dorsal/arxiv
View SchemaHilbert-Schmidt Geometry of n-Level Jakobczyk-Siennicki Two-Dimensional Quantum Systems
| Authors | Paul B. Slater |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0508227 |
| URL | https://arxiv.org/abs/quant-ph/0508227 |
Abstract
Jakobczyk and Siennicki studied two-dimensional sections of a set of (generalized) Bloch vectors corresponding to n x n density matrices of two-qubit systems (that is, the case n = 4). They found essentially five different types of (nontrivial) separability regimes. We compute the Euclidean/Hilbert-Schmidt (HS) separability probabilities assigned to these regimes, and conduct parallel two-dimensional sectional analyses for the higher-level cases n=6,8,9 and 10. Making use of the newly-introduced capability for integration over implicitly defined regions of version 5.1 of Mathematica -- also fruitfully used in our n=4 three-parameter entropy-maximization-based study quant-ph/0507203 -- we obtain a wide-ranging variety of exact HS-probabilities. For n>6, the probabilities are those of having a partial positive transpose (PPT). For the n=6 case, we also obtain biseparability probabilities; in the n=8,9 instances, bi-PPT probabilities; and for n=8, tri-PPT probabilities. By far, the most frequently recorded probability for n>4 is \pi/4 = 0.785398. We also conduct a number of related analyses, pertaining to the (one-dimensional) boundaries (both exterior and interior) of the separability and PPT domains, and attempt (with limited success) some exact calculations pertaining to the 9-dimensional (real) and 15-dimensional (complex) convex sets of two-qubit density matrices -- for which HS-separability probabilities have been conjectured, but not verified.
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"abstract": "Jakobczyk and Siennicki studied two-dimensional sections of a set of\n(generalized) Bloch vectors corresponding to n x n density matrices of\ntwo-qubit systems (that is, the case n = 4). They found essentially five\ndifferent types of (nontrivial) separability regimes. We compute the\nEuclidean/Hilbert-Schmidt (HS) separability probabilities assigned to these\nregimes, and conduct parallel two-dimensional sectional analyses for the\nhigher-level cases n=6,8,9 and 10. Making use of the newly-introduced\ncapability for integration over implicitly defined regions of version 5.1 of\nMathematica -- also fruitfully used in our n=4 three-parameter\nentropy-maximization-based study quant-ph/0507203 -- we obtain a wide-ranging\nvariety of exact HS-probabilities. For n\u003e6, the probabilities are those of\nhaving a partial positive transpose (PPT). For the n=6 case, we also obtain\nbiseparability probabilities; in the n=8,9 instances, bi-PPT probabilities; and\nfor n=8, tri-PPT probabilities. By far, the most frequently recorded\nprobability for n\u003e4 is \\pi/4 = 0.785398. We also conduct a number of related\nanalyses, pertaining to the (one-dimensional) boundaries (both exterior and\ninterior) of the separability and PPT domains, and attempt (with limited\nsuccess) some exact calculations pertaining to the 9-dimensional (real) and\n15-dimensional (complex) convex sets of two-qubit density matrices -- for which\nHS-separability probabilities have been conjectured, but not verified.",
"arxiv_id": "quant-ph/0508227",
"authors": [
"Paul B. Slater"
],
"categories": [
"quant-ph"
],
"title": "Hilbert-Schmidt Geometry of n-Level Jakobczyk-Siennicki Two-Dimensional Quantum Systems",
"url": "https://arxiv.org/abs/quant-ph/0508227"
},
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