Abstract

A quantum system composed of p1 subsystems, each of which is described with a 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

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  1. J. M. Pollard, “The FFT in a finite field,” Math. Comput. 25, 365-374 (1971).
    [CrossRef]
  2. J. H. McClellan and C. M. Rader, Number Theory in Digital Signal Processing (Prentice Hall, 1979).
  3. R. E. Blahut, Fast Algorithms for Digital Signal Processing (Addison Wesley, 1985).
  4. D. F. Elliott and K. R. Rao, Fast Transforms (Academic, 1982).
  5. A. Vourdas, “Quantum systems with finite Hilbert space,” Rep. Prog. Phys. 67, 267-320 (2004).
    [CrossRef]
  6. A. Vourdas, “Quantum systems with finite Hilbert space: Galois fields in quantum mechanics,” J. Phys. A 40, R285-R331 (2007).
    [CrossRef]
  7. 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]
  8. 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]
  9. 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]
  10. 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]
  11. 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]
  12. 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]
  13. 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]
  14. 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]
  15. W. P. Schleich, J. P. Dowling, and R. J. Horowicz, “Exponential decrease in phase uncertainty,” Phys. Rev. A 44, 3365-3368 (1991).
    [CrossRef] [PubMed]
  16. 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]
  17. 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]
  18. 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]
  19. P. Torma, S. Stenholm, and I. Jex, “Hamiltonian theory of symmetric optical network transforms,” Phys. Rev. A 52, 4853-4860 (1995).
    [CrossRef] [PubMed]
  20. I. Jex, S. Stenholm, and A. Zeilinger, “Hamiltonian theory of a symmetric multiport,” Opt. Commun. 117, 95-101 (1995).
    [CrossRef]
  21. A. Vourdas and J. A. Dunningham, “Fourier multiport devices,” Phys. Rev. A 71, 013809 (2005).
    [CrossRef]
  22. J. Dunningham and A. Vourdas, “Efficient comparison of path-lengths using Fourier multiport devices,” J. Phys. B 39, 1579-1586 (2006).
    [CrossRef]
  23. S. Zhang, C. Lei, A. Vourdas, and J. Dunningham, “Applications and implementation of Fourier multiport devices,” J. Phys. B 39, 1625-1637 (2006).
    [CrossRef]

2007 (1)

A. Vourdas, “Quantum systems with finite Hilbert space: Galois fields in quantum mechanics,” J. Phys. A 40, R285-R331 (2007).
[CrossRef]

2006 (2)

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]

2005 (2)

A. Vourdas and J. A. Dunningham, “Fourier multiport devices,” Phys. Rev. A 71, 013809 (2005).
[CrossRef]

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]

2004 (1)

A. Vourdas, “Quantum systems with finite Hilbert space,” Rep. Prog. Phys. 67, 267-320 (2004).
[CrossRef]

2003 (1)

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]

2002 (2)

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. 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]

1998 (2)

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. P. Dowling, “Correlated input-port, matter-wave interferometer: Quantum-noise limits to the atom-laser gyroscope,” Phys. Rev. A 57, 4736-4746 (1998).
[CrossRef]

1996 (1)

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]

1995 (2)

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]

1994 (1)

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]

1993 (2)

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]

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]

1991 (1)

W. P. Schleich, J. P. Dowling, and R. J. Horowicz, “Exponential decrease in phase uncertainty,” Phys. Rev. A 44, 3365-3368 (1991).
[CrossRef] [PubMed]

1989 (1)

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]

1971 (1)

J. M. Pollard, “The FFT in a finite field,” Math. Comput. 25, 365-374 (1971).
[CrossRef]

Barnett, S. M.

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]

Bernstein, H. J.

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]

Bertani, P.

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]

Blahut, R. E.

R. E. Blahut, Fast Algorithms for Digital Signal Processing (Addison Wesley, 1985).

Braunstein, S. L.

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]

Burnett, K.

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]

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 S. M. Barnett, “Interferometry below the standard quantum limit with Bose-Einstein condensates,” Phys. Rev. Lett. 89, 150401 (2002).
[CrossRef] [PubMed]

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]

Buzek, V.

