Abstract

We describe a unified framework of phase-covariant multiuser quantum transformations for d-dimensional quantum systems. We derive the optimal phase-covariant cloning and transposition transformations for multiphase states. We show that for some particular relations between the input and output number of copies, they correspond to economical transformations, which can be achieved without the need of auxiliary systems. We prove a relation between the optimal phase-covariant cloning and transposition maps and optimal estimation of multiple phases for equatorial states.

© 2007 Optical Society of America

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  1. H. Bechmann-Pasquinucci and A. Peres, "Quantum cryptography with 3-state systems," Phys. Rev. Lett. 85, 3313-3316 (2000).
    [CrossRef] [PubMed]
  2. D. Bruß and C. Macchiavello, "Optimal eavesdropping in cryptography with three-dimensional quantum states," Phys. Rev. Lett. 88, 127901 (2002).
    [CrossRef] [PubMed]
  3. N. Cerf, M. Bourennane, A. Karlsson, and N. Gisin, "Security of quantum key distribution using d-level systems," Phys. Rev. Lett. 88, 127902 (2002).
    [CrossRef] [PubMed]
  4. M. Fitzi, N. Gisin, and U. Maurer, "Quantum solution to the Byzantine agreement problem," Phys. Rev. Lett. 87, 217901 (2002).
    [CrossRef]
  5. G. Molina-Terriza, A. Vaziri, J. Rehacek, Z. Hradila, and A. Zeilinger, "Triggered qutrits for quantum communication protocols," Phys. Rev. Lett. 92, 167903 (2004).
    [CrossRef] [PubMed]
  6. R. T. Thew, A. Acin, H. Zbinden, and N. Gisin, "Bell-type test of energy-time entangled qutrits," Phys. Rev. Lett. 93, 010503 (2004).
    [CrossRef]
  7. R. Das, A. Mitra, V. Kumar, and A. Kumar, "Quantum information processing by NMR: preparation of pseudopure states and implementation of unitary operations in a single-qutrit system," http:arxiv.orglabs/quant-ph/0307240.
  8. R. Cleve, A. Ekert, C. Macchiavello, and M. Mosca, "Quantum algorithms revisited," Proc. R. Soc. London, Ser. A 454, 339-354 (1998).
    [CrossRef]
  9. W. K. Wootters and W. H. Zurek, "A single quantum cannot be cloned," Nature 299, 802-803 (1982).
    [CrossRef]
  10. C. H. Bennett and G. Brassard, "Quantum cryptography: public key distribution and coin tossing," in Proceedings of the IEEE International Conference on Computers, Systems, and Signal Processing (IEEE Press, 1984), pp. 175-179.
  11. A. Ekert, "Quantum cryptography based on Bell's theorem," Phys. Rev. Lett. 67, 661-663 (1991).
    [CrossRef] [PubMed]
  12. For a review, see, for example, V. Scarani, S. Iblisdir, N. Gisin, and A. Acín, "Quantum cloning," Rev. Mod. Phys. 77, 1225-1256 (2005).
    [CrossRef]
  13. D. Bruß, M. Cinchetti, G. M. D'Ariano, and C. Macchiavello, "Phase-covariant quantum cloning," Phys. Rev. A 62, 012302 (2000).
    [CrossRef]
  14. F. Caruso, H. Bechmann Pasquinucci, and C. Macchiavello, "Robustness of a quantum key distribution with two and three mutually unbiased bases," Phys. Rev. A 72, 032340 (2005).
    [CrossRef]
  15. V. Scarani and N. Gisin, "Spectral decomposition of Bell's operators for qubits," J. Phys. A 34, 6043-6053 (2001).
    [CrossRef]
  16. A. Peres, "Separability criterion for density matrices," Phys. Rev. Lett. 77, 1413-1415 (1996).
    [CrossRef] [PubMed]
  17. P. Horodecki, "Separability criterion and inseparable mixed states with positive partial transposition," Phys. Lett. A 232, 333-339 (1997).
    [CrossRef]
  18. V. Buzek, M. Hillery, and R. F. Werner, "Optimal manipulations with qubits: universal-NOT gate," Phys. Rev. A 60, R2626-R2629 (1999).
    [CrossRef]
  19. F. Buscemi, G. M. D'Ariano, P. Perinotti, and M. F. Sacchi, "Optimal realization of the transposition maps," Phys. Lett. A 314, 374-379 (2003).
    [CrossRef]
  20. F. Buscemi, G. M. D'Ariano, and C. Macchiavello, "Optimal time-reversal of multi-phase equatorial states," Phys. Rev. A 72, 062311 (2005).
    [CrossRef]
  21. K. Kraus, States, Effects, and Operations: Fundamental Notions in Quantum Theory, Lecture Notes in Physics (Springer-Verlag, 1983), Vol. 190.
    [CrossRef]
  22. A. Jamiolkowski, "Linear transformations which preserve trace and positive semidefiniteness of operators," Rep. Math. Phys. 3, 275-278 (1972).
    [CrossRef]
  23. M.-D. Choi, "Completely positive linear maps on complex matrices," Linear Algebr. Appl. 10, 285-290 (1975).
    [CrossRef]
  24. G. M. D'Ariano and P. Lo Presti, "Optimal nonuniversally covariant cloning," Phys. Rev. A 64, 042308 (2001).
    [CrossRef]
  25. The group U(1)×(d−1) is commutative, hence its irreducible representations are all one dimensional.
  26. W. F. Stinespring, "Positive functions on C*-algebras," Proc. Am. Math. Soc. 6, 211-216 (1955).
  27. M. Ozawa, "Quantum measuring processes of continuous observables," J. Math. Phys. 25, 79-87 (1984).
    [CrossRef]
  28. F. Buscemi, G. M. D'Ariano, and M. F. Sacchi, "Physical realizations of quantum operations," Phys. Rev. A 68, 042113 (2003).
    [CrossRef]
  29. M. Keyl and R. F. Werner, "Optimal cloning of pure states, judging single clones," J. Math. Phys. 40, 3283-3299 (1999).
    [CrossRef]
  30. G. M. D'Ariano and C. Macchiavello, "Optimal phase-covariant cloning for qubits and qutrits," Phys. Rev. A 67, 042306 (2003).
    [CrossRef]
  31. F. Buscemi, G. M. D'Ariano, and C. Macchiavello, "Economical phase-covariant cloning of qudits," Phys. Rev. A 71, 042327 (2005).
    [CrossRef]
  32. Actually, U(1)×(d−1) is a proper subgroup of SU(d).
  33. T. Durt, J. Fiurasek, and N. J. Cerf, "Economical quantum cloning in any dimension," Phys. Rev. A 72, 052322 (2005).
    [CrossRef]
  34. C. Macchiavello, "Optimal estimation of multiple phases," Phys. Rev. A 67, 062302 (2003).
    [CrossRef]
  35. Recently, it has been proved that cloning channels, in the limit of infinite output copies, tend to measure-and-prepare channels. Here, we are able to explicitly show how fast this limit is reached, for every finite M.
  36. J. Bae and A. Acín, "Asymptotic quantum cloning is state estimation," http://arxiv.orglabs/quant-ph/0603078.
  37. R. Derka, V. Buzek, and A. K. Ekert, "Universal algorithm for optimal estimation of quantum states from finite ensembles via realizable generalized measurement," Phys. Rev. Lett. 80, 1571-1575 (1998).
    [CrossRef]
  38. D. Bruß and C. Macchiavello, "Optimal state estimation for d-dimensional quantum systems," Phys. Lett. A 253, 249-251 (1999).
    [CrossRef]
  39. This holds by linearity, since every symmetric operator O can be written as a linear combination of N-fold tensor product pure states, namely, O=summation i lambda i∣psi i›‹psi i∣big dot times N.
  40. G. M. D'Ariano, V. Giovannetti, and P. Perinotti, "Optimal estimation of quantum observables," J. Math. Phys. 47, 022102 (2006).
    [CrossRef]
  41. L. Susskind and J. Glogower, "Quantum mechanical phase and time operator," Physics (Long Island City, N.Y.) 1, 49-61 (1964).

