dorsal/arxiv
View SchemaOn Approximately Symmetric Informationally Complete Positive Operator-Valued Measures and Related Systems of Quantum States
| Authors | Andreas Klappenecker, Martin Roetteler, Igor Shparlinski, Arne Winterhof |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0503239 |
| URL | https://arxiv.org/abs/quant-ph/0503239 |
| DOI | 10.1063/1.1998831 |
| Journal | Journal of Mathematical Physics, 46:082104, 2005 |
Abstract
We address the problem of constructing positive operator-valued measures (POVMs) in finite dimension $n$ consisting of $n^2$ operators of rank one which have an inner product close to uniform. This is motivated by the related question of constructing symmetric informationally complete POVMs (SIC-POVMs) for which the inner products are perfectly uniform. However, SIC-POVMs are notoriously hard to construct and despite some success of constructing them numerically, there is no analytic construction known. We present two constructions of approximate versions of SIC-POVMs, where a small deviation from uniformity of the inner products is allowed. The first construction is based on selecting vectors from a maximal collection of mutually unbiased bases and works whenever the dimension of the system is a prime power. The second construction is based on perturbing the matrix elements of a subset of mutually unbiased bases. Moreover, we construct vector systems in $\C^n$ which are almost orthogonal and which might turn out to be useful for quantum computation. Our constructions are based on results of analytic number theory.
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"abstract": "We address the problem of constructing positive operator-valued measures\n(POVMs) in finite dimension $n$ consisting of $n^2$ operators of rank one which\nhave an inner product close to uniform. This is motivated by the related\nquestion of constructing symmetric informationally complete POVMs (SIC-POVMs)\nfor which the inner products are perfectly uniform. However, SIC-POVMs are\nnotoriously hard to construct and despite some success of constructing them\nnumerically, there is no analytic construction known. We present two\nconstructions of approximate versions of SIC-POVMs, where a small deviation\nfrom uniformity of the inner products is allowed. The first construction is\nbased on selecting vectors from a maximal collection of mutually unbiased bases\nand works whenever the dimension of the system is a prime power. The second\nconstruction is based on perturbing the matrix elements of a subset of mutually\nunbiased bases.\n Moreover, we construct vector systems in $\\C^n$ which are almost orthogonal\nand which might turn out to be useful for quantum computation. Our\nconstructions are based on results of analytic number theory.",
"arxiv_id": "quant-ph/0503239",
"authors": [
"Andreas Klappenecker",
"Martin Roetteler",
"Igor Shparlinski",
"Arne Winterhof"
],
"categories": [
"quant-ph",
"cs.ET"
],
"doi": "10.1063/1.1998831",
"journal_ref": "Journal of Mathematical Physics, 46:082104, 2005",
"title": "On Approximately Symmetric Informationally Complete Positive Operator-Valued Measures and Related Systems of Quantum States",
"url": "https://arxiv.org/abs/quant-ph/0503239"
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