Abstract

Mutually unbiased bases and discrete Wigner functions are closely but not uniquely related. Such a connection becomes more interesting when the Hilbert space has a dimension that is a power of a prime N=dn, which describes a composite system of n qudits. Hence, entanglement naturally enters the picture. Although our results are general, we concentrate on the simplest nontrivial example of dimension N=8=23. It is shown that the number of fundamentally different Wigner functions is severely limited if one simultaneously imposes translational covariance and that the generating operators consist of rotations around two orthogonal axes, acting on the individual qubits only.

© 2007 Optical Society of America

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    [Crossref]
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  3. K. Husimi, "Some formal properties of the density matrix," Proc. Phys. Math. Soc. Jpn. 22, 264-314 (1940).
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  5. A. Vourdas, "Quantum systems with finite Hilbert space," Rep. Prog. Phys. 67, 267-248 (2004).
    [Crossref]
  6. J. H. Hannay and M. V. Berry, "Quantization of linear maps on a torus--Fresnel diffraction by a periodic grating," Physica D 1, 267-290 (1980).
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  7. U. Leonhardt, "Quantum-state tomography and discrete Wigner function," Phys. Rev. Lett. 74, 4101-4105 (1995).
    [Crossref] [PubMed]
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  9. F. A. Buot, "Method for calculating TrHn in solid-state theory," Phys. Rev. B 10, 3700-3705 (1974).
    [Crossref]
  10. W. K. Wootters, "A Wigner-function formulation of finite-state quantum mechanics," Ann. Phys. (N.Y.) 176, 1-21 (1987).
    [Crossref]
  11. D. Galetti and A. F. R. De Toledo Piza, "An extended Weyl-Wigner transformation for special finite spaces," Physica A 149, 267-282 (1988).
    [Crossref]
  12. O. Cohendet, Ph. Combe, M. Sirugue, and M. Sirugue-Collin, "A stochastic treatment of the dynamics of an integer spin," J. Phys. A 21, 2875-2884 (1988).
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  14. K. S. Gibbons, M. J. Hoffman, and W. K. Wootters, "Discrete phase space based on finite fields," Phys. Rev. A 70, 062101 (2004).
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  15. R. Asplund and G. Björk, "Reconstructing the discrete Wigner function and some properties of the measurement bases," Phys. Rev. A 64, 012106 (2001).
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  16. M. Koniorczyk, V. Buzek, and J. Janszky, "Wigner-function description of quantum teleportation in arbitrary dimensions and a continuous limit," Phys. Rev. A 64, 034301 (2001).
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  17. J. P. Paz, "Discrete Wigner functions and the phase-space representation of quantum teleportation," Phys. Rev. A 65, 062311 (2002).
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  18. J. A. Vaccaro and D. T. Pegg, "Wigner function for number and phase," Phys. Rev. A 41, 5156-5163 (1990).
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  20. Y. Aharonov and B.-G. Englert, "The mean king's problem: spin 1," Z. Naturforsch., A: Phys. Sci. 56, 16-19 (2001).
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    [Crossref]
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  27. W. K. Wootters, "Quantum mechanics without probability amplitudes," Found. Phys. 16, 391-405 (1986).
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  28. W. K. Wootters and B. D. Fields, "Optimal state-determination by mutually unbiased measurements," Ann. Phys. (N.Y.) 191, 363-381 (1989).
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    [Crossref] [PubMed]
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    [Crossref]
  34. J. Lawrence, C. Brukner, and A. Zeilinger, "Mutually unbiased binary observable sets on N qubits," Phys. Rev. A 65, 032320 (2002).
    [Crossref]
  35. J. L. Romero, G. Björk, A. B. Klimov, and L. L. Sánchez-Soto, "Structure of the sets of mutually unbiased bases for N qubits," Phys. Rev. A 72, 062310 (2005).
    [Crossref]
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    [Crossref]
  37. A. R. Calderbank, P. J. Cameron, W. M. Cantor, and J. J. Seidel, "Z4-Kerdock codes, orthogonal spreads, and extremal Euclidean line-sets," Proc. London Math. Soc. 75, 436-480 (1997).
    [Crossref]
  38. S. Chaturvedi, "Aspects of mutually unbiased bases in odd-prime-power dimensions," Phys. Rev. A 65, 044301 (2002).
    [Crossref]
  39. A. O. Pittenger and M. H. Rubin, "Mutually unbiased bases, generalized spin matrices and separability," Linear Algebr. Appl. 390, 255-278 (2004).
    [Crossref]
  40. T. Durt, "About mutually unbiased bases in even and odd prime power dimensions," J. Phys. A 38, 5267-5284 (2005).
    [Crossref]
  41. S. Bandyopadhyay, P. O. Boykin, V. Roychowdhury, and F. Vatan, "A new proof for the existence of mutually unbiased bases," Algorithmica 34, 512-528 (2002).
    [Crossref]
  42. A. B. Klimov, L. L. Sánchez-Soto, and H. de Guise, "Multicomplementary operators via finite Fourier transform," J. Phys. A 38, 2747-2760 (2005).
    [Crossref]
  43. M. Grassl, "On SIC-POVMs and MUBs in dimension 6," arXiv.org e-Print archive, quant-ph/0406175, June 23, 2004, http://arxiv.org/abs/quant-ph/0406175.
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  45. C. Archer, "There is no generalization of known formulas for mutually unbiased bases," J. Math. Phys. 46, 022106 (2005).
    [Crossref]
  46. A. Klappenecker and M. Rötteler, "Constructions of mutually unbiased bases," Lect. Notes Comput. Sci. 2948, 137-144 (2004).
    [Crossref]
  47. W. K. Wootters, "Quantum measurements and finite geometry," Found. Phys. 36, 112-126 (2006).
    [Crossref]
  48. I. Bengtsson and A. Ericsson, "Mutually unbiased bases and the complementarity polytope," Open Syst. Inf. Dyn. 12, 107-120 (2005).
    [Crossref]
  49. M. Saniga, M. Planat, and H. Rosu, "Mutually unbiased bases and finite projective planes," J. Opt. B 6, L19-L20 (2004).
    [Crossref]
  50. T. Durt, "Tomography of one and two qubit states and factorisation of the Wigner distribution in prime power dimensions," arXiv.org e-Print archive, quant-ph/0604117, April 17, 2006, http://arxiv.org/abs/quant-ph/0604117.
  51. A. O. Pittenger and M. H. Rubin, "Wigner function and separability for finite systems," J. Phys. A 38, 6005-6036 (2005).
    [Crossref]

