dorsal/arxiv
View SchemaHall Normalization Constants for the Bures Volumes of the n-State Quantum Systems
| Authors | Paul B. Slater |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/9904101 |
| URL | https://arxiv.org/abs/quant-ph/9904101 |
| DOI | 10.1088/0305-4470/32/47/303 |
| Journal | J.Phys.A32:8231-8246,1999 |
Abstract
We report the results of certain integrations of quantum-theoretic interest, relying, in this regard, upon recently developed parameterizations of Boya et al of the n x n density matrices, in terms of squared components of the unit (n-1)-sphere and the n x n unitary matrices. Firstly, we express the normalized volume elements of the Bures (minimal monotone) metric for n = 2 and 3, obtaining thereby "Bures prior probability distributions" over the two- and three-state systems. Then, as an essential first step in extending these results to n > 3, we determine that the "Hall normalization constant" (C_{n}) for the marginal Bures prior probability distribution over the (n-1)-dimensional simplex of the n eigenvalues of the n x n density matrices is, for n = 4, equal to 71680/pi^2. Since we also find that C_{3} = 35/pi, it follows that C_{4} is simply equal to 2^{11} C_{3}/pi. (C_{2} itself is known to equal 2/pi.) The constant C_{5} is also found. It too is associated with a remarkably simple decompositon, involving the product of the eight consecutive prime numbers from 2 to 23. We also preliminarily investigate several cases, n > 5, with the use of quasi-Monte Carlo integration. We hope that the various analyses reported will prove useful in deriving a general formula (which evidence suggests will involve the Bernoulli numbers) for the Hall normalization constant for arbitrary n. This would have diverse applications, including quantum inference and universal quantum coding.
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"abstract": "We report the results of certain integrations of quantum-theoretic interest,\nrelying, in this regard, upon recently developed parameterizations of Boya et\nal of the n x n density matrices, in terms of squared components of the unit\n(n-1)-sphere and the n x n unitary matrices. Firstly, we express the normalized\nvolume elements of the Bures (minimal monotone) metric for n = 2 and 3,\nobtaining thereby \"Bures prior probability distributions\" over the two- and\nthree-state systems. Then, as an essential first step in extending these\nresults to n \u003e 3, we determine that the \"Hall normalization constant\" (C_{n})\nfor the marginal Bures prior probability distribution over the\n(n-1)-dimensional simplex of the n eigenvalues of the n x n density matrices\nis, for n = 4, equal to 71680/pi^2. Since we also find that C_{3} = 35/pi, it\nfollows that C_{4} is simply equal to 2^{11} C_{3}/pi. (C_{2} itself is known\nto equal 2/pi.) The constant C_{5} is also found. It too is associated with a\nremarkably simple decompositon, involving the product of the eight consecutive\nprime numbers from 2 to 23.\n We also preliminarily investigate several cases, n \u003e 5, with the use of\nquasi-Monte Carlo integration. We hope that the various analyses reported will\nprove useful in deriving a general formula (which evidence suggests will\ninvolve the Bernoulli numbers) for the Hall normalization constant for\narbitrary n. This would have diverse applications, including quantum inference\nand universal quantum coding.",
"arxiv_id": "quant-ph/9904101",
"authors": [
"Paul B. Slater"
],
"categories": [
"quant-ph",
"math-ph",
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],
"doi": "10.1088/0305-4470/32/47/303",
"journal_ref": "J.Phys.A32:8231-8246,1999",
"title": "Hall Normalization Constants for the Bures Volumes of the n-State Quantum Systems",
"url": "https://arxiv.org/abs/quant-ph/9904101"
},
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