dorsal/arxiv
View SchemaQuantum phase uncertainty in mutually unbiased measurements and Gauss sums
| Authors | Michel Planat, Haret Rosu |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0502167 |
| URL | https://arxiv.org/abs/quant-ph/0502167 |
| DOI | 10.1117/12.608976 |
| Journal | Fluctuations and Noise in Photonics and Quantum Optics (2005) 183-193 |
Abstract
Mutually unbiased bases (MUBs), which are such that the inner product between two vectors in different orthogonal bases is constant equal to the inverse $1/\sqrt{d}$, 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 (Wootters in 1989, Klappenecker in 2003). Presumably, for non prime dimensions the cardinality is much less. The bases can be reinterpreted as quantum phase states, i.e. as eigenvectors of Hermitean phase operators generalizing those introduced by Pegg & Barnett in 1989. The MUB states are related to additive characters of Galois fields (in odd characteristic p) and of 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 physical states and find them 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 our quest of minimal uncertainty in quantum information primitives.
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"abstract": "Mutually unbiased bases (MUBs), which are such that the inner product between\ntwo vectors in different orthogonal bases is constant equal to the inverse\n$1/\\sqrt{d}$, with $d$ the dimension of the finite Hilbert space, are becoming\nmore and more studied for applications such as quantum tomography and\ncryptography, and in relation to entangled states and to the Heisenberg-Weil\ngroup of quantum optics. Complete sets of MUBs of cardinality $d+1$ have been\nderived for prime power dimensions $d=p^m$ using the tools of abstract algebra\n(Wootters in 1989, Klappenecker in 2003). Presumably, for non prime dimensions\nthe cardinality is much less. The bases can be reinterpreted as quantum phase\nstates, i.e. as eigenvectors of Hermitean phase operators generalizing those\nintroduced by Pegg \u0026 Barnett in 1989. The MUB states are related to additive\ncharacters of Galois fields (in odd characteristic p) and of Galois rings (in\ncharacteristic 2). Quantum Fourier transforms of the components in vectors of\nthe bases define a more general class of MUBs with multiplicative characters\nand additive ones altogether. We investigate the complementary properties of\nthe above phase operator with respect to the number operator. We also study the\nphase probability distribution and variance for physical states and find them\nrelated to the Gauss sums, which are sums over all elements of the field (or of\nthe ring) of the product of multiplicative and additive characters. Finally we\nrelate the concepts of mutual unbiasedness and maximal entanglement. This\nallows to use well studied algebraic concepts as efficient tools in our quest\nof minimal uncertainty in quantum information primitives.",
"arxiv_id": "quant-ph/0502167",
"authors": [
"Michel Planat",
"Haret Rosu"
],
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"doi": "10.1117/12.608976",
"journal_ref": "Fluctuations and Noise in Photonics and Quantum Optics (2005)\n 183-193",
"title": "Quantum phase uncertainty in mutually unbiased measurements and Gauss sums",
"url": "https://arxiv.org/abs/quant-ph/0502167"
},
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