2006 (1)

W. K. Wootters, "Quantum measurements and finite geometry," Found. Phys. 36, 112-126 (2006).
[Crossref]

2005 (8)

I. Bengtsson and A. Ericsson, "Mutually unbiased bases and the complementarity polytope," Open Syst. Inf. Dyn. 12, 107-120 (2005).
[Crossref]

A. O. Pittenger and M. H. Rubin, "Wigner function and separability for finite systems," J. Phys. A 38, 6005-6036 (2005).
[Crossref]

T. Durt, "About mutually unbiased bases in even and odd prime power dimensions," J. Phys. A 38, 5267-5284 (2005).
[Crossref]

A. B. Klimov, L. L. Sánchez-Soto, and H. de Guise, "Multicomplementary operators via finite Fourier transform," J. Phys. A 38, 2747-2760 (2005).
[Crossref]

P. Wocjan and T. Beth, "New construction of mutually unbiased bases in square dimensions," Quantum Inf. Comput. 5, 93-101 (2005).

C. Archer, "There is no generalization of known formulas for mutually unbiased bases," J. Math. Phys. 46, 022106 (2005).
[Crossref]

A. Hayashi, M. Horibe, and T. Hashimoto, "Mean king's problem with mutually unbiased bases and orthogonal Latin squares," Phys. Rev. A 71, 052331 (2005).
[Crossref]

J. L. Romero, G. Björk, A. B. Klimov, and L. L. Sánchez-Soto, "Structure of the sets of mutually unbiased bases for N qubits," Phys. Rev. A 72, 062310 (2005).
[Crossref]

2004 (6)

A. Vourdas, "Quantum systems with finite Hilbert space," Rep. Prog. Phys. 67, 267-248 (2004).
[Crossref]

W. K. Wootters, "Picturing qubits in phase space," IBM J. Res. Dev. 48, 99-110 (2004).
[Crossref]

K. S. Gibbons, M. J. Hoffman, and W. K. Wootters, "Discrete phase space based on finite fields," Phys. Rev. A 70, 062101 (2004).
[Crossref]

A. Klappenecker and M. Rötteler, "Constructions of mutually unbiased bases," Lect. Notes Comput. Sci. 2948, 137-144 (2004).
[Crossref]

A. O. Pittenger and M. H. Rubin, "Mutually unbiased bases, generalized spin matrices and separability," Linear Algebr. Appl. 390, 255-278 (2004).
[Crossref]

M. Saniga, M. Planat, and H. Rosu, "Mutually unbiased bases and finite projective planes," J. Opt. B 6, L19-L20 (2004).
[Crossref]

2003 (2)

P. K. Aravind, "Solution to the king's problem in prime power dimensions," Z. Naturforsch., A: Phys. Sci. 58, 85-92 (2003).

P. K. Aravind, "Best conventional solutions to the king's problem," Z. Naturforsch., A: Phys. Sci. 58, 682-690 (2003).

2002 (6)

J. P. Paz, "Discrete Wigner functions and the phase-space representation of quantum teleportation," Phys. Rev. A 65, 062311 (2002).
[Crossref]

C. Miquel, J. P. Paz, and M. Saraceno, "Quantum computers in phase space," Phys. Rev. A 65, 062309 (2002).
[Crossref]

D. Bruß and C. Macchiavello, "Optimal eavesdropping in cryptography with three-dimensional quantum states," Phys. Rev. Lett. 88, 127901 (2002).
[Crossref] [PubMed]

S. Bandyopadhyay, P. O. Boykin, V. Roychowdhury, and F. Vatan, "A new proof for the existence of mutually unbiased bases," Algorithmica 34, 512-528 (2002).
[Crossref]

J. Lawrence, C. Brukner, and A. Zeilinger, "Mutually unbiased binary observable sets on N qubits," Phys. Rev. A 65, 032320 (2002).
[Crossref]

S. Chaturvedi, "Aspects of mutually unbiased bases in odd-prime-power dimensions," Phys. Rev. A 65, 044301 (2002).
[Crossref]

2001 (5)

R. Asplund, G. Björk, and M. Bourennane, "An expectation value expansion of Hermitian operators in a discrete Hilbert space," J. Opt. B 3, 163-170 (2001).
[Crossref]

Y. Aharonov and B.-G. Englert, "The mean king's problem: spin 1," Z. Naturforsch., A: Phys. Sci. 56, 16-19 (2001).

B.-G. Englert and Y. Aharonov, "The mean king's problem: prime degrees of freedom," Phys. Lett. A 284, 1-5 (2001).
[Crossref]

R. Asplund and G. Björk, "Reconstructing the discrete Wigner function and some properties of the measurement bases," Phys. Rev. A 64, 012106 (2001).
[Crossref]

M. Koniorczyk, V. Buzek, and J. Janszky, "Wigner-function description of quantum teleportation in arbitrary dimensions and a continuous limit," Phys. Rev. A 64, 034301 (2001).
[Crossref]

2000 (1)

H. Bechmann-Pasquinucci and A. Peres, "Quantum cryptography with 3-state systems," Phys. Rev. Lett. 85, 3313-3316 (2000).
[Crossref] [PubMed]

1997 (2)

A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane, "Quantum error correction and orthogonal geometry," Phys. Rev. Lett. 78, 405-408 (1997).
[Crossref]

A. R. Calderbank, P. J. Cameron, W. M. Cantor, and J. J. Seidel, "Z4-Kerdock codes, orthogonal spreads, and extremal Euclidean line-sets," Proc. London Math. Soc. 75, 436-480 (1997).
[Crossref]

1996 (1)

U. Leonhardt, "Discrete Wigner function and quantum-state tomography," Phys. Rev. A 53, 2998-3013 (1996).
[Crossref] [PubMed]