2006 (1)

G. M. D'Ariano, V. Giovannetti, and P. Perinotti, "Optimal estimation of quantum observables," J. Math. Phys. 47, 022102 (2006).
[CrossRef]

2005 (5)

F. Buscemi, G. M. D'Ariano, and C. Macchiavello, "Economical phase-covariant cloning of qudits," Phys. Rev. A 71, 042327 (2005).
[CrossRef]

T. Durt, J. Fiurasek, and N. J. Cerf, "Economical quantum cloning in any dimension," Phys. Rev. A 72, 052322 (2005).
[CrossRef]

For a review, see, for example, V. Scarani, S. Iblisdir, N. Gisin, and A. Acín, "Quantum cloning," Rev. Mod. Phys. 77, 1225-1256 (2005).
[CrossRef]

F. Caruso, H. Bechmann Pasquinucci, and C. Macchiavello, "Robustness of a quantum key distribution with two and three mutually unbiased bases," Phys. Rev. A 72, 032340 (2005).
[CrossRef]

F. Buscemi, G. M. D'Ariano, and C. Macchiavello, "Optimal time-reversal of multi-phase equatorial states," Phys. Rev. A 72, 062311 (2005).
[CrossRef]

2004 (2)

G. Molina-Terriza, A. Vaziri, J. Rehacek, Z. Hradila, and A. Zeilinger, "Triggered qutrits for quantum communication protocols," Phys. Rev. Lett. 92, 167903 (2004).
[CrossRef] [PubMed]

R. T. Thew, A. Acin, H. Zbinden, and N. Gisin, "Bell-type test of energy-time entangled qutrits," Phys. Rev. Lett. 93, 010503 (2004).
[CrossRef]

2003 (4)

C. Macchiavello, "Optimal estimation of multiple phases," Phys. Rev. A 67, 062302 (2003).
[CrossRef]

F. Buscemi, G. M. D'Ariano, P. Perinotti, and M. F. Sacchi, "Optimal realization of the transposition maps," Phys. Lett. A 314, 374-379 (2003).
[CrossRef]

F. Buscemi, G. M. D'Ariano, and M. F. Sacchi, "Physical realizations of quantum operations," Phys. Rev. A 68, 042113 (2003).
[CrossRef]

G. M. D'Ariano and C. Macchiavello, "Optimal phase-covariant cloning for qubits and qutrits," Phys. Rev. A 67, 042306 (2003).
[CrossRef]

2002 (3)

