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and there is a big redundancy among the measurement results there. To remove this redundancy, we will propose an optimal tomography process for qutrits states. This optimal quantum tomography process is the MUBs-based qutrit states tomog- raphy, and the number of required measurements is greatly reduced. A d -dimensional quantum system is represented by a positive semidefinite Hermitian matrix ρ with unit trace in d -dimensional Hilbert space, which is specified by d 2 − 1 real parameters. A nondegenerate measurement performed on such a system provides d − 1 independent probabilities. So, in general, one requires at least d + 1 different orthogonal mea- surements for fully determining an unknown ρ . For n -qutrit tomography it only needs 3 n + 1 measurements. Through analysis, one can find that the tomography process for qubit system can not be generalized to qudit system case in a trivial way. The MUBs-based tomography process for qubit system proposed by Klimov et al can not be directly applied to qutrit system [37]. This is because the entanglement feature of the MUBs of qubits system is totally different from that of qutrits system. So, we will study the physical implementation of an optimal tomographic scheme for the case of determining the states of multi-qutrit system based on the MUMs. 

From the experimental point of view, the physical complex- ity is a key point for the implementation of a scheme. Here, for multi-qutrit quantum tomography, the physical complex- ity mainly comes from the entanglement bases, i.e. the num- ber of the two-qutrit conditional operation needed for the de- composition of these entangled bases. In addition, there ex- ists many different MUBs with different entanglement prop- erties in multi-qutrit system. So the physical complexity of the quantum tomography process here depends on the entan- glement structure of the MUBs used, and it becomes very im- portant to optimize the quantum tomography process over all the possible MUBs entanglement structures of the system. 

This paper is arranged as follows. In the next section we introduce the MUBs in a d -dimensional system ( d = p n ( p ̸ = 2) ) and show how to reconstruct an unknown state by MUMs. Here p is a prime. Section III briefly reviews the general method to reconstruct a qutrit state, where the number of measurements is 8 . However the MUBs-based qutrit to- mography proposed here only needs 4 measurements, which means the number of measurements is reduced. Here the mea- surement reduction for single qutrit case is not obvious, so in section IV we will extend the one-qutrit case to two-qutrit sys- tem. For the two-qutrit system the number of measurements is only 10 for determining all the elements of the density op- erator rather than 3 4 − 1 = 80 measurements in the scheme proposed in Ref. [12]. It means a great reduction of the ex- perimental complexity. So we conclude that the optimal mea- surements on the unknown qutrit states are the MUMs. In sec- tion V we discuss the physical complexity for implementing the MUMs in three-qutrit system, and give the optimal MUB for the qutrit system quantum tomography process with mini- mized physical complexity. The last section is the conclusion. 

## II. MUTUALLY UNBIASED BASES AND MUTUALLY UNBIASED MEASUREMENTS 

As shown by Wootters, Fields [14] and Klappenecker, R¨otteler [19], in finite field language, the first MUB in a d = p n ( p ̸ = 2) -dimensional quantum system is the standard basis B 0 given by the vector ( a (0) k )$_{l}$ = δ$_{kl}$, k, l ∈ ϝ p n , where the superscript denotes the basis, k the vector in the basis, l the component and ϝ p n is the field with p n elements. The other d MUBs are denoted by B r which consists of vectors ( a ( r ) k )$_{l}$ defined by[14]: ( a ( r ) k )$_{l}$ = (1 $_{/}$√$_{d}$$_{)}$$_{ω}$ T r ( r $^{·}$l $^{2}$+ k $^{·}$l $^{)}$, r, k, l ∈ ϝ p n , r ̸ = 0 . Here ω = exp (2 πi/p ) and T r θ = θ + θ p + θ p 2 + · · · + θ p n$_{−}$ $^{1}$. The set of mutually unbiased projec- tors can be given by P ( r ) k = | a ( r ) k 〉〈 a ( r ) k | . It is worth notic- ing that | a ( r ) k 〉 contains the computational basis B $^{0}$. Here T r ( P ( s ) j P ( r ) k ) = (1 /d )(1 − δ$_{sr}$ + dδ$_{sr}$δ$_{jk}$ ) . Then the measure- ment probabilities given by ω ( r ) k = T r ( P ( r ) k ρ ) completely determine the unknown density operator of a d -dimensional system [13]: ρ = Σ d r $_{=0}$Σ d − 1 k $_{=0}$ω ( r ) k P ( r ) k − I. For instance, in a qutrit system, there are three MUBs be- 

side the computational basis B 0 = {| 0 〉 , | 1 〉 , | 2 〉} , in the fol- lowing form with ω = exp (2 πi/ 3) : 

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## III. RECONSTRUCTION PROCESS FOR AN ARBITRARY SINGLE QUTRIT STATE 

An unknown single qutrit state can be expressed as [12, 40]: ρ = (1 / 3) ∑ 8 j =0 r$_{j}$λ$_{j}$ , where λ$_{0}$ is an identity operator and the other λ$_{j}$ are the SU(3) generators [41]. The general method to reconstruct the qutrit state is to measure the ex- pectation values of the λ operators [12], where r$_{j}$ = 〈 λ$_{j}$ 〉 = T r [ ρλ$_{j}$ ] . Thus one will find that the number of required mea- surements is 8 . However, if we choose the MUMs to deter- mine the qutrit state, the number of needed MUMs is only 4 rather than 8 of Ref. [12]. The four optimal set of MUBs have been presented by Eqs.(1a,1b,1c) plus the standard computa- tional basis in the preceding section. Each of the three MUBs in Eqs.(1a,1b,1c) is related with the standard computational 

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