Quantum covariance, quantum fisher information, and the uncertainty relations

IEEE Transactions on Information Theory

Volumn 55, Issue 1, 2009, Pages 439-443

  Gibilisco, Paolo ^a   Hiai, Funio ^b   Petz, Dénes ^c  

^a UNIVERSITY OF ROME TOR VERGATA   (Italy)

^b TOHOKU UNIVERSITY   (Japan)

^c ALFRÉD RÉNYI INSTITUTE OF MATHEMATICS   (Hungary)

Author keywords

Generalized variance; Operator monotone functions; Quantum covariance; Quantum Fisher information; Uncertainty principle

Indexed keywords

GENERALIZED VARIANCE; OPERATOR MONOTONE FUNCTIONS; QUANTUM COVARIANCE; QUANTUM FISHER INFORMATION; UNCERTAINTY PRINCIPLE;

FISHER INFORMATION MATRIX;

EID: 58249132448     PISSN: 00189448     EISSN: None     Source Type: Journal    
DOI: 10.1109/TIT.2008.2008142     Document Type: Article

References (28)

1. S. Amari and H. Nagaoka, Methods of Information Geometry. Oxford, U.K.: AMS/Oxford Univ. Press, 2000, vol. 191, Translations of Mathematical Monographs.
2. A. Andai, "Uncertainty principle with quantum Fisher information," J. Math. Phys., vol. 49, 2008, 012106.
3. N. N. Cencov, Statistical Decision Rules and Optimal Inferences. Providence, RI: Amer. Math. Society, 1982, Translations of Mathematical Monographs 53.
4. P. Gibilisco and T. Isola, "Uncertainty principle and quantum Fisher information," Ann. Inst. Statist. Math., vol. 59, pp. 147-159, 2007.
5. P. Gibilisco, D. Imparato, and T. Isola, "A volume inequality for quantum Fisher information and the uncertainty principle," J. Statist. Phys., vol. 130, pp. 545-559, 2008.
6. P. Gibilisco, D. Imparato, and T. Isola, "A Robertson-type uncertainty principle and quantum Fisher information," Linear Algebra Appl., vol. 428, pp. 1706-1724, 2008.
7. P. Gibilisco, D. Imparato, and T. Isola, "Uncertainty principle and quantum Fisher information II," J. Math. Phys., vol. 48, 2007, 072109.
8. F. Hansen, "Metric adjusted skew information," in Proc. Nat. Acad. Sci. USA, 2008, vol. 105, pp. 9909-9916.
9. C. W. Helstrom, Quantum Detection and Estimation Theory. New York: Academic, 1976.
10. A. S. Holevo, Probabilistic and Statistical aspects of Quantum Theory. Amsterdam, The Netherlands: North-Holland, 1982.
11. H. Kosaki, "Matrix trace inequality related to uncertainty principle," Int. J. Math., vol. 16, pp. 629-645, 2005.
12. F. Kubo and T. Ando, "Means of positive linear operators," Math. Ann., vol. 246, pp. 205-224, 1979/80.
13. S. Luo, "Quantum Fisher information and uncertainty relations," Lett. Math. Phys., vol. 53, pp. 243-251, 2000.
14. S. Luo, "Wigner-Yanase skew information and uncertainty relations," Phys. Rev. Lett., vol. 91, 2003, 180403.
15. S. Luo and Z. Zhang, "An informational characterization of Schrödinger's uncertainty relations," J. Stat. Phys., vol. 114, pp. 1557-1576, 2004.
16. S. Luo and Q. Zhang, "On skew information," IEEE Trans. Inf. Theory vol. 50, no. 8, pp. 1778-1782, Aug. 2004.
17. S. Luo and Q. Zhang, "Correction to on skew information.," IEEE Trans. Inf. Theory, vol. 51, no. 12, pp. 4432-, Dec. 2005.
18. E. A. Morozova and N. N. Chentsov, "Markov invariant geometry on state manifolds," (in Russian) Itogi Nauki i Tehniki, vol. 36, pp. 69-102, 1990.
19. M. Ohya and D. Petz, Quantum Entropy and Its Use, 2nd ed. Heidelberg, Germany: Springer-Verlag, 1993.
20. D. Petz, "Quasi-entropies for finite quantum systems," Rep. Math. Phys., vol. 23, pp. 57-65, 1986.
21. D. Petz, "Monotone metrics on matrix spaces," Linear Algebra Appl. vol. 244, pp. 81-96, 1996.
22. D. Petz, "Covariance and Fisher information in quantum mechanics.," J. Phys. A, Math. Gen., vol. 35, pp. 79-91, 2003.
23. D. Petz, Quantum Information Theory and Quantum Statistics. Berlin, Germany: Springer-Verlag, 2007.
24. D. Petz and Cs. Sudár, "Geometries of quantum states," J. Math. Phys., vol. 37, pp. 2662-2673, 1996.
25. H. P. Robertson, "An indeterminacy relation for several observables and its classical interpretation.," Phys. Rev., vol. 46, pp. 794-801, 1934.
26. E. P. Wigner and M. M. Yanase, "Information content of distributions.," Proc. Nat. Acad. Sci. USA, vol. 49, pp. 910-918, 1963.
27. K. Yanagi, S. Furuichi, and K. Kuriyama, "A generalized skew information and uncertainty relation," IEEE Trans. Inf. Theory, vol. 51, no. 12, pp. 4401-4404, Dec. 2005.
28. H. P. Yuen and M. Lax, "Multiple-parameter quantum estimation and measurement of nonselfadjoint observables," IEEE Trans. Inf. Theory vol. IT-19, no. 6, pp. 740-750, Nov. 1973.