Numerical and exact analyses of Bures and Hilbert–Schmidt separability and PPT probabilities

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 \(\frac{25}{341} =\frac{5^2}{11 \cdot 31} \approx 0.073313783\). However, 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 \(\big (\frac{16}{12375} =\frac{4^2}{3^2 \cdot 5^3 \cdot 11}\) and \(\frac{625}{109531136}=\frac{5^4}{2^{12} \cdot 11^2 \cdot 13 \cdot 17}\big )\) 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–qutrit and rebit–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 \(1 -\frac{256}{27 \pi ^2}\).