We employ a quasirandom methodology, recently developed by Martin Roberts, to estimate the separability probabilities, with respect to the Bures (minimal monotone/statistical distinguishability) measure, of generic two-qubit and two-rebit states. This procedure, based on generalized properties of the golden ratio, yielded, in the course of almost seventeen billion iterations (recorded at intervals of five million), two-qubit estimates repeatedly close to nine decimal places to $\frac{ 25}{ 341} =\frac{ 5^2}{ 11 \cdot 31} \approx 0.073313783$. Howeer, despite the use of over twenty-three billion iterations, we do not presently perceive an exact value (rational or otherwise) for an estimate of 0.15709623 for the Bures two-rebit separability probability. The Bures qubit-qutrit case--for which Khvedelidze and Rogojin gave an estimate of 0.0014--is analyzed too. The value of $\frac{ 1}{ 715}=\frac{ 1}{ 5 \cdot 11 \cdot 13} \approx 0.00139860$ is a well-fitting value to an estimate of 0.00139884. Interesting values ($\frac{ 16}{ 12375} =\frac{ 4^2}{ 3^2 \cdot 5^3 \cdot 11}$ and $\frac{ 625}{ 109531136}=\frac{ 5^4}{ 2^{ 12} \cdot 11^2 \cdot 13 \cdot 17}$) are conjectured for the Hilbert-Schmidt (HS) and Bures qubit-qudit ($2 \times 4$) positive-partial-transpose (PPT)-probabilities. We re-examine, strongly supporting, conjectures that the HS qubit-{ \it qutrit} and rebit-{ \it retrit} separability probabilities are $\frac{ 27}{ 1000}=\frac{ 3^3}{ 2^3 \cdot 5^3}$ and $\frac{ 860}{ 6561}= \frac{ 2^2 \cdot 5 \cdot 43}{ 3^8}$, respectively. Prior studies have demonstrated that the HS two-rebit separability probability is $\frac{ 29}{ 64}$ and strongly pointed to the HS two-qubit counterpart being $\frac{ 8}{ 33}$, and a certain operator monotone one (other than the Bures) being $1 -\frac{ 256}{ 27 \pi^2}$.
Categories: quant-ph math-ph math.MP math.PR physics.comp-ph