Extremal covariant quantum operations and positive operator valued measures

Journal of Mathematical Physics

Volumn 45, Issue 9, 2004, Pages 3620-3635

  D'Ariano, Giacomo Mauro ^a,b  

^a UNIVERSITY OF PAVIA   (Italy)

^b Northwestern University   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 5644243198     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.1777813     Document Type: Article

References (28)

1. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2000).
2. A. S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory, Series in Statistics and Probability (North-Holland, Amsterdam, 1982).
3. C. W. Helstrom, Quantum Detection and Estimation Theory (Academic, New York, 1976).
4. E. B. Davies, Quantum Theory of Open Systems (Academic, London, 1976).
5. P. Busch, P. J. Lahti, and P. Mittelstaedt, The Quantum Theory of Measurement, Lecture Notes in Physics Vol. 2 (Springer, Berlin, 1991).
6. K. Kraus, States, Effects, and Operations (Springer-Verlag, Berlin, 1983).
7. E. B. Davies, Quantum Theory of Open Systems (Academic, London, 1976).
8. W. F. Stinespring, Proc. Am. Math. Soc. 6, 211 (1955).
9. E. B. Davies, J. Funct. Anal. 6, 318 (1970).
10. It must be noticed that in the present paper we are focusing only on the statistical description of the measurement process - not on the change of state due to the measurement, nor on its realization scheme - and for this reason we identify the measurement covariance with the covariance of the POVM. The state change due to the measurement is described by the so-called " instrument" (Ref. 11), and the measuring scheme is given in terms of a unitary interaction with an ancilla and an orthogonal measurement on it. Clearly, the POVM covariance does not imply the covariance of all of its instruments, and the covarance of an instrument does not imply the covariance of all of the measurement realizations of the instrument.
11. E. B. Davies and J. T. Lewis, Commun. Math. Phys. 17, 239 (1970).
12. H. P. Yuen and J. H. Shapiro, IEEE Trans. Inf. Theory 24, 657 (1978); 25, 179 (1979); 26, 78 (1980).
13. H. P. Yuen and J. H. Shapiro, IEEE Trans. Inf. Theory 24, 657 (1978); 25, 179 (1979); 26, 78 (1980).
14. H. P. Yuen and J. H. Shapiro, IEEE Trans. Inf. Theory 24, 657 (1978); 25, 179 (1979); 26, 78 (1980).
15. G. M. D'Ariano, "Quantum estimation theory and optical detection," in Concepts and Advances in Quantum Optics and Spectroscopy of Solids, edited by T. Hakioglu and A. S. Shumovsky (Kluwer Academic, Amsterdam, 1997), pp. 139-174.
16. C. H. Bennett and G. Brassard, Proceedings of the IEEE International conference on computers, systems, and signal processing, Bangalore, India, 1984, pp. 175-179.
17. N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, Rev. Mod. Phys. 74, 145 (2002).
18. G. M. D'Ariano and C. Macchiavello, Phys. Rev. A 67, 042306 (2003).
19. M.-D. Choi, Linear Algebr. Appl. 10, 285 (1975).
20. K. R. Parthasaraty, Infinite Dimen. Anal., Quantum Probab., Relat. Top. 2, 557 (1999).
21. G. M. D'Ariano and P. Lo Presti, quant-ph/0301110.
22. G. M. D'Ariano, C. Macchiavello, and M. F. Sacchi, Phys. Lett. A 248, 103 (1998).
23. G. M. D'Ariano, C. Macchiavello, P. Perinotti, and M. F. Sacchi, Phys. Lett. A 268, 241 (2000).
24. G. M. D'Ariano and P. Lo Presti, Phys. Rev. A 64, 042308 (2001).
25. R. F. Werner, Phys. Rev. A 58, 1827 (1998).
26. D. P. Zhelobenko, Compact Lie Groups and Their Representations (American Mathematical Society, Providence, 1973).
27. C.-K. Li and B.-S. Tam, SIAM J. Matrix Anal. Appl. 15, 903 (1994).
28. F. Buscemi, G. M. D'Ariano, P. Perinotti, and M. F. Sacchi, Phys. Lett. A 314, 374 (2003).