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

N. Cerf, M. Bourennane, A. Karlsson, and N. Gisin, "Security of quantum key distribution using d-level systems," Phys. Rev. Lett. 88, 127902 (2002).
[CrossRef] [PubMed]

M. Fitzi, N. Gisin, and U. Maurer, "Quantum solution to the Byzantine agreement problem," Phys. Rev. Lett. 87, 217901 (2002).
[CrossRef]

2001 (2)

V. Scarani and N. Gisin, "Spectral decomposition of Bell's operators for qubits," J. Phys. A 34, 6043-6053 (2001).
[CrossRef]

G. M. D'Ariano and P. Lo Presti, "Optimal nonuniversally covariant cloning," Phys. Rev. A 64, 042308 (2001).
[CrossRef]

2000 (2)

D. Bruß, M. Cinchetti, G. M. D'Ariano, and C. Macchiavello, "Phase-covariant quantum cloning," Phys. Rev. A 62, 012302 (2000).
[CrossRef]

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

1999 (3)

V. Buzek, M. Hillery, and R. F. Werner, "Optimal manipulations with qubits: universal-NOT gate," Phys. Rev. A 60, R2626-R2629 (1999).
[CrossRef]

M. Keyl and R. F. Werner, "Optimal cloning of pure states, judging single clones," J. Math. Phys. 40, 3283-3299 (1999).
[CrossRef]

D. Bruß and C. Macchiavello, "Optimal state estimation for d-dimensional quantum systems," Phys. Lett. A 253, 249-251 (1999).
[CrossRef]

1998 (2)

R. Derka, V. Buzek, and A. K. Ekert, "Universal algorithm for optimal estimation of quantum states from finite ensembles via realizable generalized measurement," Phys. Rev. Lett. 80, 1571-1575 (1998).
[CrossRef]

R. Cleve, A. Ekert, C. Macchiavello, and M. Mosca, "Quantum algorithms revisited," Proc. R. Soc. London, Ser. A 454, 339-354 (1998).
[CrossRef]

1997 (1)

P. Horodecki, "Separability criterion and inseparable mixed states with positive partial transposition," Phys. Lett. A 232, 333-339 (1997).
[CrossRef]

1996 (1)

A. Peres, "Separability criterion for density matrices," Phys. Rev. Lett. 77, 1413-1415 (1996).
[CrossRef] [PubMed]

1991 (1)

A. Ekert, "Quantum cryptography based on Bell's theorem," Phys. Rev. Lett. 67, 661-663 (1991).
[CrossRef] [PubMed]

1984 (1)

M. Ozawa, "Quantum measuring processes of continuous observables," J. Math. Phys. 25, 79-87 (1984).
[CrossRef]

1982 (1)

W. K. Wootters and W. H. Zurek, "A single quantum cannot be cloned," Nature 299, 802-803 (1982).
[CrossRef]

1975 (1)

M.-D. Choi, "Completely positive linear maps on complex matrices," Linear Algebr. Appl. 10, 285-290 (1975).
[CrossRef]

1972 (1)

A. Jamiolkowski, "Linear transformations which preserve trace and positive semidefiniteness of operators," Rep. Math. Phys. 3, 275-278 (1972).
[CrossRef]

1964 (1)

L. Susskind and J. Glogower, "Quantum mechanical phase and time operator," Physics (Long Island City, N.Y.) 1, 49-61 (1964).

1955 (1)

W. F. Stinespring, "Positive functions on C*-algebras," Proc. Am. Math. Soc. 6, 211-216 (1955).

Acin, A.

R. T. Thew, A. Acin, H. Zbinden, and N. Gisin, "Bell-type test of energy-time entangled qutrits," Phys. Rev. Lett. 93, 010503 (2004).
[CrossRef]

Acín, A.

For a review, see, for example, V. Scarani, S. Iblisdir, N. Gisin, and A. Acín, "Quantum cloning," Rev. Mod. Phys. 77, 1225-1256 (2005).
[CrossRef]

J. Bae and A. Acín, "Asymptotic quantum cloning is state estimation," http://arxiv.orglabs/quant-ph/0603078.

Bae, J.

J. Bae and A. Acín, "Asymptotic quantum cloning is state estimation," http://arxiv.orglabs/quant-ph/0603078.

Bechmann Pasquinucci, H.

F. Caruso, H. Bechmann Pasquinucci, and C. Macchiavello, "Robustness of a quantum key distribution with two and three mutually unbiased bases," Phys. Rev. A 72, 032340 (2005).
[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]

Bennett, C. H.

C. H. Bennett and G. Brassard, "Quantum cryptography: public key distribution and coin tossing," in Proceedings of the IEEE International Conference on Computers, Systems, and Signal Processing (IEEE Press, 1984), pp. 175-179.

Bourennane, M.

N. Cerf, M. Bourennane, A. Karlsson, and N. Gisin, "Security of quantum key distribution using d-level systems," Phys. Rev. Lett. 88, 127902 (2002).
[CrossRef] [PubMed]

Brassard, G.

C. H. Bennett and G. Brassard, "Quantum cryptography: public key distribution and coin tossing," in Proceedings of the IEEE International Conference on Computers, Systems, and Signal Processing (IEEE Press, 1984), pp. 175-179.

Bruß, D.