1995 (1)

U. Leonhardt, "Quantum-state tomography and discrete Wigner function," Phys. Rev. Lett. 74, 4101-4105 (1995).
[Crossref] [PubMed]

1990 (1)

J. A. Vaccaro and D. T. Pegg, "Wigner function for number and phase," Phys. Rev. A 41, 5156-5163 (1990).
[Crossref] [PubMed]

1989 (1)

W. K. Wootters and B. D. Fields, "Optimal state-determination by mutually unbiased measurements," Ann. Phys. (N.Y.) 191, 363-381 (1989).
[Crossref]

1988 (2)

D. Galetti and A. F. R. De Toledo Piza, "An extended Weyl-Wigner transformation for special finite spaces," Physica A 149, 267-282 (1988).
[Crossref]

O. Cohendet, Ph. Combe, M. Sirugue, and M. Sirugue-Collin, "A stochastic treatment of the dynamics of an integer spin," J. Phys. A 21, 2875-2884 (1988).
[Crossref]

1987 (1)

W. K. Wootters, "A Wigner-function formulation of finite-state quantum mechanics," Ann. Phys. (N.Y.) 176, 1-21 (1987).
[Crossref]

1986 (1)

W. K. Wootters, "Quantum mechanics without probability amplitudes," Found. Phys. 16, 391-405 (1986).
[Crossref]

1981 (1)

I. D. Ivanovic, "Geometrical description of quantal state determination," J. Phys. A 14, 3241-3246 (1981).
[Crossref]

1980 (1)

J. H. Hannay and M. V. Berry, "Quantization of linear maps on a torus--Fresnel diffraction by a periodic grating," Physica D 1, 267-290 (1980).
[Crossref]

1975 (1)

P. Delsarte, J. M. Goethals, and J. J. Seidel, "Bounds for systems of lines and Jacobi polynomials," Philips Res. Rep. 30, 91-105 (1975).

1974 (1)

F. A. Buot, "Method for calculating TrHn in solid-state theory," Phys. Rev. B 10, 3700-3705 (1974).
[Crossref]

1963 (2)

R. J. Glauber, "Coherent and incoherent states of the radiation field," Phys. Rev. 131, 2766-2788 (1963).
[Crossref]

E. C. G. Sudarshan, "Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams," Phys. Rev. Lett. 10, 277-279 (1963).
[Crossref]

1940 (1)

K. Husimi, "Some formal properties of the density matrix," Proc. Phys. Math. Soc. Jpn. 22, 264-314 (1940).

1932 (1)

E. P. Wigner, "On the quantum correction for thermodynamic equilibrium," Phys. Rev. 40, 749-759 (1932).
[Crossref]

Aharonov, Y.

B.-G. Englert and Y. Aharonov, "The mean king's problem: prime degrees of freedom," Phys. Lett. A 284, 1-5 (2001).
[Crossref]

Y. Aharonov and B.-G. Englert, "The mean king's problem: spin 1," Z. Naturforsch., A: Phys. Sci. 56, 16-19 (2001).

Aravind, P. K.

P. K. Aravind, "Solution to the king's problem in prime power dimensions," Z. Naturforsch., A: Phys. Sci. 58, 85-92 (2003).

P. K. Aravind, "Best conventional solutions to the king's problem," Z. Naturforsch., A: Phys. Sci. 58, 682-690 (2003).

Archer, C.

C. Archer, "There is no generalization of known formulas for mutually unbiased bases," J. Math. Phys. 46, 022106 (2005).
[Crossref]

Asplund, R.

R. Asplund and G. Björk, "Reconstructing the discrete Wigner function and some properties of the measurement bases," Phys. Rev. A 64, 012106 (2001).
[Crossref]

R. Asplund, G. Björk, and M. Bourennane, "An expectation value expansion of Hermitian operators in a discrete Hilbert space," J. Opt. B 3, 163-170 (2001).
[Crossref]

Bandyopadhyay, S.

S. Bandyopadhyay, P. O. Boykin, V. Roychowdhury, and F. Vatan, "A new proof for the existence of mutually unbiased bases," Algorithmica 34, 512-528 (2002).
[Crossref]

Bechmann-Pasquinucci, H.

H. Bechmann-Pasquinucci and A. Peres, "Quantum cryptography with 3-state systems," Phys. Rev. Lett. 85, 3313-3316 (2000).
[Crossref] [PubMed]

Bengtsson, I.

I. Bengtsson and A. Ericsson, "Mutually unbiased bases and the complementarity polytope," Open Syst. Inf. Dyn. 12, 107-120 (2005).
[Crossref]

Bennett, C. H.

C. H. Bennett and G. Brassard, in Proceedings of the IEEE International Conference on Computers, Systems, and Signal Processing, Bangalore, India (IEEE, 1984), pp. 175-179.
[PubMed]

Berry, M. V.

J. H. Hannay and M. V. Berry, "Quantization of linear maps on a torus--Fresnel diffraction by a periodic grating," Physica D 1, 267-290 (1980).
[Crossref]

Beth, T.

P. Wocjan and T. Beth, "New construction of mutually unbiased bases in square dimensions," Quantum Inf. Comput. 5, 93-101 (2005).

Björk, G.

J. L. Romero, G. Björk, A. B. Klimov, and L. L. Sánchez-Soto, "Structure of the sets of mutually unbiased bases for N qubits," Phys. Rev. A 72, 062310 (2005).
[Crossref]

R. Asplund, G. Björk, and M. Bourennane, "An expectation value expansion of Hermitian operators in a discrete Hilbert space," J. Opt. B 3, 163-170 (2001).
[Crossref]

R. Asplund and G. Björk, "Reconstructing the discrete Wigner function and some properties of the measurement bases," Phys. Rev. A 64, 012106 (2001).
[Crossref]

Bourennane, M.

R. Asplund, G. Björk, and M. Bourennane, "An expectation value expansion of Hermitian operators in a discrete Hilbert space," J. Opt. B 3, 163-170 (2001).
[Crossref]

Boykin, P. O.

