Abstract
For the absolute value |C|=(C*C)1/2 and the Hilbert-Schmidt norm ∥C∥HS=(trC*C)1/2 of an operatorC, the following inequality is proved for any bounded linear operatorsA andB on a Hilbert space
. The corresponding inequality for two normal state ϕ and ψ of a von Neumann algebraM is also proved in the following form:
. Here ξ(χ) denotes the unique vector representative of a state χ in a natural positive coneP ♯ forM, andd(ϕ, ψ) denotes the Bures distance defined as the infimum (which is also the minimum) of the distance of vector representatives of ϕ and ψ. In particular,
for any vector representatives ξ j of ϕ j ,j=1, 2.
Similar content being viewed by others
References
Araki, H.: Publ. RIMS Kyoto Univ.6, 385–442 (1971)
Araki, H.: Pac. J. Math.50, 309–354 (1974)
von Neumann, J.: Actual. Sci. Ind.299, (1935)
Powers, R.T. and Størmer, E.: Commun. Math. Phys.16, 1–33 (1970)
Author information
Authors and Affiliations
Additional information
Communicated by H. Araki
Rights and permissions
About this article
Cite this article
Araki, H., Yamagami, S. An inequality for Hilbert-Schmidt norm. Commun.Math. Phys. 81, 89–96 (1981). https://doi.org/10.1007/BF01941801
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01941801