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

D. Bruß, M. Cinchetti, G. M. D'Ariano, and C. Macchiavello, "Phase-covariant quantum cloning," Phys. Rev. A 62, 012302 (2000).
[CrossRef]

D. Bruß and C. Macchiavello, "Optimal state estimation for d-dimensional quantum systems," Phys. Lett. A 253, 249-251 (1999).
[CrossRef]

Buscemi, F.

F. Buscemi, G. M. D'Ariano, and C. Macchiavello, "Optimal time-reversal of multi-phase equatorial states," Phys. Rev. A 72, 062311 (2005).
[CrossRef]

F. Buscemi, G. M. D'Ariano, and C. Macchiavello, "Economical phase-covariant cloning of qudits," Phys. Rev. A 71, 042327 (2005).
[CrossRef]

F. Buscemi, G. M. D'Ariano, and M. F. Sacchi, "Physical realizations of quantum operations," Phys. Rev. A 68, 042113 (2003).
[CrossRef]

F. Buscemi, G. M. D'Ariano, P. Perinotti, and M. F. Sacchi, "Optimal realization of the transposition maps," Phys. Lett. A 314, 374-379 (2003).
[CrossRef]

Buzek, V.

V. Buzek, M. Hillery, and R. F. Werner, "Optimal manipulations with qubits: universal-NOT gate," Phys. Rev. A 60, R2626-R2629 (1999).
[CrossRef]

R. Derka, V. Buzek, and A. K. Ekert, "Universal algorithm for optimal estimation of quantum states from finite ensembles via realizable generalized measurement," Phys. Rev. Lett. 80, 1571-1575 (1998).
[CrossRef]

Caruso, F.

F. Caruso, H. Bechmann Pasquinucci, and C. Macchiavello, "Robustness of a quantum key distribution with two and three mutually unbiased bases," Phys. Rev. A 72, 032340 (2005).
[CrossRef]

Cerf, N.

N. Cerf, M. Bourennane, A. Karlsson, and N. Gisin, "Security of quantum key distribution using d-level systems," Phys. Rev. Lett. 88, 127902 (2002).
[CrossRef] [PubMed]

Cerf, N. J.

T. Durt, J. Fiurasek, and N. J. Cerf, "Economical quantum cloning in any dimension," Phys. Rev. A 72, 052322 (2005).
[CrossRef]

Choi, M.-D.

M.-D. Choi, "Completely positive linear maps on complex matrices," Linear Algebr. Appl. 10, 285-290 (1975).
[CrossRef]

Cinchetti, M.

D. Bruß, M. Cinchetti, G. M. D'Ariano, and C. Macchiavello, "Phase-covariant quantum cloning," Phys. Rev. A 62, 012302 (2000).
[CrossRef]

Cleve, R.

R. Cleve, A. Ekert, C. Macchiavello, and M. Mosca, "Quantum algorithms revisited," Proc. R. Soc. London, Ser. A 454, 339-354 (1998).
[CrossRef]

D'Ariano, G. M.

G. M. D'Ariano, V. Giovannetti, and P. Perinotti, "Optimal estimation of quantum observables," J. Math. Phys. 47, 022102 (2006).
[CrossRef]

F. Buscemi, G. M. D'Ariano, and C. Macchiavello, "Economical phase-covariant cloning of qudits," Phys. Rev. A 71, 042327 (2005).
[CrossRef]

F. Buscemi, G. M. D'Ariano, and C. Macchiavello, "Optimal time-reversal of multi-phase equatorial states," Phys. Rev. A 72, 062311 (2005).
[CrossRef]

F. Buscemi, G. M. D'Ariano, and M. F. Sacchi, "Physical realizations of quantum operations," Phys. Rev. A 68, 042113 (2003).
[CrossRef]

F. Buscemi, G. M. D'Ariano, P. Perinotti, and M. F. Sacchi, "Optimal realization of the transposition maps," Phys. Lett. A 314, 374-379 (2003).
[CrossRef]

G. M. D'Ariano and C. Macchiavello, "Optimal phase-covariant cloning for qubits and qutrits," Phys. Rev. A 67, 042306 (2003).
[CrossRef]

G. M. D'Ariano and P. Lo Presti, "Optimal nonuniversally covariant cloning," Phys. Rev. A 64, 042308 (2001).
[CrossRef]

D. Bruß, M. Cinchetti, G. M. D'Ariano, and C. Macchiavello, "Phase-covariant quantum cloning," Phys. Rev. A 62, 012302 (2000).
[CrossRef]

Das, R.

R. Das, A. Mitra, V. Kumar, and A. Kumar, "Quantum information processing by NMR: preparation of pseudopure states and implementation of unitary operations in a single-qutrit system," http:arxiv.orglabs/quant-ph/0307240.

Derka, R.

R. Derka, V. Buzek, and A. K. Ekert, "Universal algorithm for optimal estimation of quantum states from finite ensembles via realizable generalized measurement," Phys. Rev. Lett. 80, 1571-1575 (1998).
[CrossRef]

Durt, T.

T. Durt, J. Fiurasek, and N. J. Cerf, "Economical quantum cloning in any dimension," Phys. Rev. A 72, 052322 (2005).
[CrossRef]

Ekert, A.

