Abstract
Information encoded in quantum system has to obey rules of quantum physics which impose strict bounds on state estimation and on possible manipulations with quantum information. That is, an unknown state of a qubit cannot be precisely determined from a measurement performed on a finite ensemble of identically prepared qubits. As a consequence the perfect, universal cloning map is not allowed. Another map which cannot be performed perfectly on an unknown state of a qubit is the “spin-flip” i.e. the universal-NOT gate. Within this significant framework the contextual experimental realization of the optimal 1 to 2 universal cloning and the optimal universal-NOT (U-NOT) gate by quantum injected optical parametric amplification (QIOPA) through a modified quantum state teleportation protocol is reported. In addition, the realization of a multi-photon “all optical” Schroedinger cat state is investigated by exploiting the information preserving property of the parametric amplification. Finally, as a significant demonstration of the perspectives offered by quantum entanglement to modern measurement theory, we shall present how the total parallelism of a bipartite entangled state can be adopted to extract efficiently the full information that characterizes any unknown “quantum operations”.
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References
1. K. Kraus: States, Effects and Operations (Springer-Verlag, Berlin, 1983); J. Preskill: Lecture Notes on Quantum Computation, http://www.theory.caltech.edu/people/ph229/#lecture. A completely-positive map (CP-map) Λ(ρ ) preserves positivity for: (a) any local state in the Hilbert space H, (b) when tensor-multiplied with the identical map I acting on any Hilbert space K, the extended map Λ (ρ ) otimes I is positive for any state in the entangled space H otimes K, for any extension of K. A positive map (P-map) only satisfies property (a).
2. A. S. Holevo: Probabilistic and Statistical Aspects of Quantum Theory (North-Holland, Amsterdam, 1982)
3. S. Massar and S. Popescu: Phys. Rev. Lett. 74, 1259 (1995); N. Gisin and S. Popescu: Phys. Rev. Lett. 83, 432 (1999); R.G. Sachs: The Physics of Time Reversal (U. of Chicago Press, 1987).
4. R. Derka, V. Buzek and A. Ekert: Phys. Rev. Lett. 80, 1571 (1998); V. Buzek, M. Hillery, R.F. Werner: Phys. Rev. A 60, R2626 (1999); R.F. Werner: Phys. Rev. A 58, 1827 (1998).
5. W.K. Wootters and W.K. Zurek: Nature (London) 299, 802 (1982).
6. N. Herbert: Founds. Phys. 12, 1171 (1982). The first precise formulation of the no-cloning theorem based on the linear structure of quantum theory was indeed expressed in 1981 by G. Ghirardi in a Referee Report for Foundations of Physics to the paper by N. Herbert (G. Ghirardi: private communication).
7. N. Gisin: Phys. Lett. A 242, 1 (1998); D. Bruss, G.M. D’Ariano, C. Macchiavello, and M.F. Sacchi: Phys. Rev. A 62, 062302 (2000).
8. C. Simon, V. Buzek, and N. Gisin: Phys. Rev. Lett. 87, 170405 (2001).
9. F. De Martini, V. Buzek, F. Sciarrino, and C. Sias: Nature (London) 419, 815 (2002).
10. A. Peres: Quantum Theory: Concepts and Methods (Kluwer, Dordrecht, 1993).
11. A. Alber et al: Quantum Information: An Introduction to Basic Theoretical Concepts and Experiments (Springer-Verlag, Berlin, 2001).
12. F. De Martini: Phys. Rev. Lett. 81, 2842 (1998); F. De Martini, V. Mussi, and F. Bovino: Opt. Comm. 179, 581 (2000).
13. C. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. Wootters: Phys. Rev. Lett. 70, 1895 (1993).
14. A. Perelomov: Generalized Coherent States and Their Applications (Springer-Verlag, Berlin, 1985).
15. M.J. Collett: Phys. Rev. A 38, 2233 (1988).
16. F. De Martini, G. Di Giuseppe and S. Padua: Phys. Rev. Lett. 87, 150401 (2001).
17. C. Simon, G. Weihs and A. Zeilinger: Phys. Rev. Lett. 84, 2993 (2000).
18. F.A. Bovino, F. De Martini and V. Mussi: arXiv:quant-ph/9905048 (1999).
19. D.F. Walls and G.I. Milburn: Quantum Optics (Springer Verlag, Berlin, 1994).
20. E. Schrödinger: Naturwissenshaften 23, 807 (1935).
21. A. Caldeira, A. Leggett: Physica A 121, 587 (1983).
22. A. Leggett, A. Garg: Phys. Rev. Lett. 54, 857 (1985).
23. W. Zurek: Physics Today, October 1991, p 36.
24. C. Monroe, D. Meekhof, B. King, D.Wineland: Science 272, 1131 (1996).
25. M. Brune, E. Hagley, J. Dreyer, X. Maistre, A. Maali, C. Wunderlich, J.M. Raimond, S. Haroche: Phys. Rev. Lett. 77, 4887 (1996); L. Davidovich, M. Brune, J.M. Raimond and S. Haroche: Phys. Rev. A 53, 1295 (1996); J.R. Friedman, V. Patel, W. Chen, S.K. Tolpygo, and J.E. Lukens: Nature (London) 406, 43 (2000); M. Grenier, O. Mandel, T. Hänsch, I. Bloch: Nature (London) 419, 51 (2002).
