dorsal/arxiv
View SchemaA priori probability that two qubits are unentangled
| Authors | Paul B. Slater |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0207181 |
| URL | https://arxiv.org/abs/quant-ph/0207181 |
| Journal | Quantum Info. Proc. vol. 1, no. 5, Oct. 2002, pp. 397-408 |
Abstract
In a previous study (quant-ph/9911058), several remarkably simple exact results were found, in certain specialized m-dimensional scenarios (m<5), for the a priori probability that a pair of qubits is unentangled/separable. The measure used was the volume element of the Bures metric (identically one-fourth the statistical distinguishability [SD] metric). Here, making use of a newly-developed (Euler angle) parameterization of the 4 x 4 density matrices (math-ph/0202002), we extend the analysis to the complete 15-dimensional convex set (C) of arbitrarily paired qubits -- the total SD volume of which is known to be \pi^8 / 1680 = \pi^8 / (2^4 3 5 7) = 5.64794. Using advanced quasi-Monte Carlo procedures (scrambled Halton sequences) for numerical integration in this high-dimensional space, we approximately (5.64851) reproduce that value, while obtaining an estimate of .416302 for the SD volume of separable states. We conjecture that this is but an approximation to \pi^6 /2310 = \pi^6 / (2 3 5 7 11) = .416186. The ratio of the two volumes, 8 / (11 \pi^2) = .0736881, would then constitute the exact Bures/SD probability of separability. The SD area of the 14-dimensional boundary of C is 142 \pi^7 / 12285 = 142 \pi^7 /(3^3 5 7 13) = 34.911, while we obtain a numerical estimate of 1.75414 for the SD area of the boundary of separable states.
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"abstract": "In a previous study (quant-ph/9911058), several remarkably simple exact\nresults were found, in certain specialized m-dimensional scenarios (m\u003c5), for\nthe a priori probability that a pair of qubits is unentangled/separable. The\nmeasure used was the volume element of the Bures metric (identically one-fourth\nthe statistical distinguishability [SD] metric). Here, making use of a\nnewly-developed (Euler angle) parameterization of the 4 x 4 density matrices\n(math-ph/0202002), we extend the analysis to the complete 15-dimensional convex\nset (C) of arbitrarily paired qubits -- the total SD volume of which is known\nto be \\pi^8 / 1680 = \\pi^8 / (2^4 3 5 7) = 5.64794. Using advanced quasi-Monte\nCarlo procedures (scrambled Halton sequences) for numerical integration in this\nhigh-dimensional space, we approximately (5.64851) reproduce that value, while\nobtaining an estimate of .416302 for the SD volume of separable states. We\nconjecture that this is but an approximation to \\pi^6 /2310 = \\pi^6 / (2 3 5 7\n11) = .416186. The ratio of the two volumes, 8 / (11 \\pi^2) = .0736881, would\nthen constitute the exact Bures/SD probability of separability. The SD area of\nthe 14-dimensional boundary of C is 142 \\pi^7 / 12285 = 142 \\pi^7 /(3^3 5 7 13)\n= 34.911, while we obtain a numerical estimate of 1.75414 for the SD area of\nthe boundary of separable states.",
"arxiv_id": "quant-ph/0207181",
"authors": [
"Paul B. Slater"
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"journal_ref": "Quantum Info. Proc. vol. 1, no. 5, Oct. 2002, pp. 397-408",
"title": "A priori probability that two qubits are unentangled",
"url": "https://arxiv.org/abs/quant-ph/0207181"
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