R. Cleve, A. Ekert, C. Macchiavello, and M. Mosca, "Quantum algorithms revisited," Proc. R. Soc. London, Ser. A 454, 339-354 (1998).
[CrossRef]

A. Ekert, "Quantum cryptography based on Bell's theorem," Phys. Rev. Lett. 67, 661-663 (1991).
[CrossRef] [PubMed]

Ekert, A. K.

R. Derka, V. Buzek, and A. K. Ekert, "Universal algorithm for optimal estimation of quantum states from finite ensembles via realizable generalized measurement," Phys. Rev. Lett. 80, 1571-1575 (1998).
[CrossRef]

Fitzi, M.

M. Fitzi, N. Gisin, and U. Maurer, "Quantum solution to the Byzantine agreement problem," Phys. Rev. Lett. 87, 217901 (2002).
[CrossRef]

Fiurasek, J.

T. Durt, J. Fiurasek, and N. J. Cerf, "Economical quantum cloning in any dimension," Phys. Rev. A 72, 052322 (2005).
[CrossRef]

Giovannetti, V.

G. M. D'Ariano, V. Giovannetti, and P. Perinotti, "Optimal estimation of quantum observables," J. Math. Phys. 47, 022102 (2006).
[CrossRef]

Gisin, N.

For a review, see, for example, V. Scarani, S. Iblisdir, N. Gisin, and A. Acín, "Quantum cloning," Rev. Mod. Phys. 77, 1225-1256 (2005).
[CrossRef]

R. T. Thew, A. Acin, H. Zbinden, and N. Gisin, "Bell-type test of energy-time entangled qutrits," Phys. Rev. Lett. 93, 010503 (2004).
[CrossRef]

M. Fitzi, N. Gisin, and U. Maurer, "Quantum solution to the Byzantine agreement problem," Phys. Rev. Lett. 87, 217901 (2002).
[CrossRef]

N. Cerf, M. Bourennane, A. Karlsson, and N. Gisin, "Security of quantum key distribution using d-level systems," Phys. Rev. Lett. 88, 127902 (2002).
[CrossRef] [PubMed]

V. Scarani and N. Gisin, "Spectral decomposition of Bell's operators for qubits," J. Phys. A 34, 6043-6053 (2001).
[CrossRef]

Glogower, J.

L. Susskind and J. Glogower, "Quantum mechanical phase and time operator," Physics (Long Island City, N.Y.) 1, 49-61 (1964).

Hillery, M.

V. Buzek, M. Hillery, and R. F. Werner, "Optimal manipulations with qubits: universal-NOT gate," Phys. Rev. A 60, R2626-R2629 (1999).
[CrossRef]

Horodecki, P.

P. Horodecki, "Separability criterion and inseparable mixed states with positive partial transposition," Phys. Lett. A 232, 333-339 (1997).
[CrossRef]

Hradila, Z.

G. Molina-Terriza, A. Vaziri, J. Rehacek, Z. Hradila, and A. Zeilinger, "Triggered qutrits for quantum communication protocols," Phys. Rev. Lett. 92, 167903 (2004).
[CrossRef] [PubMed]

Iblisdir, S.

For a review, see, for example, V. Scarani, S. Iblisdir, N. Gisin, and A. Acín, "Quantum cloning," Rev. Mod. Phys. 77, 1225-1256 (2005).
[CrossRef]

Jamiolkowski, A.

A. Jamiolkowski, "Linear transformations which preserve trace and positive semidefiniteness of operators," Rep. Math. Phys. 3, 275-278 (1972).
[CrossRef]

Karlsson, A.

N. Cerf, M. Bourennane, A. Karlsson, and N. Gisin, "Security of quantum key distribution using d-level systems," Phys. Rev. Lett. 88, 127902 (2002).
[CrossRef] [PubMed]

Keyl, M.

M. Keyl and R. F. Werner, "Optimal cloning of pure states, judging single clones," J. Math. Phys. 40, 3283-3299 (1999).
[CrossRef]

Kraus, K.

K. Kraus, States, Effects, and Operations: Fundamental Notions in Quantum Theory, Lecture Notes in Physics (Springer-Verlag, 1983), Vol. 190.
[CrossRef]

Kumar, A.

R. Das, A. Mitra, V. Kumar, and A. Kumar, "Quantum information processing by NMR: preparation of pseudopure states and implementation of unitary operations in a single-qutrit system," http:arxiv.orglabs/quant-ph/0307240.

Kumar, V.

R. Das, A. Mitra, V. Kumar, and A. Kumar, "Quantum information processing by NMR: preparation of pseudopure states and implementation of unitary operations in a single-qutrit system," http:arxiv.orglabs/quant-ph/0307240.

Lo Presti, P.

G. M. D'Ariano and P. Lo Presti, "Optimal nonuniversally covariant cloning," Phys. Rev. A 64, 042308 (2001).
[CrossRef]

Macchiavello, C.

