dorsal/arxiv
View SchemaMutually unbiased phase states, phase uncertainties, and Gauss sums
| Authors | M. Planat, H. C. Rosu |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0506128 |
| URL | https://arxiv.org/abs/quant-ph/0506128 |
| DOI | 10.1140/epjd/e2005-00208-4 |
| Journal | Eur. Phys. J. D 36, 133-139 (Oct. 2005) |
Abstract
Mutually unbiased bases (MUBs), which are such that the inner product between two vectors in different orthogonal bases is a constant equal to 1/sqrt{d), with d the dimension of the finite Hilbert space, are becoming more and more studied for applications such as quantum tomography and cryptography, and in relation to entangled states and to the Heisenberg-Weil group of quantum optics. Complete sets of MUBs of cardinality d+1 have been derived for prime power dimensions d=p^m using the tools of abstract algebra. Presumably, for non prime dimensions the cardinality is much less. Here we reinterpret MUBs as quantum phase states, i.e. as eigenvectors of Hermitean phase operators generalizing those introduced by Pegg & Barnett in 1989. We relate MUB states to additive characters of Galois fields (in odd characteristic p) and to Galois rings (in characteristic 2). Quantum Fourier transforms of the components in vectors of the bases define a more general class of MUBs with multiplicative characters and additive ones altogether. We investigate the complementary properties of the above phase operator with respect to the number operator. We also study the phase probability distribution and variance for general pure quantum electromagnetic states and find them to be related to the Gauss sums, which are sums over all elements of the field (or of the ring) of the product of multiplicative and additive characters. Finally, we relate the concepts of mutual unbiasedness and maximal entanglement. This allows to use well studied algebraic concepts as efficient tools in the study of entanglement and its information aspects
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"abstract": "Mutually unbiased bases (MUBs), which are such that the inner product between\ntwo vectors in different orthogonal bases is a constant equal to 1/sqrt{d),\nwith d the dimension of the finite Hilbert space, are becoming more and more\nstudied for applications such as quantum tomography and cryptography, and in\nrelation to entangled states and to the Heisenberg-Weil group of quantum\noptics. Complete sets of MUBs of cardinality d+1 have been derived for prime\npower dimensions d=p^m using the tools of abstract algebra. Presumably, for non\nprime dimensions the cardinality is much less. Here we reinterpret MUBs as\nquantum phase states, i.e. as eigenvectors of Hermitean phase operators\ngeneralizing those introduced by Pegg \u0026 Barnett in 1989. We relate MUB states\nto additive characters of Galois fields (in odd characteristic p) and to Galois\nrings (in characteristic 2). Quantum Fourier transforms of the components in\nvectors of the bases define a more general class of MUBs with multiplicative\ncharacters and additive ones altogether. We investigate the complementary\nproperties of the above phase operator with respect to the number operator. We\nalso study the phase probability distribution and variance for general pure\nquantum electromagnetic states and find them to be related to the Gauss sums,\nwhich are sums over all elements of the field (or of the ring) of the product\nof multiplicative and additive characters. Finally, we relate the concepts of\nmutual unbiasedness and maximal entanglement. This allows to use well studied\nalgebraic concepts as efficient tools in the study of entanglement and its\ninformation aspects",
"arxiv_id": "quant-ph/0506128",
"authors": [
"M. Planat",
"H. C. Rosu"
],
"categories": [
"quant-ph"
],
"doi": "10.1140/epjd/e2005-00208-4",
"journal_ref": "Eur. Phys. J. D 36, 133-139 (Oct. 2005)",
"title": "Mutually unbiased phase states, phase uncertainties, and Gauss sums",
"url": "https://arxiv.org/abs/quant-ph/0506128"
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