S. Bandyopadhyay, P. O. Boykin, V. Roychowdhury, and F. Vatan, "A new proof for the existence of mutually unbiased bases," Algorithmica 34, 512-528 (2002).
[Crossref]

Brassard, G.

C. H. Bennett and G. Brassard, in Proceedings of the IEEE International Conference on Computers, Systems, and Signal Processing, Bangalore, India (IEEE, 1984), pp. 175-179.
[PubMed]

Brukner, C.

J. Lawrence, C. Brukner, and A. Zeilinger, "Mutually unbiased binary observable sets on N qubits," Phys. Rev. A 65, 032320 (2002).
[Crossref]

Bruß, D.

D. Bruß and C. Macchiavello, "Optimal eavesdropping in cryptography with three-dimensional quantum states," Phys. Rev. Lett. 88, 127901 (2002).
[Crossref] [PubMed]

Buot, F. A.

F. A. Buot, "Method for calculating TrHn in solid-state theory," Phys. Rev. B 10, 3700-3705 (1974).
[Crossref]

Buzek, V.

M. Koniorczyk, V. Buzek, and J. Janszky, "Wigner-function description of quantum teleportation in arbitrary dimensions and a continuous limit," Phys. Rev. A 64, 034301 (2001).
[Crossref]

Calderbank, A. R.

A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane, "Quantum error correction and orthogonal geometry," Phys. Rev. Lett. 78, 405-408 (1997).
[Crossref]

A. R. Calderbank, P. J. Cameron, W. M. Cantor, and J. J. Seidel, "Z4-Kerdock codes, orthogonal spreads, and extremal Euclidean line-sets," Proc. London Math. Soc. 75, 436-480 (1997).
[Crossref]

Cameron, P. J.

A. R. Calderbank, P. J. Cameron, W. M. Cantor, and J. J. Seidel, "Z4-Kerdock codes, orthogonal spreads, and extremal Euclidean line-sets," Proc. London Math. Soc. 75, 436-480 (1997).
[Crossref]

Cantor, W. M.

A. R. Calderbank, P. J. Cameron, W. M. Cantor, and J. J. Seidel, "Z4-Kerdock codes, orthogonal spreads, and extremal Euclidean line-sets," Proc. London Math. Soc. 75, 436-480 (1997).
[Crossref]

Chaturvedi, S.

S. Chaturvedi, "Aspects of mutually unbiased bases in odd-prime-power dimensions," Phys. Rev. A 65, 044301 (2002).
[Crossref]

Cohendet, O.

O. Cohendet, Ph. Combe, M. Sirugue, and M. Sirugue-Collin, "A stochastic treatment of the dynamics of an integer spin," J. Phys. A 21, 2875-2884 (1988).
[Crossref]

Combe, Ph.

O. Cohendet, Ph. Combe, M. Sirugue, and M. Sirugue-Collin, "A stochastic treatment of the dynamics of an integer spin," J. Phys. A 21, 2875-2884 (1988).
[Crossref]

de Guise, H.

A. B. Klimov, L. L. Sánchez-Soto, and H. de Guise, "Multicomplementary operators via finite Fourier transform," J. Phys. A 38, 2747-2760 (2005).
[Crossref]

De Toledo Piza, A. F. R.

D. Galetti and A. F. R. De Toledo Piza, "An extended Weyl-Wigner transformation for special finite spaces," Physica A 149, 267-282 (1988).
[Crossref]

Delsarte, P.

P. Delsarte, J. M. Goethals, and J. J. Seidel, "Bounds for systems of lines and Jacobi polynomials," Philips Res. Rep. 30, 91-105 (1975).

Durt, T.

T. Durt, "About mutually unbiased bases in even and odd prime power dimensions," J. Phys. A 38, 5267-5284 (2005).
[Crossref]

T. Durt, "Tomography of one and two qubit states and factorisation of the Wigner distribution in prime power dimensions," arXiv.org e-Print archive, quant-ph/0604117, April 17, 2006, http://arxiv.org/abs/quant-ph/0604117.

T. Durt, "Bell states, mutually unbiased bases, and the mean king's problem," arXiv.org e-Print archive, quant-ph/0401037, June 27, 2005, http://arxiv.org/abs/quant-ph/0401037.

Englert, B.-G.

Y. Aharonov and B.-G. Englert, "The mean king's problem: spin 1," Z. Naturforsch., A: Phys. Sci. 56, 16-19 (2001).

B.-G. Englert and Y. Aharonov, "The mean king's problem: prime degrees of freedom," Phys. Lett. A 284, 1-5 (2001).
[Crossref]

Ericsson, A.

I. Bengtsson and A. Ericsson, "Mutually unbiased bases and the complementarity polytope," Open Syst. Inf. Dyn. 12, 107-120 (2005).
[Crossref]

Fields, B. D.

W. K. Wootters and B. D. Fields, "Optimal state-determination by mutually unbiased measurements," Ann. Phys. (N.Y.) 191, 363-381 (1989).
[Crossref]

Galetti, D.

D. Galetti and A. F. R. De Toledo Piza, "An extended Weyl-Wigner transformation for special finite spaces," Physica A 149, 267-282 (1988).
[Crossref]

Gibbons, K. S.

K. S. Gibbons, M. J. Hoffman, and W. K. Wootters, "Discrete phase space based on finite fields," Phys. Rev. A 70, 062101 (2004).
[Crossref]

Glauber, R. J.

R. J. Glauber, "Coherent and incoherent states of the radiation field," Phys. Rev. 131, 2766-2788 (1963).
[Crossref]

Goethals, J. M.

P. Delsarte, J. M. Goethals, and J. J. Seidel, "Bounds for systems of lines and Jacobi polynomials," Philips Res. Rep. 30, 91-105 (1975).

Grassl, M.

M. Grassl, "On SIC-POVMs and MUBs in dimension 6," arXiv.org e-Print archive, quant-ph/0406175, June 23, 2004, http://arxiv.org/abs/quant-ph/0406175.

Hannay, J. H.

J. H. Hannay and M. V. Berry, "Quantization of linear maps on a torus--Fresnel diffraction by a periodic grating," Physica D 1, 267-290 (1980).
[Crossref]

Hashimoto, T.

