Less eavesdropping losses induce more eavesdropping information gain

Abstract

The eavesdropping scheme proposed by W\'{o}jcik [Phys. Rev. Lett. {\bf 90},157901(2003)] on the quantum communication protocol of Bostr\"{o}m and Felbinger [Phys. Rev. Lett. {\bf 89}, 187902(2002)] is improved by constituting a new set of attack operations. The improved scheme only induces half of the eavesdropping losses in W\'{o}jcik's scheme, therefore, in a larger domain of the quantum channel transmission efficiency $\eta$, i.e., [0,75%], the eavesdropper Eve can attack all the transmitted bits. Comparing to W\'{o}jcik's scheme, in the improved scheme the eavesdropping (legitimate) information gain does not vary in the $\eta$ domain of [0, 50%], while in the $\eta$ domain of (50%, 75%] the less eavesdropping losses induce more eavesdropping information gains, for Eve can attack {\it all} the transmitted bits and accordingly eavesdropping information gains do {\it not} decrease. Moreover, for the Bostr\"{o}m-Felbinger protocol, the insecurity upper bound of $\eta$ presented by W\'{o}jcik is pushed up in the this paper, that is, according to W\'{o}jcik's eavesdropping scheme, the Bostr\"{o}m-Felbinger protocol is not secure for transmission efficiencies lower than almost 60%, while according to the improved scheme, it is not secure for transmission efficiencies lower than almost 80%.