F. Buscemi, G. M. D'Ariano, and C. Macchiavello, "Economical phase-covariant cloning of qudits," Phys. Rev. A 71, 042327 (2005).
[CrossRef]

F. Buscemi, G. M. D'Ariano, and C. Macchiavello, "Optimal time-reversal of multi-phase equatorial states," Phys. Rev. A 72, 062311 (2005).
[CrossRef]

F. Caruso, H. Bechmann Pasquinucci, and C. Macchiavello, "Robustness of a quantum key distribution with two and three mutually unbiased bases," Phys. Rev. A 72, 032340 (2005).
[CrossRef]

G. M. D'Ariano and C. Macchiavello, "Optimal phase-covariant cloning for qubits and qutrits," Phys. Rev. A 67, 042306 (2003).
[CrossRef]

C. Macchiavello, "Optimal estimation of multiple phases," Phys. Rev. A 67, 062302 (2003).
[CrossRef]

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

D. Bruß, M. Cinchetti, G. M. D'Ariano, and C. Macchiavello, "Phase-covariant quantum cloning," Phys. Rev. A 62, 012302 (2000).
[CrossRef]

D. Bruß and C. Macchiavello, "Optimal state estimation for d-dimensional quantum systems," Phys. Lett. A 253, 249-251 (1999).
[CrossRef]

R. Cleve, A. Ekert, C. Macchiavello, and M. Mosca, "Quantum algorithms revisited," Proc. R. Soc. London, Ser. A 454, 339-354 (1998).
[CrossRef]

Maurer, U.

M. Fitzi, N. Gisin, and U. Maurer, "Quantum solution to the Byzantine agreement problem," Phys. Rev. Lett. 87, 217901 (2002).
[CrossRef]

Mitra, A.

R. Das, A. Mitra, V. Kumar, and A. Kumar, "Quantum information processing by NMR: preparation of pseudopure states and implementation of unitary operations in a single-qutrit system," http:arxiv.orglabs/quant-ph/0307240.

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G. Molina-Terriza, A. Vaziri, J. Rehacek, Z. Hradila, and A. Zeilinger, "Triggered qutrits for quantum communication protocols," Phys. Rev. Lett. 92, 167903 (2004).
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[CrossRef]

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G. Molina-Terriza, A. Vaziri, J. Rehacek, Z. Hradila, and A. Zeilinger, "Triggered qutrits for quantum communication protocols," Phys. Rev. Lett. 92, 167903 (2004).
[CrossRef] [PubMed]

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For a review, see, for example, V. Scarani, S. Iblisdir, N. Gisin, and A. Acín, "Quantum cloning," Rev. Mod. Phys. 77, 1225-1256 (2005).
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[CrossRef]

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G. Molina-Terriza, A. Vaziri, J. Rehacek, Z. Hradila, and A. Zeilinger, "Triggered qutrits for quantum communication protocols," Phys. Rev. Lett. 92, 167903 (2004).
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R. T. Thew, A. Acin, H. Zbinden, and N. Gisin, "Bell-type test of energy-time entangled qutrits," Phys. Rev. Lett. 93, 010503 (2004).
[CrossRef]

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G. Molina-Terriza, A. Vaziri, J. Rehacek, Z. Hradila, and A. Zeilinger, "Triggered qutrits for quantum communication protocols," Phys. Rev. Lett. 92, 167903 (2004).
[CrossRef] [PubMed]

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[CrossRef]

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[CrossRef]

V. Buzek, M. Hillery, and R. F. Werner, "Optimal manipulations with qubits: universal-NOT gate," Phys. Rev. A 60, R2626-R2629 (1999).
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[CrossRef]

F. Buscemi, G. M. D'Ariano, and M. F. Sacchi, "Physical realizations of quantum operations," Phys. Rev. A 68, 042113 (2003).
[CrossRef]

T. Durt, J. Fiurasek, and N. J. Cerf, "Economical quantum cloning in any dimension," Phys. Rev. A 72, 052322 (2005).
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[CrossRef] [PubMed]

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

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G. Molina-Terriza, A. Vaziri, J. Rehacek, Z. Hradila, and A. Zeilinger, "Triggered qutrits for quantum communication protocols," Phys. Rev. Lett. 92, 167903 (2004).
[CrossRef] [PubMed]

R. T. Thew, A. Acin, H. Zbinden, and N. Gisin, "Bell-type test of energy-time entangled qutrits," Phys. Rev. Lett. 93, 010503 (2004).
[CrossRef]

Physics (Long Island City, N.Y.) (1)

L. Susskind and J. Glogower, "Quantum mechanical phase and time operator," Physics (Long Island City, N.Y.) 1, 49-61 (1964).

Proc. Am. Math. Soc. (1)

W. F. Stinespring, "Positive functions on C*-algebras," Proc. Am. Math. Soc. 6, 211-216 (1955).

Proc. R. Soc. London, Ser. A (1)

R. Cleve, A. Ekert, C. Macchiavello, and M. Mosca, "Quantum algorithms revisited," Proc. R. Soc. London, Ser. A 454, 339-354 (1998).
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[CrossRef]

Other (8)

C. H. Bennett and G. Brassard, "Quantum cryptography: public key distribution and coin tossing," in Proceedings of the IEEE International Conference on Computers, Systems, and Signal Processing (IEEE Press, 1984), pp. 175-179.

R. Das, A. Mitra, V. Kumar, and A. Kumar, "Quantum information processing by NMR: preparation of pseudopure states and implementation of unitary operations in a single-qutrit system," http:arxiv.orglabs/quant-ph/0307240.

K. Kraus, States, Effects, and Operations: Fundamental Notions in Quantum Theory, Lecture Notes in Physics (Springer-Verlag, 1983), Vol. 190.
[CrossRef]

The group U(1)×(d−1) is commutative, hence its irreducible representations are all one dimensional.