A. Hayashi, M. Horibe, and T. Hashimoto, "Mean king's problem with mutually unbiased bases and orthogonal Latin squares," Phys. Rev. A 71, 052331 (2005).
[Crossref]

Hayashi, A.

A. Hayashi, M. Horibe, and T. Hashimoto, "Mean king's problem with mutually unbiased bases and orthogonal Latin squares," Phys. Rev. A 71, 052331 (2005).
[Crossref]

Hoffman, M. J.

K. S. Gibbons, M. J. Hoffman, and W. K. Wootters, "Discrete phase space based on finite fields," Phys. Rev. A 70, 062101 (2004).
[Crossref]

Horibe, M.

A. Hayashi, M. Horibe, and T. Hashimoto, "Mean king's problem with mutually unbiased bases and orthogonal Latin squares," Phys. Rev. A 71, 052331 (2005).
[Crossref]

Husimi, K.

K. Husimi, "Some formal properties of the density matrix," Proc. Phys. Math. Soc. Jpn. 22, 264-314 (1940).

Ivanovic, I. D.

I. D. Ivanovic, "Geometrical description of quantal state determination," J. Phys. A 14, 3241-3246 (1981).
[Crossref]

Janszky, J.

M. Koniorczyk, V. Buzek, and J. Janszky, "Wigner-function description of quantum teleportation in arbitrary dimensions and a continuous limit," Phys. Rev. A 64, 034301 (2001).
[Crossref]

Klappenecker, A.

A. Klappenecker and M. Rötteler, "Constructions of mutually unbiased bases," Lect. Notes Comput. Sci. 2948, 137-144 (2004).
[Crossref]

Klimov, A. B.

A. B. Klimov, L. L. Sánchez-Soto, and H. de Guise, "Multicomplementary operators via finite Fourier transform," J. Phys. A 38, 2747-2760 (2005).
[Crossref]

J. L. Romero, G. Björk, A. B. Klimov, and L. L. Sánchez-Soto, "Structure of the sets of mutually unbiased bases for N qubits," Phys. Rev. A 72, 062310 (2005).
[Crossref]

Koniorczyk, M.

M. Koniorczyk, V. Buzek, and J. Janszky, "Wigner-function description of quantum teleportation in arbitrary dimensions and a continuous limit," Phys. Rev. A 64, 034301 (2001).
[Crossref]

Lawrence, J.

J. Lawrence, C. Brukner, and A. Zeilinger, "Mutually unbiased binary observable sets on N qubits," Phys. Rev. A 65, 032320 (2002).
[Crossref]

Leonhardt, U.

U. Leonhardt, "Discrete Wigner function and quantum-state tomography," Phys. Rev. A 53, 2998-3013 (1996).
[Crossref] [PubMed]

U. Leonhardt, "Quantum-state tomography and discrete Wigner function," Phys. Rev. Lett. 74, 4101-4105 (1995).
[Crossref] [PubMed]

Macchiavello, C.

D. Bruß and C. Macchiavello, "Optimal eavesdropping in cryptography with three-dimensional quantum states," Phys. Rev. Lett. 88, 127901 (2002).
[Crossref] [PubMed]

Miquel, C.

C. Miquel, J. P. Paz, and M. Saraceno, "Quantum computers in phase space," Phys. Rev. A 65, 062309 (2002).
[Crossref]

Paz, J. P.

C. Miquel, J. P. Paz, and M. Saraceno, "Quantum computers in phase space," Phys. Rev. A 65, 062309 (2002).
[Crossref]

J. P. Paz, "Discrete Wigner functions and the phase-space representation of quantum teleportation," Phys. Rev. A 65, 062311 (2002).
[Crossref]

Pegg, D. T.

J. A. Vaccaro and D. T. Pegg, "Wigner function for number and phase," Phys. Rev. A 41, 5156-5163 (1990).
[Crossref] [PubMed]

Peres, A.

H. Bechmann-Pasquinucci and A. Peres, "Quantum cryptography with 3-state systems," Phys. Rev. Lett. 85, 3313-3316 (2000).
[Crossref] [PubMed]

Pittenger, A. O.

A. O. Pittenger and M. H. Rubin, "Wigner function and separability for finite systems," J. Phys. A 38, 6005-6036 (2005).
[Crossref]

A. O. Pittenger and M. H. Rubin, "Mutually unbiased bases, generalized spin matrices and separability," Linear Algebr. Appl. 390, 255-278 (2004).
[Crossref]

Planat, M.

M. Saniga, M. Planat, and H. Rosu, "Mutually unbiased bases and finite projective planes," J. Opt. B 6, L19-L20 (2004).
[Crossref]

Rains, E. M.

A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane, "Quantum error correction and orthogonal geometry," Phys. Rev. Lett. 78, 405-408 (1997).
[Crossref]

Romero, J. L.

J. L. Romero, G. Björk, A. B. Klimov, and L. L. Sánchez-Soto, "Structure of the sets of mutually unbiased bases for N qubits," Phys. Rev. A 72, 062310 (2005).
[Crossref]

Rosu, H.

M. Saniga, M. Planat, and H. Rosu, "Mutually unbiased bases and finite projective planes," J. Opt. B 6, L19-L20 (2004).
[Crossref]

Rötteler, M.

A. Klappenecker and M. Rötteler, "Constructions of mutually unbiased bases," Lect. Notes Comput. Sci. 2948, 137-144 (2004).
[Crossref]

Roychowdhury, V.

S. Bandyopadhyay, P. O. Boykin, V. Roychowdhury, and F. Vatan, "A new proof for the existence of mutually unbiased bases," Algorithmica 34, 512-528 (2002).
[Crossref]

Rubin, M. H.

A. O. Pittenger and M. H. Rubin, "Wigner function and separability for finite systems," J. Phys. A 38, 6005-6036 (2005).
[Crossref]

A. O. Pittenger and M. H. Rubin, "Mutually unbiased bases, generalized spin matrices and separability," Linear Algebr. Appl. 390, 255-278 (2004).
[Crossref]

Sánchez-Soto, L. L.

