An inequality for Hilbert-Schmidt norm

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

$$|| |A|---|B| ||_{HS} \leqq 2^{1/2} ||A - B||_{HS} $$

. The corresponding inequality for two normal state ϕ and ψ of a von Neumann algebraM is also proved in the following form:

$$d(\varphi ,\psi ) \leqq ||\xi (\varphi ) - \xi (\psi )|| \leqq 2^{1/2} d(\varphi ,\psi )$$

. 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,

$$||\xi (\varphi _1 ) - \xi (\varphi _2 )|| \leqq 2^{1/2} ||\xi _1 - \xi _2 ||$$

for any vector representatives ξ _j of ϕ _j ,j=1, 2.