Comparison of Uhlmann's transition probability with the one induced by the natural positive cone of von Neumann algebras in standard form

Abstract

It is shown that for normal states ρ and φ of a W ^*-algebra $A, P(\rho ,\phi ) \leqslant (\xi (\rho ),\xi (\phi )) \leqslant P(\rho ,\phi )^{1/2}$, where P(.,.) is the transition probability considered by Uhlmann [1], and ζ(ω) is the vector in the natural positive cone of some standard faithful representation of A, associated with the normal state ω. The above inequality is equivalent to: $d(\rho ,\phi ) \leqslant ||\xi (\rho ) - \xi (\phi )|| \leqslant \leqslant d(\rho ,\phi )(2(1 - \frac{1}{4}d(\rho ,\phi )^2 ))^{1/2} \leqslant 2^{1/2} d(\rho ,\phi )$, where d(.,.) is the Bures distance function [5].