26. M.W. Noel, C.R. Stroud: Phys. Rev. Lett. 77, 1913 (1996).
27. D. Pelliccia, V. Schettini, F. Sciarrino, C. Sias and F. De Martini: Phys. Rev. A 68, 042306 (2003); F. De Martini, D. Pelliccia, and F. Sciarrino: Phys. Rev. Lett. 92, 067901 (2004).
28. C.H. Bennett, D.P. Di Vincenzo, J.A. Smolin, and W.K. Wootters: Phys. Rev. A 54, 3824 (1996).
29. P.G. Kwiat, E. Waks, A.G. White, I. Appelbaum and P.H. Eberhard: Phys. Rev. A 60, R773 (1999)
30. M. Horodecki, P. Horodecki and R. Horodecki. In: Quantum Information: An Introduction to Basic Theoretical Concepts and Experiments (Springer, Berlin, 2001).
31. M.D. Reid, W.L. Munro, and F. De Martini: Phys. Rev. A 66, 033801 (2002).
32. M. Redhead: Incompleteness, Nonlocality and Realism (Clarendon Press, Oxford, 1987).
33. A. Lamas-Linares, C. Simon, J. C. Howell and D. Bouwmeester: Science 296, 712 (2002) first reported a N=1,M=2 cloning experiment with a semi-classical strongly attenuated coherent injection state reaching a UOQCM fidelity F≈ 0.81.
34. V. Buzek: private communication to FDM; V. Buzek and M. Hillery: Phys. Rev. A 62, 022303 (2000); M. Hillery, V. Buzek, and M. Ziman: Phys. Rev. A 65, 022301 (2002).
35. F. De Martini: Phys. Lett. A 250, 15 (1998). The collinear QIOPA is found to be an efficient generator of multiphoton Schrödinger Cat π-states. The no-interaction term in (9.23) is easily discarded by a simple 2-detector coincidence measurement on the mode k.
36. M. Ricci, F. Sciarrino, C. Sias, and F. De Martini: Phys. Rev. Lett. 92, 047901 (2004); F. Sciarrino, C. Sias, M. Ricci and F. De Martini: Phys. Lett. A 323, 34 (2004).
37. C.H. Bennett, and D. Di Vincenzo: Nature (London) 404, 247 (2000).
38. D. Gottesman, and I.L. Chuang: Nature (London) 402, 390 (1999).
39. D. Boschi, S. Branca, F. De Martini, L. Hardy, S. Popescu: arXiv:quant-ph/9710013 (1997); Phys. Rev. Lett. 80, 1121 (1998); D. Bouwmeester et al: Nature (London) 390, 575 (1997); E. Lombardi, F. Sciarrino, S. Popescu, and F. De Martini: Phys. Rev. Lett. 88, 070402 (2002).
40. S. Giacomini, F. Sciarrino, E. Lombardi, and F. De Martini: Phys. Rev. A 66, 030302(R) (2002).
41. C.K. Hong, Z.Y. Ou and L. Mandel: Phys. Rev. Lett. 59, 2044 (1987).
42. I.L. Chuang, M.A. Nielsen: Quantum Information and Quantum Computation (Cambridge Univ. Press, Cambridge, 2000).
43. G.M. D’Ariano. In: Experimental Quantum Computation and Information, Proceedings of the Intl.School “E. Fermi,” Varenna, ed by F. De Martini and C. Monroe (Editrice Compositori, Bologna, 2001).
44. G.M. D’Ariano and P. Lo Presti: Phys. Rev. Lett. 86, 4195 (2001).
45. G.M. D’Ariano and H.P. Yuen: Phys. Rev. Lett. 76, 2832 (1996).
46. F. De Martini, A. Mazzei, M. Ricci, and G.M. D’Ariano: Phys. Rev. A 67, 062307 (2003).
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De Martini, F., Ricci, M., Sciarrino, F. 9 Quantum Operations on Qubitsand Their Characterization. In: Paris, M., Řeháček, J. (eds) Quantum State Estimation. Lecture Notes in Physics, vol 649. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-44481-7_9
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DOI: https://doi.org/10.1007/978-3-540-44481-7_9
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