basis by a unitary transformation. These transformations have been listed in Table.I. Here, F denotes the Fourier transfor- mation: R denotes a phase operation: and the Controlled gate is Where ⊖ denotes the difference j − i modulo 3.If there are n qutrits, the number of MUM is 3 n + 1 , which is far less than 3 2 n − 1 in Ref. [12]. That is to say the use of MUMs can represent a considerable reduction in the operations and time required for performing the full state determination [37]. Now if we further extend one-qutrit case to two-qutrit case, the density matrix can be expressed as: ρ$_{12}$ = (1 / 9) ∑ 8 j,k =0 r$_{jk}$λ$_{j}$ ⊗ λ$_{k}$ , where r$_{jk}$ = 〈 λ$_{j}$ ⊗ λ$_{k}$ 〉 . If we use the general method in Ref. [12] to fully determine the state, d 2 n − 1 = 3 4 − 1 = 80 measurements will be needed. So much measurements will inevitably introduce redundant information of the state, which is obviously a resource waste. So here we will take advantage of the MUMs to reconstruct the two-qutrit state. It is easy to find that the nine elementsof ϝ 9 (finite field) are { 0 , α, 2 α, 1 , 1+ α, 1+2 α, 2 , 2+ α, 2+2 α } by using the irreducible polynomials θ $^{2}$+ θ +2 = 0 [14]. Here we use the representation {| 0 〉 , | α 〉 , | 2 α 〉 · · · | 2 + 2 α 〉} as the standard basis. One can find that there will be only d 2 + 1 = 3 $^{2}$+ 1 = 10 MUMs to be done, which is much less than 80 of Ref. [12]. It means that the operations and time needed for the whole state determination is great reduced. The decompositions for all the MUBs of the two-qutrit system have been listed in Table. II. In general, the fidelity of single logic gates can be greater than 99% , but nonlocal gates have a relatively lower fidelity. We have explicitly presented an optimal tomographic scheme for the single-qutrit states, two-qutrit states and three- qutrit states based on the MUMs. Because the MUBs based state reconstruction process is free of information waste, the minimal number of required conditional operations are needed. So we call our qutrits tomographic scheme the op- For the multi-qutrit system( n > 3 ), it is not easy to get all the sets of MUBs, and it is even more difficult to get the ex- plicit decompositions of the optimal set of MUBs. Neverthe- less, the results for the two-qutrit and three-qutrit cases have provided the experimentalists valuable references. However in three-qutrit case, there are five sets of MUBs with different structures, namely { (0 , 12 , 16) , (1 , 9 , 18) , (2 , 6 , 20) , (3 , 3 , 22) , (4 , 0 , 24) } [21, 44]. It is easy to see that the (0 , 12 , 16) set of MUBs has the minimum physical complexity. We say that the optimal set of the MUBs is (0,12,16). The decompositions of the MUBs in the three-qutrit case for (0,12,16) are listed in Table.III. So the set of MUBs (0,12,16) has a complexity C ∝ 44 , which is a very important value in experimental realization of it. The fidelity of a practical CNOT gate can reach a value up to 0.926 for trapped ions system in Lab [43]. Klimov et al have introduced the concept of the physical complexity of each set of MUBs as a function of the number of nonlocal gates needed for implementing the MUMs [37]. Here the fidelity value of the CNOT gates for qubits systems also can be used to eval- uate the physical complexity of the MUBs of qutrits systems. Why we can say so is because of the following point. Al- though the systems involved here are three-state ones, all the operations used in our reconstruction process can be decom- posed into effective two-state operations. So the complexity of the current tomography scheme is proportional to the num- ber of the nonlocal gates used( C ∝ 6 ) for two qutrits system. As shown in Ref[44], the only MUB structure for a two-qutrit system is (4 , 6) , where 4 is the number of the separable bases and 6 is the number of the bipartite entangled bases. F | j 〉 = (1 $_{/}$√$_{3)}$ 2 ∑ l =0 exp (2 πilj/ 3) | l 〉 , j = 0 , 1 , 2 , (2) R = | 0 〉〈 0 | + ω | 1 〉〈 1 | + ω | 2 〉〈 2 | , (3) X | i 〉| j 〉 = | i 〉| j ⊖ i 〉 . (4) V. THE PHYSICAL COMPLEXITY FOR IMPLEMENTING THE MUMS IN THE THREE-QUTRIT SYSTEM VI. CONCLUSION IV. RECONSTRUCTION PROCESS FOR AN ARBITRARY TWO-QUTRIT STATE TABLE I: The transformations connecting the MUBs with the stan- dard computational basis for a qutrit system based on Fourier trans- forms and the phase operations. TABLE II: The decompositions of MUBs for the two-qutrit system based on Fourier transformations, phase operations and controlled- NOT gates ( X 12 ) [42] with the first particle as source and the second one as target. The subscript denotes the i th particle, i = 1 , 2 . 3 Basis Transformation 2 F − 1 3 F − $^{1}$R 4 F − $^{1}$R − 1 Basis Decompositions 2 F − 1 1 F − 1 2 3 F − 1 1 R 1 F − 1 2 R 2 4 F − 1 1 X 12 F − 1 2 R − 1 2 5 F − 1 1 X − 1 12 R 1 F − 1 2 R − 1 2 6 F − 1 1 X − 1 12 F − 1 2 R − 1 1 7 F − 1 1 R − 1 1 F − 1 2 R − 1 2 8 F − 1 1 X − 1 12 F − 1 2 R 2 9 F − 1 1 R − 1 1 X 12 F − 1 2 R 2 10 F − 1 1 R 1 X 12 F − 1 2