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]

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]

Caves, C. M.

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]

Dowling, J. P.

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]

W. P. Schleich, J. P. Dowling, and R. J. Horowicz, “Exponential decrease in phase uncertainty,” Phys. Rev. A 44, 3365-3368 (1991).
[CrossRef] [PubMed]

Dunningham, J.

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]

Dunningham, J. A.

A. Vourdas and J. A. Dunningham, “Fourier multiport devices,” Phys. Rev. A 71, 013809 (2005).
[CrossRef]

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]

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]

Elliott, D. F.

D. F. Elliott and K. R. Rao, Fast Transforms (Academic, 1982).

Hall, J. L.

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]

Hillery, M.

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]

Holland, M. J.

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]

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]

Horowicz, R. J.

W. P. Schleich, J. P. Dowling, and R. J. Horowicz, “Exponential decrease in phase uncertainty,” Phys. Rev. A 44, 3365-3368 (1991).
[CrossRef] [PubMed]

Jex, I.

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]

Kim, M. S.

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]

Kim, T.

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]

Knight, P. L.

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]

Lane, A. S.

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]

Lei, C.

S. Zhang, C. Lei, A. Vourdas, and J. Dunningham, “Applications and implementation of Fourier multiport devices,” J. Phys. B 39, 1625-1637 (2006).
[CrossRef]

McClellan, J. H.

J. H. McClellan and C. M. Rader, Number Theory in Digital Signal Processing (Prentice Hall, 1979).

Noh, J.

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]

Pfister, O.

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]

Phillips, W. D.

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]

Pollard, J. M.

J. M. Pollard, “The FFT in a finite field,” Math. Comput. 25, 365-374 (1971).
[CrossRef]

Rader, C. M.

J. H. McClellan and C. M. Rader, Number Theory in Digital Signal Processing (Prentice Hall, 1979).

Rao, K. R.

D. F. Elliott and K. R. Rao, Fast Transforms (Academic, 1982).

Reck, M.

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]

Roberts, D. C.

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]

Schleich, W. P.

W. P. Schleich, J. P. Dowling, and R. J. Horowicz, “Exponential decrease in phase uncertainty,” Phys. Rev. A 44, 3365-3368 (1991).
[CrossRef] [PubMed]

Shapiro, J. H.

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]

Shepard, S. R.

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]

Son, W.

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]

Stenholm, S.

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]

Torma, P.

P. Torma, S. Stenholm, and I. Jex, “Hamiltonian theory of symmetric optical network transforms,” Phys. Rev. A 52, 4853-4860 (1995).
[CrossRef] [PubMed]

Vourdas, A.

A. Vourdas, “Quantum systems with finite Hilbert space: Galois fields in quantum mechanics,” J. Phys. A 40, R285-R331 (2007).
[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]

A. Vourdas and J. A. Dunningham, “Fourier multiport devices,” Phys. Rev. A 71, 013809 (2005).
[CrossRef]

A. Vourdas, “Quantum systems with finite Hilbert space,” Rep. Prog. Phys. 67, 267-320 (2004).
[CrossRef]

Wong, N. C.

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]

Zeilinger, A.

I. Jex, S. Stenholm, and A. Zeilinger, “Hamiltonian theory of a symmetric multiport,” Opt. Commun. 117, 95-101 (1995).
[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]

Zhang, S.

S. Zhang, C. Lei, A. Vourdas, and J. Dunningham, “Applications and implementation of Fourier multiport devices,” J. Phys. B 39, 1625-1637 (2006).
[CrossRef]

Zou, M.

