# Optimal reconstruction of the states in qutrits system 

$^{1}$Key Laboratory of Opto-electronic Information Acquisition and Manipulation, Ministry of Education, School of Physics and Material Science, Anhui University, Hefei 230039, People’s Republic of China 

Fei Yan, 1 Ming Yang $^{∗}$, 1 and Zhuo-Liang Cao 2 

$^{2}$Department of Physics & Electronic Engineering, Hefei Normal University, Hefei 230061, People’s Republic of China 

Based on mutually unbiased measurements, an optimal tomographic scheme for the multiqutrit states is pre- sented explicitly. Because the reconstruction process of states based on mutually unbiased states is free of information waste, we refer to our scheme as the optimal scheme. By optimal we mean that the number of the required conditional operations reaches the minimum in this tomographic scheme for the states of qutrit systems. Special attention will be paid to how those different mutually unbiased measurements are realized; that is, how to decompose each transformation that connects each mutually unbiased basis with the standard computational basis. It is found that all those transformations can be decomposed into several basic implementable single- and two-qutrit unitary operations. For the three-qutrit system, there exist five different mutually unbiased-bases structures with different entanglement properties, so we introduce the concept of physical complexity to min- imize the number of nonlocal operations needed over the five d ifferent structures. This scheme is helpful for experimental scientists to realize the most economical reconstruction of quantum states in qutrit systems. 

The quantum state of a system is a fundamental concept in quantum mechanics, and a quantum state can be described by a density matrix, which contains all the information one can obtain about that system. A main task for implementing quan- tum computation is to reconstruct the density matrix of an un- known state, which is called quantum state reconstruction or quantum state tomography[1, 2]. The technique was first de- veloped by Stokes to determine the polarization state of a light beam [3]. Recently, Minimal qubit tomography process has been proposed by ˇ Reh´aˇcek et al , where only four measure- ment probabilities are needed for fully determining a single qubit state, rather than the six probabilities in the standard pro- cedure [4]. But the implementation of this tomography pro- cess requires measurements of N-particle correlations[5].The statistical reconstruction of biphotons states based on mu- tually complementary measurements has been proposed by Bogdanov et al [6, 7]. Ivanov et al proposed a method to determine an unknown mixed qutrit state from nine indepen- dent fluorescence signals[8]. Moreva et al paid attention to experimental problem of the realization of the optimal proto- col for polarization ququarts state tomography [9]. In 2009, Taguchi et al developed the single scan tomography of spatial three-dimensional (qutrits) state based on the effect of realis- tic measurement operators[10]. Allevi et al studied the imple- mentation of the reconstruction of the Wigner function and the density matrix for coherent and thermal states by by switching on/off single photon avalanche photodetectors[11]. 

In order to obtain the full information about the system we need to perform a series of measurements on a large num- ber of identically prepared copies of the system. These mea- surement results are not independent of each other, so there 

PACS numbers: 03.65.Wj, 03.65.Ta, 03.65.Ud, 03.67.Mn 

## I. INTRODUCTION 

is redundancy in these results in the previously used quantum tomography processes [12], which causes a resources waste. If we remove this redundancy completely, the reconstruction process will become an optimal one. So, to design an opti- mal set of measurements for removing the redundancy is of fundamental significance in quantum information processin g. 

Mutually unbiased bases (MUBs) have been used in a va- riety of topics in quantum mechanics [13–36]. MUBs are de- fined by the property that the squared overlap between a vecto r in one basis and all basis vectors in the other bases are equal. That is to say the detection over a particular basis state does not give any information about the state if it is measured in an- other basis. Ivanovi´c first introduced the concept of MUBs t o the problem of quantum state determination [13], and proved the existence of such bases in the prime-dimension system by an explicit construction. Then it has been shown by Wootters and Fields that measurements in this special class of bases, i.e. mutually unbiased measurements (MUMs) provide a minimal as well as optimal way of complete specification of an un- known density matrix [14]. They proved that the maximal numbers of MUBs is d + 1 in prime-dimension system. This result also applies to the prime-power-dimension system. 

MUBs play a special role in determining quantum states, such as it forms a minimal set of measurement bases and pro- vides an optimal way for determining a quantum state [13–16] etc. Recently an optimal tomographic reconstruction scheme was proposed by Klimov et al for the case of determining a state of multiqubit quantum system based on MUMs in trapped ions system [37]. However, the use of three-level systems instead of two-level systems has been proven to be securer against a symmetric attack on a quantum key distri- bution protocol with MUMs than the currently existing mea- surement protocol [38, 39]. Quantum tomography in high di- mensional (qudit) systems has been proposed and the num- ber of required measurements is d 2 n − 1 with d being the di- mension of the qudit system and n being the number of the qudits [12]. This tomography process is not an optimal one, 

