dorsal/arxiv
View SchemaTwo-Qubit Separability Probabilities and Beta Functions
| Authors | Paul B. Slater |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0609006 |
| URL | https://arxiv.org/abs/quant-ph/0609006 |
| DOI | 10.1103/PhysRevA.75.032326 |
| Journal | Phys. Rev. A 75, 032326 (March, 2007) (7 pages) |
Abstract
Due to recent important work of Zyczkowski and Sommers (quant-ph/0302197 and quant-ph/0304041), exact formulas are available (both in terms of the Hilbert-Schmidt and Bures metrics) for the (n^2-1)-dimensional and (n(n-1)/2-1)-dimensional volumes of the complex and real n x n density matrices. However, no comparable formulas are available for the volumes (and, hence, probabilities) of various separable subsets of them. We seek to clarify this situation for the Hilbert-Schmidt metric for the simplest possible case of n=4, that is, the two-qubit systems. Making use of the density matrix (rho) parameterization of Bloore (J. Phys. A 9, 2059 [1976]), we are able to reduce each of the real and complex volume problems to the calculation of a one-dimensional integral, the single relevant variable being a certain ratio of diagonal entries, nu = (rho_{11} rho_{44})/{rho_{22} rho_{33})$. The associated integrand in each case is the product of a known (highly oscillatory near nu=1) jacobian and a certain unknown univariate function, which our extensive numerical (quasi-Monte Carlo) computations indicate is very closely proportional to an (incomplete) beta function B_{nu}(a,b), with a=1/2, b=sqrt{3}in the real case, and a=2 sqrt{6}/5, b =3/sqrt{2} in the complex case. Assuming the full applicability of these specific incomplete beta functions, we undertake separable volume calculations.
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"abstract": "Due to recent important work of Zyczkowski and Sommers (quant-ph/0302197 and\nquant-ph/0304041), exact formulas are available (both in terms of the\nHilbert-Schmidt and Bures metrics) for the (n^2-1)-dimensional and\n(n(n-1)/2-1)-dimensional volumes of the complex and real n x n density\nmatrices. However, no comparable formulas are available for the volumes (and,\nhence, probabilities) of various separable subsets of them. We seek to clarify\nthis situation for the Hilbert-Schmidt metric for the simplest possible case of\nn=4, that is, the two-qubit systems. Making use of the density matrix (rho)\nparameterization of Bloore (J. Phys. A 9, 2059 [1976]), we are able to reduce\neach of the real and complex volume problems to the calculation of a\none-dimensional integral, the single relevant variable being a certain ratio of\ndiagonal entries, nu = (rho_{11} rho_{44})/{rho_{22} rho_{33})$. The associated\nintegrand in each case is the product of a known (highly oscillatory near nu=1)\njacobian and a certain unknown univariate function, which our extensive\nnumerical (quasi-Monte Carlo) computations indicate is very closely\nproportional to an (incomplete) beta function B_{nu}(a,b), with a=1/2,\nb=sqrt{3}in the real case, and a=2 sqrt{6}/5, b =3/sqrt{2} in the complex case.\nAssuming the full applicability of these specific incomplete beta functions, we\nundertake separable volume calculations.",
"arxiv_id": "quant-ph/0609006",
"authors": [
"Paul B. Slater"
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"doi": "10.1103/PhysRevA.75.032326",
"journal_ref": "Phys. Rev. A 75, 032326 (March, 2007) (7 pages)",
"title": "Two-Qubit Separability Probabilities and Beta Functions",
"url": "https://arxiv.org/abs/quant-ph/0609006"
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