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]

J. Phys. A (1)

A. Vourdas, “Quantum systems with finite Hilbert space: Galois fields in quantum mechanics,” J. Phys. A 40, R285-R331 (2007).
[CrossRef]

J. Phys. B (2)

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]

Math. Comput. (1)

J. M. Pollard, “The FFT in a finite field,” Math. Comput. 25, 365-374 (1971).
[CrossRef]

Opt. Commun. (1)

I. Jex, S. Stenholm, and A. Zeilinger, “Hamiltonian theory of a symmetric multiport,” Opt. Commun. 117, 95-101 (1995).
[CrossRef]

Philos. Trans. R. Soc. London, Ser. A (1)

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]

Phys. Rev. A (7)

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]

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]

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]

A. Vourdas and J. A. Dunningham, “Fourier multiport devices,” Phys. Rev. A 71, 013809 (2005).
[CrossRef]

P. Torma, S. Stenholm, and I. Jex, “Hamiltonian theory of symmetric optical network transforms,” Phys. Rev. A 52, 4853-4860 (1995).
[CrossRef] [PubMed]

Phys. Rev. Lett. (5)

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]

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]

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]

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]

Quantum Semiclassic. Opt. (1)

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]

Rep. Prog. Phys. (1)

A. Vourdas, “Quantum systems with finite Hilbert space,” Rep. Prog. Phys. 67, 267-320 (2004).
[CrossRef]

Other (3)

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).

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Equations (100)