A. B. Klimov, L. L. Sánchez-Soto, and H. de Guise, "Multicomplementary operators via finite Fourier transform," J. Phys. A 38, 2747-2760 (2005).
[Crossref]

J. L. Romero, G. Björk, A. B. Klimov, and L. L. Sánchez-Soto, "Structure of the sets of mutually unbiased bases for N qubits," Phys. Rev. A 72, 062310 (2005).
[Crossref]

Saniga, M.

M. Saniga, M. Planat, and H. Rosu, "Mutually unbiased bases and finite projective planes," J. Opt. B 6, L19-L20 (2004).
[Crossref]

Saraceno, M.

C. Miquel, J. P. Paz, and M. Saraceno, "Quantum computers in phase space," Phys. Rev. A 65, 062309 (2002).
[Crossref]

Seidel, J. J.

A. R. Calderbank, P. J. Cameron, W. M. Cantor, and J. J. Seidel, "Z4-Kerdock codes, orthogonal spreads, and extremal Euclidean line-sets," Proc. London Math. Soc. 75, 436-480 (1997).
[Crossref]

P. Delsarte, J. M. Goethals, and J. J. Seidel, "Bounds for systems of lines and Jacobi polynomials," Philips Res. Rep. 30, 91-105 (1975).

Shor, P. W.

A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane, "Quantum error correction and orthogonal geometry," Phys. Rev. Lett. 78, 405-408 (1997).
[Crossref]

Sirugue, M.

O. Cohendet, Ph. Combe, M. Sirugue, and M. Sirugue-Collin, "A stochastic treatment of the dynamics of an integer spin," J. Phys. A 21, 2875-2884 (1988).
[Crossref]

Sirugue-Collin, M.

O. Cohendet, Ph. Combe, M. Sirugue, and M. Sirugue-Collin, "A stochastic treatment of the dynamics of an integer spin," J. Phys. A 21, 2875-2884 (1988).
[Crossref]

Sloane, N. J. A.

A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane, "Quantum error correction and orthogonal geometry," Phys. Rev. Lett. 78, 405-408 (1997).
[Crossref]

Sudarshan, E. C. G.

E. C. G. Sudarshan, "Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams," Phys. Rev. Lett. 10, 277-279 (1963).
[Crossref]

Vaccaro, J. A.

J. A. Vaccaro and D. T. Pegg, "Wigner function for number and phase," Phys. Rev. A 41, 5156-5163 (1990).
[Crossref] [PubMed]

Vatan, F.

S. Bandyopadhyay, P. O. Boykin, V. Roychowdhury, and F. Vatan, "A new proof for the existence of mutually unbiased bases," Algorithmica 34, 512-528 (2002).
[Crossref]

Vourdas, A.

A. Vourdas, "Quantum systems with finite Hilbert space," Rep. Prog. Phys. 67, 267-248 (2004).
[Crossref]

Wigner, E. P.

E. P. Wigner, "On the quantum correction for thermodynamic equilibrium," Phys. Rev. 40, 749-759 (1932).
[Crossref]

Wocjan, P.

P. Wocjan and T. Beth, "New construction of mutually unbiased bases in square dimensions," Quantum Inf. Comput. 5, 93-101 (2005).

Wootters, W. K.

W. K. Wootters, "Quantum measurements and finite geometry," Found. Phys. 36, 112-126 (2006).
[Crossref]

W. K. Wootters, "Picturing qubits in phase space," IBM J. Res. Dev. 48, 99-110 (2004).
[Crossref]

K. S. Gibbons, M. J. Hoffman, and W. K. Wootters, "Discrete phase space based on finite fields," Phys. Rev. A 70, 062101 (2004).
[Crossref]

W. K. Wootters and B. D. Fields, "Optimal state-determination by mutually unbiased measurements," Ann. Phys. (N.Y.) 191, 363-381 (1989).
[Crossref]

W. K. Wootters, "A Wigner-function formulation of finite-state quantum mechanics," Ann. Phys. (N.Y.) 176, 1-21 (1987).
[Crossref]

W. K. Wootters, "Quantum mechanics without probability amplitudes," Found. Phys. 16, 391-405 (1986).
[Crossref]

Zeilinger, A.

J. Lawrence, C. Brukner, and A. Zeilinger, "Mutually unbiased binary observable sets on N qubits," Phys. Rev. A 65, 032320 (2002).
[Crossref]

Algorithmica (1)

S. Bandyopadhyay, P. O. Boykin, V. Roychowdhury, and F. Vatan, "A new proof for the existence of mutually unbiased bases," Algorithmica 34, 512-528 (2002).
[Crossref]

Ann. Phys. (N.Y.) (2)

W. K. Wootters, "A Wigner-function formulation of finite-state quantum mechanics," Ann. Phys. (N.Y.) 176, 1-21 (1987).
[Crossref]

W. K. Wootters and B. D. Fields, "Optimal state-determination by mutually unbiased measurements," Ann. Phys. (N.Y.) 191, 363-381 (1989).
[Crossref]

Found. Phys. (2)

W. K. Wootters, "Quantum mechanics without probability amplitudes," Found. Phys. 16, 391-405 (1986).
[Crossref]

W. K. Wootters, "Quantum measurements and finite geometry," Found. Phys. 36, 112-126 (2006).
[Crossref]

IBM J. Res. Dev. (1)

W. K. Wootters, "Picturing qubits in phase space," IBM J. Res. Dev. 48, 99-110 (2004).
[Crossref]

J. Math. Phys. (1)

C. Archer, "There is no generalization of known formulas for mutually unbiased bases," J. Math. Phys. 46, 022106 (2005).
[Crossref]

J. Opt. B (2)

M. Saniga, M. Planat, and H. Rosu, "Mutually unbiased bases and finite projective planes," J. Opt. B 6, L19-L20 (2004).
[Crossref]

R. Asplund, G. Björk, and M. Bourennane, "An expectation value expansion of Hermitian operators in a discrete Hilbert space," J. Opt. B 3, 163-170 (2001).
[Crossref]

