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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 

<img src='content_image/374269.jpg'>

<img src='content_image/374270.jpg'>

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 

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. 

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. □ 

This bound can give rise to further 

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. 

<img src='content_image/374271.jpg'>

<img src='content_image/374272.jpg'>

