Numerical and Exact Analyses of Bures and Hilbert-Schmidt Separability and PPT-Probabilities

DOI10.1007/S11128-019-2431-2arXiv1901.09889WikidataQ125930174 ScholiaQ125930174MaRDI QIDQ6313188

Author name not available (Why is that?)

Publication date: 25 January 2019

Abstract: 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

f r a c 25341 = f r a c 5^{2} 11 c d o t 31 a p p r o x 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

f r a c 1715 = f r a c 1 5 c d o t 11 c d o t 13 a p p r o x 0.00139860

is a well-fitting value to an estimate of 0.00139884. Interesting values (

f r a c 1612375 = f r a c 4^{2} 3^{2} c d o t 5^{3} c d o t 11

and

f r a c 625109531136 = f r a c 5^{4} 2^{12} c d o t 1 1^{2} c d o t 13 c d o t 17

) are conjectured for the Hilbert-Schmidt (HS) and Bures qubit-qudit (

2 i m e s 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

f r a c 271000 = f r a c 3^{3} 2^{3} c d o t 5^{3}

and

f r a c 8606561 = f r a c 2^{2} c d o t 5 c d o t 43 3^{8}

, respectively. Prior studies have demonstrated that the HS two-rebit separability probability is

f r a c 2964

and strongly pointed to the HS two-qubit counterpart being

f r a c 833

, and a certain operator monotone one (other than the Bures) being

1 - f r a c 256 27 p i^{2}

.

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