content_image/368830.jpg content_image/368832.jpg
λ$_{i}$ , i = 1 , ..., d 2 − 1, are the generators of the SU ( d ) algebra with T r { λ$_{i}$λ$_{j}$ } = 2 δ$_{ij}$ , r$_{i}$ ( ρ ) = 1 $_{2}$T r { ρλ$_{i}$ (1) ⊗ I } , s$_{j}$ ( ρ ) = 1 $_{2}$T r { ρI ⊗ λ$_{j}$ (2) } , n$_{ij}$ ( ρ ) = 1 $_{4}$T r { ρλ$_{i}$ (1) ⊗ λ$_{j}$ (2) , P$_{+}$ stands for the projection operator to | ψ$_{+}$ 〉 , N ( P$_{+}$ ) is similarly defined to N ( ρ ), N T stands for the transpose of N , || N || KF = $_{T r}$√$_{NN}$$_{†}$ is the Ky Fan norm of N . This upper bound was used to improve the distillation protocol proposed in [16]. Here we show that the upper bound in (8) is different from that in (15) by an example. 

Example 1: We consider the bound entangled state [17] 

<img src='content_image/368830.jpg'>

By using the operator norm, we have further 

<img src='content_image/368832.jpg'>