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F = p 1 2 m , n = 0 p 1 ω ( m n ) X ; m X ; n ; F 4 = 1 ,
ω = exp [ i 2 π p ] ; ω ( m ) ω m ; m Z p .
P ; m = F X ; m = p 1 2 n ω ( m n ) X ; n .
F = ϖ 0 + i ϖ 1 ϖ 2 i ϖ 3 .
ϖ λ = μ = 0 3 i λ μ F μ ; F ϖ λ = i λ ϖ λ ,
ϖ λ ϖ ν = ϖ λ δ ( λ , ν ) ; ϖ 0 + ϖ 1 + ϖ 2 + ϖ 3 = 1 .
T = F 2 = ( ϖ 0 + ϖ 2 ) ( ϖ 1 + ϖ 3 ) ; T 2 = 1 .
X = n = 0 p 1 ω ( n ) P ; n P ; n = n = 0 p 1 X ; n + 1 X ; n ,
Z = F X F = n = 0 p 1 ω ( n ) X ; n X ; n = n = 0 p 1 P ; n + 1 P ; n ,
X p = Z p = 1 ; X β Z α = Z α X β ω ( α β ) ,
D ( α , β ) = Z α X β ω ( 2 1 α β ) .
T ( α , β ) = D ( α , β ) T [ D ( α , β ) ] .
X ; ( m λ ) = X ; m 0 , , m p 2 = X ; m 0 X ; m p 2 ,
P ; ( m λ ) = P ; m 0 , , m p 2 = P ; m 0 P ; m p 2 ,
F = F F ; F F = F 4 = 1 .
F = π 0 + i π 1 π 2 i π 3 ,
π λ = 1 4 μ = 0 3 i λ μ F μ ; F π λ = i λ π λ ,
π λ π ν = π λ δ ( λ , ν ) ; π 0 + π 1 + π 2 + π 3 = 1 .
T = F 2 = ( π 0 + π 2 ) ( π 1 + π 3 ) ; T 2 = 1 .
T X ; m 0 , , m p 2 = X ; m 0 , , m p 2 ,
T P ; m 0 , , m p 2 = P ; m 0 , , m p 2 .
X κ = 1 1 X 1 1 ,
Z κ = 1 1 Z 1 1 ,
X κ p = Z κ p = 1 ; X κ n Z κ l = Z κ l X κ n ω ( l n ) ,
κ λ [ X κ , Z λ ] = 0 .
( α λ ) = ( α 0 , , α p 2 ) ; ( β λ ) = ( β 0 , , β p 2 )
X ( β λ ) = X 0 β 0 X p 2 β p 2 ,
Z ( α λ ) = Z 0 α 0 Z p 2 α p 2 = F X ( α λ ) F ,
X ( β λ ) Z ( α λ ) = Z ( α λ ) X ( β λ ) ω [ λ α λ β λ ] ;
λ Z p 1 , α λ , β λ Z p .
D ( α λ , β λ ) = Z ( α λ ) X ( β λ ) ω [ 2 1 λ α λ β λ ] = D ( α 0 , β 0 ) D ( α p 2 , β p 2 ) .
T ( α λ , β λ ) = D ( α λ , β λ ) T [ D ( α λ , β λ ) ] .
W 1 ( α λ , β λ ) = Tr [ ρ T ( α λ , β λ ) ] .
ϵ p 1 = 1 .
λ = 0 p 2 ϵ κ λ = δ ( κ , 0 ) ; κ , λ Z p 1 ,
Φ κ λ = ϵ κ λ ; κ , λ Z p 1 .
( Φ 1 ) κ λ = ϵ κ λ ; κ , λ Z p 1 .
Φ 4 = 1 .
( Φ Φ ) κ λ = δ ( κ + λ , 0 ) .
Φ = ( 1 1 1 2 ) ; Φ 1 = ( 2 2 2 1 ) .
Φ = ( 1 1 1 1 1 2 4 3 1 4 1 4 1 3 4 2 ) ; Φ 1 = ( 4 4 4 4 4 2 1 3 4 1 4 1 4 3 1 2 ) .
x = ( 0 0 1 1 0 0 0 1 0 ) ; z = Φ x Φ 1 = ( 1 0 0 0 ϵ 0 0 0 ϵ p 2 ) .
x p 1 = z p 1 = 1 ; x β z α ϵ α β = z α x β ; α , β Z p 1 .
( x α ) λ = κ x λ κ α κ = ( α p 2 , α 0 , , α p 3 ) ,
( z α ) λ = κ z λ κ α κ = ( α 0 , ϵ α 1 , , ϵ p 2 α p 2 ) .
a = ( a 0 , , a p 2 ) Φ a = ( ( Φ a ) 0 , , ( Φ a ) p 2 ) ,
( Φ a ) κ = λ = 0 p 2 ϵ κ λ a λ .
( Φ 1 a ) κ = λ = 0 p 2 ϵ κ λ a λ .
( Φ 4 a ) κ = a κ .
( Φ 2 a ) κ = a κ ; κ Z p 1 ; α κ Z p .
( a b ) κ a κ b κ , ( a b ) κ λ a κ λ b λ ,
Φ ( a b ) = ( Φ a ) ( Φ b )
κ = 0 p 2 ( Φ a ) κ ( Φ 1 b ) κ = κ = 0 p 2 ( Φ 1 a ) κ ( Φ b ) κ = κ = 0 p 2 a κ b κ ,
κ = 0 p 2 ( Φ a ) κ b κ = κ = 0 p 2 a κ ( Φ b ) κ .
δ ( Φ a , b ) = δ ( ( Φ a ) 0 , b 0 ) δ ( ( Φ a ) p 2 , b p 2 ) ,
