F ( ρ ) ≤ max i { λ$_{i}$ } (18) 〈 ψ$_{+}$ | ( I ⊗ U †$_{)}$ ρ ( I ⊗ U ) | ψ$_{+}$ 〉 ≤ ‖〈 ψ$_{+}$ |‖‖ ( I ⊗ U †$_{)}$ ρ ( I ⊗ U ) | ψ$_{+}$ 〉‖ ≤ ‖ ( I ⊗ U †$_{)}$ ρ ( I ⊗ U ) ‖‖| ψ$_{+}$ 〉‖ 2 = ‖ ρ ‖‖| ψ$_{+}$ 〉‖ 2 = ‖ ρ ‖ , (19) F ( ρ ) ≤ λ$_{max}$ = 〈 ψ | ρ | ψ 〉 = 1 d ∑ ij,kl √$_{da}$$_{∗}$ ij 〈 ij | $_{ρ}$√$_{da}$$_{kl}$ | kl 〉 = ∑ i,k 1 d 〈 ii | I ⊗ A † ρI ⊗ A | kk 〉 = 〈 ψ$_{+}$ | I ⊗ A † ρI ⊗ A | ψ$_{+}$ 〉 ≤ F ( ρ ) . ρ = 2 7 | ψ$_{+}$ 〉〈 ψ$_{+}$ | + α 7 σ$_{+}$ + 5 − α 7 $^{σ}$−, (20) 5 Observation: For any ρ ∈ H ⊗ H , the fully entangled fraction F ( ρ ) satisfies where λ$_{i}$ s are the eigenvalues of ρ . Proof: For any quantum state ρ and unitary U , we have We have used the Cauchy-Schwarz inequality to obtain the first inequality. The second inequality is due to the basic property of operator norm. The followed equality follows from the fact that unitary transformation does not change the operator norm. where ‖ ρ ‖ stands for the operator norm, ‖ ρ ‖ = sup ( ‖ ρ | x 〉‖ : ‖| x 〉‖ = 1), ‖| x 〉‖ = √ 〈 x | x 〉 . We have used the Cauchy-Schwarz inequality to obtain the first inequality. The second This bound can give rise to further From [18] ‖ ρ ‖ is an eigenvalue of ρ , actually, it is the maximal eigenvalue of ρ , i.e. ‖ ρ ‖ = max$_{i}$ { λ$_{i}$ } where λ$_{i}$ s are the eigenvalues of ρ , which ends the proof. □ Corollary: Let | ψ 〉 = ∑ ij a$_{ij}$ | ij 〉 with ||| ψ 〉|| = 1 be the normalized eigenvector of ρ with respect to the maximal eigenvalue λ$_{max}$ . If the matrix A with elements A$_{ij}$ = √$_{da}$$_{ij}$ are unitary, the upper bound derived in (18) becomes the exact value of FEF. Proof: A simple computation shows that Thus we have F ( ρ ) = λ$_{max}$ . According to the corollary, we can find out when the upper bound derived in theorem 2 becomes the exact value of FEF. Example 2: Consider the 3 ⊗ 3 state [19] where σ$_{+}$ = 1 $_{3}$( | 01 〉〈 01 | + | 12 〉〈 12 | + | 20 〉〈 20 | ) $^{, σ}$− = 1 $_{3}$( | 10 〉〈 10 | + | 21 〉〈 21 | + | 02 〉〈 02 | ). ρ is entangled when 3 < α ≤ 5 . The maximal eigenvalue of ρ is 2 $_{7}$, with the corresponding normalized eigenvector { 1 √$_{3}$ , 0 , 0 , 0 , 1 √$_{3}$ , 0 , 0 , 0 , 1 √$_{3}$ } . The matrix A related to this eigenvector is just the 3 × 3 identity matrix which is obviously unitary. Thus we have for the state (20), F ( ρ ) = 2 $_{7}$. The following upper bound of FEF gives a very tight estimation of FEF for weakly mixed quantum states.