J. Phys. A (5)

A. B. Klimov, L. L. Sánchez-Soto, and H. de Guise, "Multicomplementary operators via finite Fourier transform," J. Phys. A 38, 2747-2760 (2005).
[Crossref]

T. Durt, "About mutually unbiased bases in even and odd prime power dimensions," J. Phys. A 38, 5267-5284 (2005).
[Crossref]

A. O. Pittenger and M. H. Rubin, "Wigner function and separability for finite systems," J. Phys. A 38, 6005-6036 (2005).
[Crossref]

I. D. Ivanovic, "Geometrical description of quantal state determination," J. Phys. A 14, 3241-3246 (1981).
[Crossref]

O. Cohendet, Ph. Combe, M. Sirugue, and M. Sirugue-Collin, "A stochastic treatment of the dynamics of an integer spin," J. Phys. A 21, 2875-2884 (1988).
[Crossref]

Lect. Notes Comput. Sci. (1)

A. Klappenecker and M. Rötteler, "Constructions of mutually unbiased bases," Lect. Notes Comput. Sci. 2948, 137-144 (2004).
[Crossref]

Linear Algebr. Appl. (1)

A. O. Pittenger and M. H. Rubin, "Mutually unbiased bases, generalized spin matrices and separability," Linear Algebr. Appl. 390, 255-278 (2004).
[Crossref]

Open Syst. Inf. Dyn. (1)

I. Bengtsson and A. Ericsson, "Mutually unbiased bases and the complementarity polytope," Open Syst. Inf. Dyn. 12, 107-120 (2005).
[Crossref]

Philips Res. Rep. (1)

P. Delsarte, J. M. Goethals, and J. J. Seidel, "Bounds for systems of lines and Jacobi polynomials," Philips Res. Rep. 30, 91-105 (1975).

Phys. Lett. A (1)

B.-G. Englert and Y. Aharonov, "The mean king's problem: prime degrees of freedom," Phys. Lett. A 284, 1-5 (2001).
[Crossref]

Phys. Rev. (2)

R. J. Glauber, "Coherent and incoherent states of the radiation field," Phys. Rev. 131, 2766-2788 (1963).
[Crossref]

E. P. Wigner, "On the quantum correction for thermodynamic equilibrium," Phys. Rev. 40, 749-759 (1932).
[Crossref]

Phys. Rev. A (11)

U. Leonhardt, "Discrete Wigner function and quantum-state tomography," Phys. Rev. A 53, 2998-3013 (1996).
[Crossref] [PubMed]

K. S. Gibbons, M. J. Hoffman, and W. K. Wootters, "Discrete phase space based on finite fields," Phys. Rev. A 70, 062101 (2004).
[Crossref]

R. Asplund and G. Björk, "Reconstructing the discrete Wigner function and some properties of the measurement bases," Phys. Rev. A 64, 012106 (2001).
[Crossref]

M. Koniorczyk, V. Buzek, and J. Janszky, "Wigner-function description of quantum teleportation in arbitrary dimensions and a continuous limit," Phys. Rev. A 64, 034301 (2001).
[Crossref]

J. P. Paz, "Discrete Wigner functions and the phase-space representation of quantum teleportation," Phys. Rev. A 65, 062311 (2002).
[Crossref]

J. A. Vaccaro and D. T. Pegg, "Wigner function for number and phase," Phys. Rev. A 41, 5156-5163 (1990).
[Crossref] [PubMed]

C. Miquel, J. P. Paz, and M. Saraceno, "Quantum computers in phase space," Phys. Rev. A 65, 062309 (2002).
[Crossref]

A. Hayashi, M. Horibe, and T. Hashimoto, "Mean king's problem with mutually unbiased bases and orthogonal Latin squares," Phys. Rev. A 71, 052331 (2005).
[Crossref]

J. Lawrence, C. Brukner, and A. Zeilinger, "Mutually unbiased binary observable sets on N qubits," Phys. Rev. A 65, 032320 (2002).
[Crossref]

J. L. Romero, G. Björk, A. B. Klimov, and L. L. Sánchez-Soto, "Structure of the sets of mutually unbiased bases for N qubits," Phys. Rev. A 72, 062310 (2005).
[Crossref]

S. Chaturvedi, "Aspects of mutually unbiased bases in odd-prime-power dimensions," Phys. Rev. A 65, 044301 (2002).
[Crossref]

Phys. Rev. B (1)

F. A. Buot, "Method for calculating TrHn in solid-state theory," Phys. Rev. B 10, 3700-3705 (1974).
[Crossref]

Phys. Rev. Lett. (5)

U. Leonhardt, "Quantum-state tomography and discrete Wigner function," Phys. Rev. Lett. 74, 4101-4105 (1995).
[Crossref] [PubMed]

E. C. G. Sudarshan, "Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams," Phys. Rev. Lett. 10, 277-279 (1963).
[Crossref]

A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane, "Quantum error correction and orthogonal geometry," Phys. Rev. Lett. 78, 405-408 (1997).
[Crossref]

H. Bechmann-Pasquinucci and A. Peres, "Quantum cryptography with 3-state systems," Phys. Rev. Lett. 85, 3313-3316 (2000).
[Crossref] [PubMed]

D. Bruß and C. Macchiavello, "Optimal eavesdropping in cryptography with three-dimensional quantum states," Phys. Rev. Lett. 88, 127901 (2002).
[Crossref] [PubMed]

Physica A (1)

D. Galetti and A. F. R. De Toledo Piza, "An extended Weyl-Wigner transformation for special finite spaces," Physica A 149, 267-282 (1988).
[Crossref]

Physica D (1)

J. H. Hannay and M. V. Berry, "Quantization of linear maps on a torus--Fresnel diffraction by a periodic grating," Physica D 1, 267-290 (1980).
[Crossref]

Proc. London Math. Soc. (1)

A. R. Calderbank, P. J. Cameron, W. M. Cantor, and J. J. Seidel, "Z4-Kerdock codes, orthogonal spreads, and extremal Euclidean line-sets," Proc. London Math. Soc. 75, 436-480 (1997).
[Crossref]

Proc. Phys. Math. Soc. Jpn. (1)

K. Husimi, "Some formal properties of the density matrix," Proc. Phys. Math. Soc. Jpn. 22, 264-314 (1940).

Quantum Inf. Comput. (1)

P. Wocjan and T. Beth, "New construction of mutually unbiased bases in square dimensions," Quantum Inf. Comput. 5, 93-101 (2005).

