Some Reseach On Quantum Measurements And Applications

Quantum measurement is an important part in quantum information and quantum computation, which is one of the four basic assumptions in quantum mechanics, it is linked with the classical world and the quantum world. For any quantum system, the measurement will lead to the measured system of the quantum state changed. The quantum measurement makes interactions between a series of known quantum states and measured system, in order to obtain system information through the changed states. It provides theoretical tools for the transmission, discrimination, sharing and information recovery of quantum states, attracted more and more attention by the researchers. Thus, discussing some properties of quantum measurement is very useful. The quantum measurement considered by people is generally two kinds:the positive operator valued measurement (POVM) and the projection measurement.In this article, we define the notion of (A, B)-quantum measurement by using the idea of frame, which extends the quantum measurement in quantum mechanics, and obtain some properties. Besides, achieveing the quantum discrimination and quantum secret sharing is discussed by using some special quantum measurement. There are four chapters as follows.In Chapter1, we first recall the concepts of frame, dual frame, frame operator and decomposition of frame in Hilbert space, together with the definition of operator frame in Banach space. In addition, some related conceptions in quantum information are presented including two expressions for a quantum state:the vector patterns and matrix ways; two quantum measurements in common: POVM and projective measurement. Finally, the relationship between the operator frame and quantum measurement are pointed and the main work of this paper is shown.In Chapter2, the notion of a (A,B)-quantum measurement is introduced and some properties are obtained. An equivalent relation between (A,B)-quantum measurement and g-frame is established. The measure frame operator of a (A,B)-quantum measurement is defined and its typical dual operator is computed. Finally, the reconstruction formula of arbitrary quantum states is obtained in light of a (A, B)-quantum measurement.In Chapter3, we firstly introduce the concepts of the discrimination and unambiguous discrimination of a set of quantum states, abtain the sufficient and necessary condition for a set of quantum states to be discriminable and unambiguous discriminable. Secondly, we provide the mathematic description for the minimal erroneous discrimination. Lastly, we construct the square root measurement, and obtain an upper bound of the erroneous average probability for discrimination of a set of quantum states using the square root measurement.In Chapter4, we introduce a function of a quantum measurement in quantum secret sharing, list the concrete steps of the original scheme by using Bell states and the Bell measurement and point the defects of its sheme. Then we conclude the mathematic theory for the feasible scheme, which gives a relationship of particles ownd by every participant. Fanally, we propose an improved scheme by using the Bell states and the Bell measurement and analysis its security.