Kraus representation in the presence of initial correlations

Physical Review A - Atomic, Molecular, and Optical Physics

Volumn 67, Issue 6, 2003, Pages 621091-621095

  Hayashi, Hiroyuki ^a   Kimura, Gen ^a   Ota, Yukihiro ^a  

^a WASEDA UNIVERSITY   (Japan)

Author keywords

[No Author keywords available]

Indexed keywords

FUNCTIONS; HAMILTONIANS; MATHEMATICAL OPERATORS;

OPEN QUANTUM SYSTEMS;

QUANTUM THEORY;

EID: 0041932289     PISSN: 10502947     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (26)

1. K. Kraus, States, Effects and Operations: Fundamental Notations of Quantum Theory (Springer-Verlag, Berlin, 1983).
2. R. Alicki and K. Lendi, Quantum Dynamical Semigroups and Applications (Springer-Verlag, Berlin, 1987).
3. J. Bouda and V. Bužek, Phys. Rev. A 65, 034304 (2002).
4. J. Preskill, "Lecture notes for the course, Information for Physics 219/Computer Science 219, Quantum Computation," www.theory.caltech.edu/people/preskill/ph229
5. P. Štelmachovič and V. Bužek, Phys. Rev. A 64, 062106 (2001); 67, 029902(E) (2003).
6. P. Štelmachovič and V. Bužek, Phys. Rev. A 64, 062106 (2001); 67, 029902(E) (2003).
7. V. Hakim and V. Ambegaokar, Phys. Rev. A 32, 423 (1985); L.D́. Romero and J.P. Paz, ibid. 55, 4070 (1997); I. Burghardt, J. Chem. Phys. 114, 89 (2001); V.F. Los, J. Phys. A 34, 6389 (2001).
8. V. Hakim and V. Ambegaokar, Phys. Rev. A 32, 423 (1985); L.D́. Romero and J.P. Paz, ibid. 55, 4070 (1997); I. Burghardt, J. Chem. Phys. 114, 89 (2001); V.F. Los, J. Phys. A 34, 6389 (2001).
9. V. Hakim and V. Ambegaokar, Phys. Rev. A 32, 423 (1985); L.D́. Romero and J.P. Paz, ibid. 55, 4070 (1997); I. Burghardt, J. Chem. Phys. 114, 89 (2001); V.F. Los, J. Phys. A 34, 6389 (2001).
10. V. Hakim and V. Ambegaokar, Phys. Rev. A 32, 423 (1985); L.D́. Romero and J.P. Paz, ibid. 55, 4070 (1997); I. Burghardt, J. Chem. Phys. 114, 89 (2001); V.F. Los, J. Phys. A 34, 6389 (2001).
11. Notice that the Kraus representation gives the general form of completely positive and trace preserving map [2]. The role of initial correlations in connection with the complete positive condition was first argued by Pechukas [8], and discussed by several authors [5,9-11] [see also G. Kimura, Phys. Rev. A 66, 062113 (2002)].
12. P. Pechukas, Phys. Rev. Lett. 73, 1060 (1994).
13. R. Alicki, Phys. Rev. Lett. 75, 3020 (1995); P. Pechukas, ibid. 75, 3021 (1995).
14. R. Alicki, Phys. Rev. Lett. 75, 3020 (1995); P. Pechukas, ibid. 75, 3021 (1995).
15. A. Royer, Phys. Rev, Lett. 77, 3272 (1996).
16. D. Salgado and J.L. Sánchez-Gómez, e-print quant-ph/0211164.
17. Our review, although the rigorous proof is not given, does not depend on the flniteness of dimensionality of the associated Hubert space, while the arguments in Refs. [5,11] are essentially restricted to finite-dimensional cases.
18. †(t).
19. M.A. Nielsen and I.L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2000).
20. ij are nothing but the correlation parameters which were introduced in Ref. [5].
21. G. Mahler and V. A. Weberruss, Quantum Networks (Springer, Berlin, 1995).
22. m∥)/NM≥0. See Ref. [18] for the completion of these conditions.
23. G. Kimura, e-print quant-ph/0301152.
24. For the case of the controlled-NOT gate, the initial correlation of Eq. (15) does not satisfy these requirements at m = 3, the fact of which allows nonzero interaction (12).
25. We notice that the proof here is not applied to the case of infinite-dimensional composite systems (or those with infinite degrees of freedom). Such a case has been left as an open problem.
26. W.K. Nicholson, Elementary Linear Algebra (McGraw-Hill, New York, 1987).