5^6} \approx 0.059049$, with the common factor 43 cancelling. As to the intermediate rank-5 probability, a 2006 theorem of Szarek, Bengtsson and {\.Z}ycskowski informs us that it must be one-half the rank-6 probability-itself conjectured to be $\frac{27}{1000} =\frac{3^3}{2^3 \cdot 5^3}$, while for rank 3 or less, the associated probabilities must be 0 by a 2009 result of Ruskai and Werner. We are led to re-examine a 2005 qubit-qutrit analysis of ours, in these regards, and now find evidence for a $\frac{70}{2673}=\frac{2 \cdot 5 \cdot 7}{ 3^5 \cdot 11} \approx 0.0261878$ rank-4 to rank-6 probability ratio.