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Journal of the Optical Society of America B

Journal of the Optical Society of America B


  • Editor: Henry M. Van Driel
  • Vol. 24, Iss. 2 — Feb. 1, 2007
  • pp: 363–370

Economical realization of phase-covariant devices in arbitrary dimensions (Invited)

Francesco Buscemi, Giacomo Mauro D’Ariano, and Chiara Macchiavello  »View Author Affiliations

JOSA B, Vol. 24, Issue 2, pp. 363-370 (2007)

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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

OCIS Codes
(000.1600) General : Classical and quantum physics
(000.3860) General : Mathematical methods in physics
(270.0270) Quantum optics : Quantum optics

ToC Category:
Quantum Information

Original Manuscript: May 19, 2006
Manuscript Accepted: June 5, 2006
Published: January 26, 2007

Francesco Buscemi, Giacomo Mauro D'Ariano, and Chiara Macchiavello, "Economical realization of phase-covariant devices in arbitrary dimensions (Invited)," J. Opt. Soc. Am. B 24, 363-370 (2007)

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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).

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