Actually, U(1)×(d−1) is a proper subgroup of SU(d).

Recently, it has been proved that cloning channels, in the limit of infinite output copies, tend to measure-and-prepare channels. Here, we are able to explicitly show how fast this limit is reached, for every finite M.

J. Bae and A. Acín, "Asymptotic quantum cloning is state estimation," http://arxiv.orglabs/quant-ph/0603078.

This holds by linearity, since every symmetric operator O can be written as a linear combination of N-fold tensor product pure states, namely, O=summation i lambda i∣psi i›‹psi i∣big dot times N.

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Figures (1)

Fig. 1
Fig. 1

Comparison between single-site fidelities of phase covariant 1 M optimal cloning (solid curve) and phase conjugation (dotted curve) for d = 5 . Both curves tend to the limit of 9 25 = 0.36 , that is, the fidelity of optimal phase estimation.

Equations (66)

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ψ ( { ϕ j } ) = 1 d ( 0 + e i ϕ 1 1 + e i ϕ 2 2 + + e i ϕ d 1 d 1 ) ,
U { ϕ j } ψ 0 = ψ ( { ϕ j } ) .
E ( V g ρ V g ) = W g E ( ρ ) W g ,
M R M = ( M I ) Ω Ω ,
Ω = { n i } { n i } N { n i } N .
M ( ρ ) = Tr in [ ( I out ρ * ) R M ] ,
[ R E , W g V g * ] = 0 ,
R E = α R E α ,
M ( ρ ) = Tr L [ V ρ V ] .
M ( ρ ) = Tr L [ U ( ρ a a ) U ] ,
M ( ρ ) = U ( ρ a a ) U ,
M ( ρ ) = V ρ V
M ( ρ ) = i p i U i ( ρ a a ) U i ,
M ( ρ ) = i p i V i ρ V i .
ψ ( { ϕ j } ) N ψ ( { ϕ j } ) M ,
[ R C , U { ϕ j } M ( U { ϕ j } * ) N ] = 0 .
R C = { m j } R { m j } ,
{ m 0 + n 0 , m 1 + n 1 , m 2 + n 2 , , m d 1 + n d 1 M n 0 , n 1 , n 2 , , n d 1 N } { n i } ,
R C = { m j } { n i } , { n i } r { n i } , { n i } { m j } { m j } + { n i } { m j } + { n i } M { n i } { n i } N .
F C = Tr [ ψ 0 ψ 0 ( N + M ) R C ] .
F C 1 = Tr [ ψ 0 ψ 0 I ( M 1 ) ψ 0 ψ 0 N R C ] .
R C = r { k } r { k } ,
r { k } = { n j } k + n 0 , , k + n i , M n 0 , , n i N ,
j n j = N .
F C 1 = 1 d + 1 M d N + 1 { n ¯ j } i j N ! n ¯ 0 ! n ¯ i ! n ¯ j ! ( n ¯ i + k + 1 ) ( n ¯ j + k + 1 ) ( n ¯ i + 1 ) ( n ¯ j + 1 ) , M = k d + N ,
F C 1 = 1 d + ( d 1 ) ( M + d 1 ) M d 2 , M = k d + 1 ,
C ( ρ N ) = V ρ N V ,
V n 0 , n 1 , , n i , N = n 0 + k , n 1 + k , , n i + k , M .
ψ ( { ϕ j } ) N ( ψ ( { ϕ j } ) * ) M = ψ ( { ϕ j } ) M ,
[ R N , ( U { ϕ j } * ) ( M + N ) ] = 0 .
{ m 0 n 0 , m 1 n 1 , m 2 n 2 , , m d 1 n d 1 M n 0 , n 1 , n 2 , , n d 1 N } { n i } ,
F N 1 = Tr [ ψ 0 ψ 0 I ( M 1 ) ψ 0 ψ 0 N R N ] ,
R N = { m j } { n i } , { n i } r { n i } , { n i } { m j } { m j } + { n i } { m j } + { n i } M { n i } { n i } N .
R N = r { k } r { k } ,
r { k } = { n j } k n 0 , , k n i , M n 0 , , n i N , j n j = N ,
N ( ρ N ) = V ρ N V ,
V n 0 , n 1 , , n i , N = k n 0 , k n 1 , , k n i , M .
F N 1 = 1 d + 1 M d N + 1 { n ¯ j } i j N ! n ¯ 0 ! n ¯ i ! n ¯ j ! ( k n ¯ i ) ( k n ¯ j ) ( n ¯ i + 1 ) ( n ¯ j + 1 ) , M = k d N ,
