Geodesic
On Covariance and Quantum Fisher Information
Teoriâ veroâtnostej i ee primeneniâ, Tome 53 (2008) no. 2, pp. 393-397 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

For a quantum state and a set of observables, we can construct an associated covariance matrix and a natural quantum Fisher information matrix. These two matrices characterize the uncertainty and information content of the observables in the relevant state. An inequality between these two matrices is established. This inequality may be interpreted as a general quantification of the Heisenberg uncertainty principle from a statistical estimation perspective. In particular, it implies a new uncertainty relation which refines the celebrated Schrödinger uncertainty relation.
Keywords: covariance, quantum Fisher information, uncertainty relations, determinant. 
@article{TVP_2008_53_2_a15,
     author = {S. Luo},
     title = {On {Covariance} and {Quantum} {Fisher} {Information}},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {393--397},
     year = {2008},
     volume = {53},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TVP_2008_53_2_a15/}
}
TY  - JOUR
AU  - S. Luo
TI  - On Covariance and Quantum Fisher Information
JO  - Teoriâ veroâtnostej i ee primeneniâ
PY  - 2008
SP  - 393
EP  - 397
VL  - 53
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/TVP_2008_53_2_a15/
LA  - en
ID  - TVP_2008_53_2_a15
ER  - 
%0 Journal Article
%A S. Luo
%T On Covariance and Quantum Fisher Information
%J Teoriâ veroâtnostej i ee primeneniâ
%D 2008
%P 393-397
%V 53
%N 2
%U http://geodesic.mathdoc.fr/item/TVP_2008_53_2_a15/
%G en
%F TVP_2008_53_2_a15
S. Luo. On Covariance and Quantum Fisher Information. Teoriâ veroâtnostej i ee primeneniâ, Tome 53 (2008) no. 2, pp. 393-397. http://geodesic.mathdoc.fr/item/TVP_2008_53_2_a15/

[1] Belavkin V. P., “Ob obobschennykh sootnosheniyakh neopredelennosti Geizenberga i effektivnykh izmereniyakh v kvantovykh sistemakh”, Teoret. matem. fiz., 26:3 (1976), 316–329 | MR

[2] Barndorff-Nielsen O. E., Gill R. D., “Fisher information in quantum statistics”, J. Phys. A, 33:24 (2000), 4481–4490 | DOI | MR | Zbl

[3] Barndorff-Nielsen O. E., Gill R. D., Jupp P. E., “On quantum statistical inference”, J. Roy. Statist. Soc. Ser. B, 65:4 (2003), 775–816 | DOI | MR | Zbl

[4] Connes A., Noncommutative Geometry, Academic Press, San Diego, 1994, 661 pp. | MR | Zbl

[5] Kramer G., Matematicheskie metody statistiki, Mir, M., 1975, 648 pp. | MR

[6] Dirak P., Printsipy kvantovoi mekhaniki, Fizmatgiz, M., 1960, 434 pp. | MR

[7] Fisher R. A., “Theory of statistical estimation”, Proc. Cambridge Philos. Soc., 22 (1925), 700–725 | DOI | Zbl

[8] Helstrom C. W., Quantum Detection and Estimation Theory, Academic Press, New York, 1976

[9] Kholevo A. S., Veroyatnostnye i statisticheskie aspekty kvantovoi teorii, In-t kompyut. issled., M., Izhevsk, 2003, 404 pp.

[10] Kholevo A. S., “Asimptoticheskoe otsenivanie parametra sdviga kvantovogo sostoyaniya”, Teoriya veroyatn. i ee primen., 49:2 (2004), 335–350 | MR | Zbl

[11] Luo S., “Wigner–Yanase skew information vs. quantum Fisher information”, Proc. Amer. Math. Soc., 132:3 (2004), 885–890 | DOI | MR | Zbl

[12] Petz D., “Monotone metrics on matrix spaces”, Linear Algebra Appl., 244 (1996), 81–96 | DOI | MR | Zbl

[13] Sakurai J. J., Modern Quantum Mechanics, Addison-Wesley, Reading, MA, 1994

[14] Schrödinger E., “About Heisenberg uncertainty relation”, Proc. Prussian Acad. Sci., Phys.-Math. Sect., 19 (1930), 296–303 ; E. Shredinger, “K printsipu neopredelennosti Geizenberga”, Izbrannye trudy po kvantovoi mekhanike, Nauka, M., 1976, 210–217 | MR

[15] fon Neiman Dzh., Matematicheskie osnovy kvantovoi mekhaniki, Nauka, M., 1964, 367 pp. | MR