Unfortunately, precise formula of FEF has been only obtained for two qubits systems [13]. For high dimensional systems, it becomes quite difficult to derive an analytic formula for FEF. In [14] we have derived an upper bound of FEF to give an estimation of the value of FEF. In this paper, we derive more tight upper bounds for FEF. These bounds make complements on the estimation of FEF. Let H be a d -dimensional complex vector space with computational basis | i 〉 , i = 1 , ..., d . The fully entangled fraction of a density matrix ρ ∈ H ⊗ H is defined by where ϵ denotes the set of d × d -dimensional maximally entangled pure states. (2) can be also alternatively expressed as where the maximization is taken over all unitary transformations U , | ψ$_{+}$ 〉 = 1 √$_{d}$ d ∑ i $_{=1}$| ii 〉 is the maximally entangled state and I is the corresponding identity matrix. Let h and g be d × d matrices such that h | j 〉 = | ( j + 1) mod d 〉 , g | j 〉 = ω $^{j}$| j 〉 , with ω = exp { − 2 iπ d } . We can introduce d 2 linear-independent d × d -matrices U$_{st}$ = h $^{t}$g$^{s}$, which satisfy { U$_{st}$ } also satisfy the condition for bases of unitary operators in the sense of [15], i.e. From { U$_{st}$ } , we can introduce the generalized Bell-states, | Φ$_{st}$ 〉 are all maximally entangled states and form a complete orthogonal normalized basis of H ⊗ H . Theorem 1: For any quantum state ρ ∈ H ⊗ H , the fully entangled fraction defined in (2) and (3) fulfills the following inequality:  st { U$_{st}$ } form a complete basis of d × d -matrices, namely, for any d × d matrix W , W can be expressed as F ( ρ ) = max φ ∈ ϵ 〈 φ | ρ | φ 〉 , (2) F ( ρ ) = max U 〈 ψ$_{+}$ | ( I ⊗ U †$_{)}$ ρ ( I ⊗ U ) | ψ$_{+}$ 〉 , (3) U$_{st}$U$_{s}$ ′ t ′ = ω st $^{′}$− ts $^{′}$U$_{s}$ ′ t ′ U$_{st}$, Tr( U$_{st}$ ) = dδ$_{s}$$_{0}$δ$_{t}$$_{0}$. (4)   tr ( U$_{st}$U + s ′ t ′ ) = dδ$_{tt}$ ′ δ$_{ss}$ ′ , U$_{st}$U + st = I. (5) W = 1 d ∑ s,t tr ( U + $_{st}$W ) U$_{st}$. (6) | Φ$_{st}$ 〉 = ( I ⊗ U ∗ $_{st}$) | ψ$_{+}$ 〉 = 1 √$_{d}$ ∑ i,j ( U$_{st}$ $_{)}$∗ ij | ij 〉 , and | Φ$_{00}$ 〉 = | ψ$_{+}$ 〉 , (7) F ( ρ ) ≤ max j { λ$_{j}$ } , (8) 2