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

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 $\frac{25}{341}=\frac{5^2}{11\cdot 31}\approx 0.073313783$. 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 $1-\frac{256}{27{\pi }^2}$.

24 pages, 9 figures--substantially revised per referees' comments. arXiv admin note: text overlap with arXiv:1809.09040

References in corpus (12)

• Monte Carlo sampling from the quantum state space. II
• Monte Carlo sampling from the quantum state space. I
• Dyson Indices and Hilbert-Schmidt Separability Functions and Probabilities
• Bures-Hall Ensemble: Spectral Densities and Average Entropies
• Experimental Data from a Quantum Computer Verifies the Generalized Pauli Exclusion Principle
• Volumes of conditioned bipartite state spaces
• Extended Studies of Separability Functions and Probabilities and the Relevance of Dyson Indices
• The Volume of Two-Qubit States by Information Geometry
• Extensions of Generalized Two-Qubit Separability Probability Analyses to Higher Dimensions, Additional Measures and New Methodologies
• Investigating bound entangled two-qutrit states via the best separable approximation
• On the generation of random ensembles of qubits and qutrits Computing separability probabilities for fixed rank states
• Geometric quantum discord for two-qubit X-states

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• Investigating bound entangled two-qutrit states via the best separable approximation
• Quasirandom estimations of two-qubit operator-monotone-based separability probabilities