The Theory Of And Application Of Quantum State Discrimination

The birth of quantum mechanics at the beginning of the twentieth century is a milestone of modern physics. By quantum mechanics, people have got a far more comprehensive and deeper understanding of the microscopic world than before, and invented all kinds of novel techniques, such as STM, semi-conductor and so on. These new inventions and new techniques have changed people’s viewpoint of the world and life style, and thus promoted the human being’s civilization. As the rapid development of the quantum mechanics, more and more other research fields begin to interplay with it, which has led to many new interdisciplinary subjects and research hotspots, including quantum information and quantum computation as a typical one.Namely, quantum information and quantum computation is an interdiscipline of quantum mechanics and the science of computation and communication. It originated from Feynman and others’research on whether the dynamics of quantum systems can be simulated by classical computers in1980’s (and maybe can even date back to the debate over whether information can transmit faster than the light by quantum systems within the special relativity principle). As the development of quantum information and quantum computation, people surprisingly found that computation by quantum systems can be startlingly faster than computation by classical comput-ers, and can solve some problems in a polynomial time which cannot be by classical computation; also communication by quantum systems have some great advantages over classical communi-cation, for example the unconditional security of the quantum key distribution, quantum state teleportation and superdense coding with quantum entanglement, and so on. These distinct fea-tures of quantum computation and quantum information have in return aroused more people’s interest and curiosity, and stimulated more rapid development of this field.In the quantum information field, information is usually encoded in and carried by quantum states. And when one tries to extract information from encoded quantum states, a key step is to discriminate each of the encoded states. The higher the rate of success and accuracy of discrimination is, the more information can one extract from the encoded states. So, it can be seen that quantum state discrimination is critical to quantum communication, and the efficiency and reliability of quantum state discrimination decide the quality of quantum communication. Therefore, the study of optimizing quantum state discrimination, e.g. improving the rate of success and accuracy, is of great significance. Quantum state discrimination is an important and essential problem in quantum information, and has received intensive research in these years. This thesis will focus on this problem, and carry out a deep and systematic study into it.The organization of this thesis is as follows.Chapter1will review the history of quantum mechanics, quantum information and quantum computation. It will bring a brief introduction to the main directions of quantum information and computation, and also the basic principle of quantum mechanics, along with the basic concepts in quantum information.Chapter2will first describe the von Neumann measurement (i.e. the orthogonal projective measurement), the generalized quantum measurement and POVM. Then it will give a detailed introduction to the two major categories of quantum state discrimination strategies:the minimum error discrimination and the unambiguous discrimination, and will present some basic results of them. Lastly, the recent progress of theoretical study and experimental realization of quantum state discrimination will be reviewed.Chapter3will be dedicated to the unambiguous discrimination of pure quantum states. First, it will obtain a general condition that should be satisfied by unambiguous quantum state discrimination. Then, it will make a detailed analysis of the optimal solution to the unambiguous discrimination problem, and derive a necessary and sufficient condition for it. For the conve-nience of computation, the necessary and sufficient condition will be applied to the cases of the optimal point being an interior non-singular point of the critical feasible region and a boundary point of the critical feasible region, and groups of equations that the optimal solution must satisfy will be obtained for these two cases respectively. Meanwhile, the meaing of the necessary and sufficient condition for the optimal solution will be clarified from a geometrical view, and some numerical examples will be calculated. Subsequently, the analytical structure of the optimal solution will be investigated by using the necessary and sufficient condition. After that, the op-timal unambiguous state discrimination problem will also be considered for the case that a kind of generalized equal probability measurement is employed, and an analytical optimal solution will be derived. Lastly, the case of three pure states will be studied.Chapter4will mainly consider a problem derived from quantum state discrimination, that is quantum state comparison. It will first give a rigorous defintion of quantum state compari-son, and present a basic result of unambiguous comparison of pure quantum states. Then it will turn the attention to the case of mixed quantum states. It will show the impossibility of univer- sal comparison of mixed quantum states, and give a detailed study on the feasible conditions of unambiguous comparing a finite set of mixed quantum states in different situations. In ad-dtion, a special kind of quantum state comparison problem will be investigated:is it possible to determine whether the extent to which several unknown quantum states are mutually close or orthogonal is above or below a given threshold by a quantum measurement? Such a task will be proven impossible whenever the threshold is non-trivial. At the end, it will study if it is possi-ble to determine whether several unknown quantum states are orthogonal or not by a quantum measurement, and the answer will also turn out to be negative.A new interesting quantum discrimination problem will be considered in Chapter5, that is indirect quantum state discrimination. In the indirect quantum state discrimination, one does not perform a measurement on the system to be identified directly, but rather couples the system to an ancilla via interaction followed by a measurement on the ancilla, and the output from the measurement on the ancilla will give the information of the original system to be identified. The cases of a single ancilla and multiple ancillas will be studied, and analytical results will be obtained for these two cases respectively.In the last chapter, a summary of this thesis will be given, and some open problems on quantum state discrimination will be proposed.