Rep. Prog. Phys. (1)

A. Vourdas, "Quantum systems with finite Hilbert space," Rep. Prog. Phys. 67, 267-248 (2004).
[Crossref]

Z. Naturforsch., A: Phys. Sci. (3)

Y. Aharonov and B.-G. Englert, "The mean king's problem: spin 1," Z. Naturforsch., A: Phys. Sci. 56, 16-19 (2001).

P. K. Aravind, "Solution to the king's problem in prime power dimensions," Z. Naturforsch., A: Phys. Sci. 58, 85-92 (2003).

P. K. Aravind, "Best conventional solutions to the king's problem," Z. Naturforsch., A: Phys. Sci. 58, 682-690 (2003).

Other (4)

T. Durt, "Bell states, mutually unbiased bases, and the mean king's problem," arXiv.org e-Print archive, quant-ph/0401037, June 27, 2005, http://arxiv.org/abs/quant-ph/0401037.

C. H. Bennett and G. Brassard, in Proceedings of the IEEE International Conference on Computers, Systems, and Signal Processing, Bangalore, India (IEEE, 1984), pp. 175-179.
[PubMed]

T. Durt, "Tomography of one and two qubit states and factorisation of the Wigner distribution in prime power dimensions," arXiv.org e-Print archive, quant-ph/0604117, April 17, 2006, http://arxiv.org/abs/quant-ph/0604117.

M. Grassl, "On SIC-POVMs and MUBs in dimension 6," arXiv.org e-Print archive, quant-ph/0406175, June 23, 2004, http://arxiv.org/abs/quant-ph/0406175.

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Figures (6)

Fig. 1
Fig. 1

Striation-generating curves corresponding to the MUB construction defined by Table 1.

Fig. 2
Fig. 2

Striation-generating lines corresponding to the MUB construction defined by the second and fourth sets in Table 2.

Fig. 3
Fig. 3

Striation-generating rays corresponding to the MUB construction defined by Table 3, left, and the lines resulting from the displacement of the rays by ( 0 , μ 3 ) , right.

Fig. 4
Fig. 4

Homogeneous curves corresponding to the MUB construction defined by Table 4.

Fig. 5
Fig. 5

Homogeneous lines corresponding to the MUB construction defined by Table 5.

Fig. 6
Fig. 6

Homogeneous lines corresponding to the MUB construction defined by Table 6.

Tables (6)

Tables Icon

Table 1 Nine Sets of Operators Defining a (2,3,4) MUB

Tables Icon

Table 2 Five Different Sets Resulting from a Rearranging of the Three Qubits of Table 1 as 132, 213, 312, 321, and 231, Respectively a

Tables Icon

Table 3 Table Defining a (3,0,6) MUB Structure

Tables Icon

Table 4 Another Table Defining a (3,0,6) MUB Structure

Tables Icon

Table 5 Table Defining a (1,6,2) MUB Structure

Tables Icon

Table 6 Table Defining a (0,9,0) MUB Structure

Equations (31)

Equations on this page are rendered with MathJax. Learn more.

ρ ̂ = exp [ i ( q 0 p ̂ p 0 q ̂ ) ] ρ ̂ exp [ i ( q 0 p ̂ p 0 q ̂ ) ] ,
W ( q , p ) = W ( q q 0 , p p 0 ) .
θ 3 + θ + 1 = 0 .
1 = μ 3 + μ 5 + μ 6 , μ = μ 5 + μ 6 ,
μ 2 = μ 3 + μ 5 , μ 4 = μ 3 + μ 6 .
tr ( μ i μ j ) = δ i j , i , j 3 , 5 , 6 .
tr θ = θ + θ 2 + θ 4 .
( μ 3 , 0 ) σ ̂ x 1 ̂ 1 ̂ ,
( μ 5 , 0 ) 1 ̂ σ ̂ x 1 ̂ ,
( μ 6 , 0 ) 1 ̂ 1 ̂ σ ̂ x ;
( 0 , μ 3 ) σ ̂ z 1 ̂ 1 ̂ ,
( 0 , μ 5 ) 1 ̂ σ ̂ z 1 ̂ ,
( 0 , μ 6 ) 1 ̂ 1 ̂ σ ̂ z .
α = μ 3 κ + μ 6 κ 2 + μ 6 κ 4 , β = μ 2 κ + κ 2 + μ 4 κ 4 ,
β 2 + μ β = α 2 + μ α ,
β = μ 6 α + μ 3 α 2 + μ 5 α 4 ,
A ̂ ( α , β ) = k = 1 9 ρ ̂ k , ( α , β ) 1 ̂ ,
Tr [ A ̂ ( α , β ) ] = 1 ,
Tr [ A ̂ ( α , β ) A ̂ ( α , β ) ] = 8 δ α , α δ β , β ,
W ( α , β ) = 1 8 Tr [ ρ ̂ A ̂ ( α , β ) ] .
α , β W ( α , β ) = 1 ,
ρ ̂ = α , β W ( α , β ) A ̂ ( α , β ) ,
Tr ( ρ ̂ ρ ̂ ) = 8 α , β W ( α , β ) W ( α , β ) ,
α = μ 2 κ 4 , β = μ 2 κ + κ 2 + μ κ 4 ,
β 2 + μ 4 β = μ 3 α 2 + μ 2 α
β = μ 6 α + μ 5 α 2 + μ 6 α 4 .
α ( κ ) = ν 1 κ + ν 2 κ 2 + ν 3 κ 4 , β ( κ ) = η 1 κ + η 2 κ 2 + η 3 κ 4 ,
α ( κ + κ ) = α ( κ ) + α ( κ ) ,
β ( κ + κ ) = β ( κ ) + β ( κ ) .
α = 0 or β = λ α ,
α ( κ ) = η κ , β ( κ ) = ζ κ ,

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