F N 1 = 1 d + ( d 1 ) ( M + 1 ) M d 2 , M = k d 1 .
F 1 = 1 d + 1 d N + 2 { n ¯ i } i j N ! n ¯ 0 ! 1 ( n ¯ i + 1 ) ( n ¯ j + 1 ) , i n ¯ i = N 1 .
F C 1 F N 1 F P 1 , lim M F C 1 = lim M F N 1 = F P 1 .
Tr M 1 [ C ( ψ ( { ϕ j } ) ψ ( { ϕ j } ) N ) ]
= η C ψ ( { ϕ j } ) ψ ( { ϕ j } ) + ( 1 η C ) I d ,
F P 1 ( N ) F C 1 ( N , M ) , N , M .
S ( ρ M ) = η S ρ + ( 1 η S ) I d .
S ( C ( ψ ( { ϕ j } ) ψ ( { ϕ j } ) N ) ) = η S Tr M 1 [ C ( ψ ( { ϕ j } ) ψ ( { ϕ j } ) N ) ] + ( 1 η S ) I d .
S ( C ( ψ ( { ϕ j } ) ψ ( { ϕ j } ) N ) ) = η S η C ψ ( { ϕ j } ) ψ ( { ϕ j } ) + ( η S ( 1 η C ) + ( 1 η S ) ) I d .
lim M F ¯ ( N , M ) = lim M F C ( N , M ) ;
lim M F C ( N , M ) F P ( N ) , N .
p ( { ϕ j ¯ } ) = ψ ( { ϕ j } ) N e ( { ϕ j ¯ } ) 2 ,
e ( { ϕ j ¯ } ) = U { ϕ ¯ j } N { n i } { n i } N , i n i = N ,
P ( ψ ( { ϕ j } ) ψ ( { ϕ j } ) N ) = d { ϕ j ¯ } ( 2 π ) d 1 p ( { ϕ j ¯ } ) ψ ( { ϕ j ¯ } ) ψ ( { ϕ j ¯ } ) .
P ψ 0 ψ 0 N ) = d { ϕ j ¯ } ( 2 π ) d 1 Tr [ ψ 0 ψ 0 N e ( { ϕ j ¯ } ) e ( { ϕ j ¯ } ) ] ψ ( { ϕ j ¯ } ) ψ ( { ϕ j ¯ } ) = Tr N [ I ψ 0 ψ 0 N d { ϕ j ¯ } ( 2 π ) d 1 ψ ( { ϕ j ¯ } ) ψ ( { ϕ j ¯ } ) e ( { ϕ j ¯ } ) e ( { ϕ j ¯ } ) ] .
d γ 2 π exp [ i ( m n ) γ ] = δ m n , m , n Z ,
e ( { ϕ j ¯ } ) e ( { ϕ j ¯ } ) = U ( { ϕ j ¯ } ) N [ { n i } , { n j } { n i } { n j } ] U ( { ϕ j ¯ } ) N ,
d { ϕ j ¯ } ( 2 π ) d 1 ψ ( { ϕ j ¯ } ) ψ ( { ϕ j ¯ } ) e ( { ϕ j ¯ } ) e ( { ϕ j ¯ } )
= { n i } i , j i j d { n i } n 0 , , n i 1 , , n j + 1 , .
P ( ψ 0 ψ 0 ) = I d + 1 d N + 1 { n ¯ i } i j N ! n ¯ 0 ! 1 ( n ¯ i + 1 ) ( n ¯ j + 1 ) i j , j n ¯ j = N 1 ,
Tr M 1 [ C ( ψ 0 ψ 0 N ) ]
= Tr M 1 , N [ I M ψ 0 ψ 0 N { n i } , { n j } n 0 + k , n 0 + k , M n 0 , n 0 , N ] = 1 d N { n i } , { n j } [ ( N n 0 ; n 1 ; ) ( N n 0 ; n 1 ; ) ] 1 2 Tr M 1 [ n 0 + k , n 0 + k , M ] = 1 d N { n i } , { n j } [ ( N n 0 ; n 1 ; ) ( N n 0 ; n 1 ; ) ] 1 2 [ ( M n 0 + k ; ) ( M n 0 + k ; ) ] 1 2 Tr M 1 [ n 0 + k ˜ , n 0 + k ˜ , M ] = T diag + T off - diag ,
T diag = 1 d N { n i } , { n j } i [ ( N n 0 ; n 1 ; ) ( N n 0 ; n 1 ; ) ] 1 2 [ ( M n 0 + k ; ) ( M n 0 + k ; ) ] 1 2 Tr M 1 [ i i n 0 + k , , n i + k 1 , n 0 + k , n i + k 1 , ] × [ ( M 1 n 0 + k ; ; n i + k 1 ; ) ( M 1 n 0 + k ; ; n i + k 1 ; ) ] 1 2 = 1 M d N { n i } N ! n 0 ! n 1 ! i ( n i + k ) i i ,
T off - diag = 1 M d N { n i } i j N ! n 0 ! ( n i 1 ) ! n j ! ( n i + k ) ( n j + k + 1 ) n i ( n j + 1 ) i j = 1 M d N { n ¯ i } i j N ! n ¯ 0 ! n ¯ i ! n ¯ j ! ( n ¯ i + k + 1 ) ( n ¯ j + k + 1 ) ( n ¯ i + 1 ) ( n ¯ j + 1 ) i j ,
Tr [ ψ 0 ψ 0 i n i + k M i i ] = 1 d ,
Tr [ ψ 0 ψ 0 i j ( n ¯ i + k + 1 ) ( n ¯ j + k + 1 ) ( n ¯ i + 1 ) ( n ¯ j + 1 ) i j ]
= 1 d i j ( n ¯ i + k + 1 ) ( n ¯ j + k + 1 ) ( n ¯ i + 1 ) ( n ¯ j + 1 ) .
Tr M 1 [ C ( ψ 0 ψ 0 N ) ] = η C ( N , M ) ψ 0 ψ 0 + ( 1 η C ( N , M ) ) I d .

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