δ ( ( Φ a ) , b ) = δ ( a , ( Φ 1 b ) ) .
F m X ; ( Φ m ) 0 , , ( Φ m ) p 2 X ; m 0 , , m p 2 .
F 4 = 1 ; F F F = F ; [ T , F ] = 0 .
F P ; m 0 , , m p 2 = P ; ( Φ 1 m ) 0 , , ( Φ 1 m ) p 2 ,
F X ; m 0 , , m p 2 = X ; ( Φ m ) 0 , , ( Φ m ) p 2 .
( α λ , β λ ) [ ( Φ α ) λ , ( Φ 1 β ) λ ] .
Z ( α λ ) = F Z ( α λ ) F = Z [ ( Φ α ) λ ] ,
X ( β λ ) = F X ( β λ ) F = X [ ( Φ 1 β ) λ ] .
X ( β λ ) Z ( α λ ) = Z ( α λ ) X ( β λ ) ω [ λ α λ β λ ] .
F D ( α λ , β λ ) F = D [ ( Φ α ) λ , ( Φ 1 β ) λ ] .
Z 0 = Z ( 1 , 0 , 0 , 0 ) = Z ( 1 , 1 , 1 , 1 ) = Z 0 Z 1 Z 2 Z 3 ,
Z 1 = Z ( 0 , 1 , 0 , 0 ) = Z ( 1 , 2 , 4 , 3 ) = Z 0 Z 1 2 Z 2 4 Z 3 3 ,
Z 2 = Z ( 0 , 0 , 1 , 0 ) = Z ( 1 , 4 , 1 , 4 ) = Z 0 Z 1 4 Z 2 Z 3 4 ,
Z 3 = Z ( 0 , 0 , 0 , 1 ) = Z ( 1 , 3 , 4 , 2 ) = Z 0 Z 1 3 Z 2 4 Z 3 2 ,
X 0 = X ( 1 , 0 , 0 , 0 ) = X ( 4 , 4 , 4 , 4 ) = X 0 4 X 1 4 X 2 4 X 3 4 ,
X 1 = X ( 0 , 1 , 0 , 0 ) = X ( 4 , 2 , 1 , 3 ) = X 0 4 X 1 2 X 2 X 3 3 ,
X 2 = X ( 0 , 0 , 1 , 0 ) = X ( 4 , 1 , 4 , 1 ) = X 0 4 X 1 X 2 4 X 3 ,
X 3 = X ( 0 , 0 , 0 , 1 ) = X ( 4 , 3 , 1 , 2 ) = X 0 4 X 1 3 X 2 X 3 2 ,
F = Π 0 + i Π 1 Π 2 i Π 3 .
Π λ = 1 4 μ = 0 3 i λ μ F μ ; λ = 0 , 1 , 2 , 3 ,
Π λ Π ν = Π λ δ ( λ , ν ) ; Π 0 + Π 1 + Π 2 + Π 3 = 1 .
F Π λ = i λ Π λ .
[ π 0 + π 2 , Π λ ] = [ π 1 + π 3 , Π λ ] = 0 .
R = F 2 = ( Π 0 + Π 2 ) ( Π 1 + Π 3 ) ; [ Σ , T ] = 0 .
R X ; m 0 , m 1 , , m p 2 = X ; m 0 , m 1 , m ( p 2 ) ,
R P ; m 0 , m 1 , , m p 2 = P ; m 0 , m 1 , m ( p 2 ) .
ψ in = ( a 1 X ; 0 + b 1 X ; 1 + c 1 X ; 2 ) ( a 2 X ; 0 + b 2 X ; 1 + c 2 X ; 2 ) ,
ψ out F ψ in = a 1 a 2 X ; 00 + b 1 a 2 X ; 11 + c 1 a 2 X ; 22 + a 1 b 2 X ; 12 + b 1 b 2 X ; 20 + c 1 b 2 X ; 01 + a 1 c 2 X ; 21 + b 1 c 2 X ; 02 + c 1 c 2 X ; 10 .
E = 1 Tr 1 ( ρ 1 2 ) = 1 i = 1 2 ( a i 4 + b i 4 + c i 4 ) .
U X ; 0234 = X ; 4321 ,
U X ; 1234 = X ; 0432 .
ψ = ( 1 2 0 + 1 2 1 ) 234 .
U ψ = 1 2 X ; 4321 + 1 2 X ; 0432 ,
X = m λ X ; ( ( x m ) λ ) X ; ( m λ ) ; ( x m ) λ = κ x λ κ m κ ,
Z = F X F = m λ X ; ( ( z m ) λ ) X ; ( m λ ) ; ( z m ) λ = κ z λ κ m κ ,
Ω = m λ X ; ( ϵ m λ ) X ; ( m λ ) ; [ Ω , F ] = [ Ω , Σ ] = 0 .
X = m λ P ; m p 2 , m 0 , , m p 3 P ; m 0 , m 1 , , m p 2 ,
Z = m λ P ; m 0 , , m p 2 P ; m 0 , ϵ m 1 , , ϵ p 2 m p 2 .
X p 1 = Z p 1 = 1 ; [ X , Ω ] = [ Z , Ω ] = 0 ,
X β Z α Ω α β = Z α X β ; α , β Z p 1 .
D ( α , β , γ ) = Z α X β Ω γ ; α , β , γ Z p 1 ,
D ( α 1 , β 1 , γ 1 ) D ( α 2 , β 2 , γ 2 ) = D ( α 1 + α 2 , β 1 + β 2 , γ 1 + γ 2 α 2 β 1 ) ,
[ D ( α , β , γ ) ] = D ( α , β , γ ) ,
R ( α , β ) = D ( α , β ) R [ D ( α , β ) ] .
W 2 ( α , β ) = Tr [ ρ R ( α , β ) ] .

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