Qudits of composite dimension, mutually unbiased bases and projective ring geometry
Open accessAbstract
The $d^2$ Pauli operators attached to a composite qudit in dimension $d$ may be mapped to the vectors of the symplectic module $\mathcal{Z}_d^{2}$ ($\mathcal{Z}_d$ the modular ring). As a result, perpendicular vectors correspond to commuting operators, a free cyclic submodule to a maximal commuting set, and disjoint such sets to mutually unbiased bases. For dimensions $d=6,~10,~15,~12$, and 18, the fine structure and the incidence between maximal commuting sets is found to reproduce the projective line over the rings $\mathcal{Z}_{6}$, $\mathcal{Z}_{10}$, $\mathcal{Z}_{15}$, $\mathcal{Z}_6 \times \mathbf{F}_4$ and $\mathcal{Z}_6 \times \mathcal{Z}_3$, respectively.
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The quantum transferrer o2 → π achieves the transformation |Hπ|φo2 ⇒ |φπ|0o.
“Then the orbital contribution has been analysed adopting the quantum transferrer o 2 → π, which achieves the transformation |H π |ϕ o2 ⇒ |ϕ π |0 o [44,45].”
From Experimental generation and characterization of single-photon hybrid ququarts based on polarization and orbital angular momentum encoding (2010) · cites this paper as [45]
The Summary (p. 2) describes shift and clock operators acting on qudit states, not an orbital quantum transferrer. The Discussion and conclusion (p. 8) addresses commutation structures; searches for the transferrer and the stated state transformation found no passage addressing either. (LLM, conf=0.76)
Experimental — accuracy not yet measuredIt is possible to realize two fully deterministic transferrers π → o2 and o2 → π at the price of a more complex optical layout, based on a q-plate and a Sagnac interferometer.
“Nevertheless it is possible to realize two fully deterministic transferrers π → o 2 and o 2 → π at the price of a more complex optical layout, based on a q-plate and a Sagnac interferometer, as shown in [45].”
From Experimental generation and characterization of single-photon hybrid ququarts based on polarization and orbital angular momentum encoding (2010) · cites this paper as [45]
The Summary (p. 2) and Discussion and conclusion (p. 8) address Pauli-operator commutation, mutually unbiased bases, and projective ring geometry, not optical devices. Searches for q-plates, Sagnac interferometers, deterministic transferrers, and optical transfer found no passage addressing the claim. (LLM, conf=0.76)
Experimental — accuracy not yet measuredThe transferrer presented in [44,45] is based on two waveplates, a q-plate and a polarizing beam-splitter, allowing coherent transfer of information initially encoded in polarization to the OAM degree of freedom.
“This transferrer, presented in [44,45], is based on two waveplates, a q-plate and a polarizing beam-splitter, thus allowing the coherent transfer of the information initially encoded in the polarization to the OAM degree of freedom.”
From Experimental generation and characterization of single-photon hybrid ququarts based on polarization and orbital angular momentum encoding (2010) · cites this paper as [45]
The Summary (p. 2) and Discussion and conclusion (p. 8) address Pauli-operator commutation, mutually unbiased bases, and projective ring geometry, not an optical transferrer. Searches for waveplates, a q-plate, a polarizing beam-splitter, and polarization-to-OAM transfer found no passage addressing the cited claim. (LLM, conf=0.78)
Experimental — accuracy not yet measured
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References 17 · Cited by 15
Research works printed but not linked: 4
- [5][5] Planat M, Baboin A C and Saniga M 2007 Multi-line geometry of qubit/qutrit and higher order Pauli operators Preprint 0705.2538 [quant-ph] ( Int. J. Theor. Phys. accepted) -
- [8][8] Sulc P and Tolar J 2007 Group theoretical constructions of mutually unbiased bases in Hilbert spaces of prime dimensions Preprint 0708.4114 [quant-ph] -
- [9][9] Havlicek H and Saniga M 2007 Projective ring line of a specific qudit Preprint 0708.4333 [quant-ph] ( J. Phys. A: Math. Theor. accepted). -
- [13][13] Howe 2005 R Nice error bases, mutually unbiased bases, induced representations, the Heisenberg group and finite geometries Indag Mathem, N S 16 (3-4), 553-583 -
- Authors
- Planat, M., & Baboin, A.-C.
- ORCID
- Planat, Michel
- Venue
- Journal of Physics A: Mathematical and Theoretical
- Publisher
- IOP Publishing
- Published
- 2007-10-31
- Volume
- 40
- Issue
- 46
- Pages
- F1005-F1012
- ISSN
- 1751-8113 (ISSN-L)
- 1751-8121
- OpenAlex
- W2079769339
- Cited by
- 15
- References
- 17
- Citations by year
- 2024: 1 · 2017: 1 · 2012: 2
- FWCI
- 3.55
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Journal of Physics A Mathematical and Theoretical (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- HAL (submittedVersion, other-oa)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Information and Cryptography
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Applications
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
- Keywords
- qudits, mutually unbiased bases, finite-dimensional quantum systems, quantum information
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