Research On The Structure Of Mutually Unbiased Based And Their Applications

Quantum information is a new interdisciplinary science and has been developed from 1980s based on quantum mechanics. In quantum information, state of motion of microscopic particles is called quantum state, which is the concrete manifestation of quantum information. Quantum state plays a crucial role in quantum information and can be described by a unit trace Hermitian matrix. In order to gain the complete information of one unknown state, one needs to perform a series of measurements on a large number of identically prepared copies of the state. Thus one can obtain the full information of the state from the measurements results based on different bases. However, for a quantum system, there are many different set of bases, and which is the best one is very important to us. By best we mean that, the measurement result based on one of the different bases within this set will have on overlap with the measurement results based on the other bases in this set. It seems that all the bases of this best set are homogeneously distributed in the state space. In this sense, we call this set of bases mutually unbiased bases(MUBs). In the literature, all the MUBs for prime and prime-power dimensional systems are known to us, but for the non-prime-power dimensional systems, it is not clear whether there exists the maximum number of MUBs or not. So, the purpose of this PhD dissertation is to carry out research work on the MUBs. The main content is as follows:1. An optimal quantum reconstruction scheme for the multi-qutrits states based on mutually unbiased measurements is proposed. It is not difficult to see that, in the currently existing quantum state reconstruction scheme, the measurement results are not independent of each other. The reason is that there are lots of overlaps between the measurement results in the process of quantum reconstruction, i.e. there may be large redundancy in these results. Because the reconstruction process of states based on mutually unbiased basess is free of information waste, we refer to this scheme as the optimal scheme. By using the 量子无偏纂的应用及其纠缠结构研究finite fields theory, we obtain all the MUBs of the single and multi-qutrit. Then we concetrate on the following question:how those different mutually unbiased measurements are realized, that is to say 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. The advantages of our scheme are the needed numbers of required conditional operations and the time are the minimal and all the needed unitary operators can be implemented in the laboratory. We hope these decompositions can help experimental scientists to realize the most economical reconstruction of quantum states in multi-qutrit systems in the laboratory.2. The physical complexity of a set of MUBs has been considered. Because there are different entanglement structures in the complete sets of MUBs and these structures are complicated. In the three-qutrit system there are five different sets of MUBs, each of which has a specific entanglement structure. The physical complexity is a function of the number of nonlocal gates needed for implementing the mutually unbiased measurements. So the physical complexities of the MUBs-based measurements will be different correcponding to different entanglement structures. To implement the optimal quantum state reconstruction, we need to find out the MUBs whose entanglement bases can be decomposed with the minimal number of nonlocal gates. The physical complexity is an important aspect must be considered in experimental quantum information science.3. We studied the relationships between the different MUBs with different entanglement structures. The results showed that different MUBs with different entanglement structures be transformed from one to another in three-qutrit system under some specific joint unitary operations. These relationships will help the experimental scientists to realize the mutually unbiased measurements in laboratory.