From Fig. 1 we see that for 0 ≤ a < 0 . 572, the upper bound in (8) is larger than that in (15). But for 0 . 572 < a < 1 the upper bound in (8) is always lower than that in (15), i.e. the upper bound (8) is tighter than (15) in this case. FIG. 1: Upper bound of F ( ρ ) from (8) (solid line) and upper bound from (15) (dashed line). By using the operator norm, we have further 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. ρ ( a ) = 1 8 a + 1       a 0 0 0 a 0 0 0 a 0 a 0 0 0 0 0 0 0 0 0 a 0 0 0 0 0 0 0 0 0 a 0 0 0 0 0 a 0 0 0 a 0 0 0 a 0 0 0 0 0 a 0 0 0 0 0 0 0 0 0 1+ a 2 0 √$_{1}$ − a 2 2 0 0 0 0 0 0 0 a 0 a 0 0 0 a 0 √$_{1}$ − a 2 2 0 1+ a 2      $^{.}$ (17) Example 1: We consider the bound entangled state [17] 0.5 0.6 0.7 0.8 0.9 1.0 a 0.334 0.336 0.338 0.340 0.342 Upper Bound 4