## Quantum number theoretic transforms on multipartite finite systems

JOSA A, Vol. 26, Issue 6, pp. 1360-1365 (2009)

http://dx.doi.org/10.1364/JOSAA.26.001360

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### Abstract

A quantum system composed of *p*-dimensional Hilbert space (where *p* is a prime number), is considered. A quantum number theoretic transform on this system, which has properties similar to those of a Fourier transform, is studied. A representation of the Heisenberg–Weyl group in this context is also discussed.

© 2009 Optical Society of America

**OCIS Codes**

(070.2465) Fourier optics and signal processing : Finite analogs of Fourier transforms

(270.5585) Quantum optics : Quantum information and processing

**ToC Category:**

Quantum Optics

**History**

Original Manuscript: March 4, 2009

Manuscript Accepted: April 17, 2009

Published: May 13, 2009

**Citation**

A. Vourdas and S. Zhang, "Quantum number theoretic transforms on multipartite finite systems," J. Opt. Soc. Am. A **26**, 1360-1365 (2009)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-26-6-1360

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### References

- J. M. Pollard, “The FFT in a finite field,” Math. Comput. 25, 365-374 (1971). [CrossRef]
- J. H. McClellan and C. M. Rader, Number Theory in Digital Signal Processing (Prentice Hall, 1979).
- R. E. Blahut, Fast Algorithms for Digital Signal Processing (Addison Wesley, 1985).
- D. F. Elliott and K. R. Rao, Fast Transforms (Academic, 1982).
- A. Vourdas, “Quantum systems with finite Hilbert space,” Rep. Prog. Phys. 67, 267-320 (2004). [CrossRef]
- A. Vourdas, “Quantum systems with finite Hilbert space: Galois fields in quantum mechanics,” J. Phys. A 40, R285-R331 (2007). [CrossRef]
- M. J. Holland and K. Burnett, “Interferometric detection of optical-phase shifts at the Heisenberg limit,” Phys. Rev. Lett. 71, 1355-1358 (1993). [CrossRef] [PubMed]
- T. Kim, O. Pfister, M. J. Holland, J. Noh, and J. L. Hall, “Influence of decorrelation on Heisenberg-limited interferometry with quantum correlated photons,” Phys. Rev. A 57, 4004-4013 (1998). [CrossRef]
- J. A. Dunningham, K. Burnett, and S. M. Barnett, “Interferometry below the standard quantum limit with Bose-Einstein condensates,” Phys. Rev. Lett. 89, 150401 (2002). [CrossRef] [PubMed]
- D. C. Roberts and K. Burnett, “Probing states in the Mott insulator regime in the case of coherent bosons trapped in an optical lattice,” Phys. Rev. Lett. 90, 150401 (2003). [CrossRef] [PubMed]
- J. A. Dunningham, K. Burnett, and W. D. Phillips, “Bose-Einstein condensates and precision measurements,” Philos. Trans. R. Soc. London, Ser. A 363, 2165-2175 (2005). [CrossRef]
- M. Hillery, M. Zou, and V. Buzek, “Difference-phase squeezing from amplitude squeezing by means of a beamsplitter,” Quantum Semiclassic. Opt. 8, 1041-1051 (1996). [CrossRef]
- M. S. Kim, W. Son, V. Buzek, and P. L. Knight, “Entanglement by a beam splitter: Nonclassicality as a prerequisite for entanglement,” Phys. Rev. A 65, 032323 (2002). [CrossRef]
- J. H. Shapiro, S. R. Shepard, and N. C. Wong, “Ultimate quantum limits on phase measurement,” Phys. Rev. Lett. 62, 2377-2380 (1989). [CrossRef] [PubMed]
- W. P. Schleich, J. P. Dowling, and R. J. Horowicz, “Exponential decrease in phase uncertainty,” Phys. Rev. A 44, 3365-3368 (1991). [CrossRef] [PubMed]
- A. S. Lane, S. L. Braunstein, and C. M. Caves, “Maximum-likelihood statistics of multiple quantum phase measurements,” Phys. Rev. A 47, 1667-1696 (1993). [CrossRef] [PubMed]
- J. P. Dowling, “Correlated input-port, matter-wave interferometer: Quantum-noise limits to the atom-laser gyroscope,” Phys. Rev. A 57, 4736-4746 (1998). [CrossRef]
- M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73, 58-61 (1994). [CrossRef] [PubMed]
- P. Torma, S. Stenholm, and I. Jex, “Hamiltonian theory of symmetric optical network transforms,” Phys. Rev. A 52, 4853-4860 (1995). [CrossRef] [PubMed]
- I. Jex, S. Stenholm, and A. Zeilinger, “Hamiltonian theory of a symmetric multiport,” Opt. Commun. 117, 95-101 (1995). [CrossRef]
- A. Vourdas and J. A. Dunningham, “Fourier multiport devices,” Phys. Rev. A 71, 013809 (2005). [CrossRef]
- J. Dunningham and A. Vourdas, “Efficient comparison of path-lengths using Fourier multiport devices,” J. Phys. B 39, 1579-1586 (2006). [CrossRef]
- S. Zhang, C. Lei, A. Vourdas, and J. Dunningham, “Applications and implementation of Fourier multiport devices,” J. Phys. B 39, 1625-1637 (2006). [CrossRef]

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