Estimation, Operation, And Discrimination Of Quantum States

Rapid progress has been made in Quantum Information Science (QIS) during the past two decades. As an expanding multidisciplinary field, QIS combines many interesting features both in classical information theory and quantum physics, and promotes deep understanding of traditional concepts in these fields. Due to the superposition principle in quantum mechanics, many concepts have to be reconsidered in QIS. For example, we cannot copy (or delete) an unknown state exactly because of "the no-cloning theorem"(no-deleting theorem), which is known as one of the most significant differences between classical and quantum information. Therefore to get to know what we can and cannot do in QIS or in quantum physics becomes an important and interesting problem. Particularly, in quantum communication and quantum cryptography, many novel schemes are based on the fact that nonorthogonal states cannot be discriminated determinately. Henceforth study on the discrimination of quantum states has a close relation to the security of quantum cryptographic protocols. Similarly, investigations on the information encoded in finite physical resources will also be helpful for improving the efficiency of state measurement, storage, etc..This dissertation focuses on some theoretical investigations of quantum state estimation together with discrimination process, and is organized as follows.In the first chapter, we introduce some basic concepts and related background materials, including the theory of quantum operation and unitary group representation, which is helpful for understanding the following content of this dissertation.In the second part, we consider the role played by antilinear map in QIS. In the qubit case, Gisin and Popescu pointed out that Positive Operator Value Measure (POVM) acting on |(?) can get more information than two parallel states |(?). We generalize the idea to high-dimension case. We show that, for N parallel input states, an antilinear map with respect to a specific basis is essentially a classical operation. Later, We investigate the information contained in phase-conjugate pair |(?), and prove that there is more information about a quantum state encoded in phase-conjugate pair than in parallel pair. Physical explanations about the results obtained are also provided at the end of this chapter.Thirdly, based on the fact that in the most practical cases, people often get to know some specific information about the input states set and the input usually contains only a finite number of quantum states, we consider physically accessible transformation on a finite number of input states. We treat the usual probabilistic cloning, state separation, unambiguous state discrimination, etc in a uniform framework. All these transformations can be regarded as special examples of generalized completely positive trace non-increasing maps on a finite number of input states. From the system-ancilla model we construct the corresponding unitary implementation of pure→pure, pure→mixed, mixed→pure, and mixed→mixed states transformations in the whole system and obtain the necessary and sufficient conditions on the existence of the desired maps. We also provide a numerical method to reduce this to standard semidefinite programming (SDP) problem, which thus can be solved efficiently.With the method developed in the third chapter, we consider to discriminate mixed states unambiguously. By constructing the specific unitary realization of the desired transformation, we can reproduce almost all known results in the area. In the two-mixed-state case, we introduce the canonical vectors and partly reduced the original problem to the unambiguous discriminate between pairs of canonical vectors. We present a series of new upper bounds on the total success probability which depends on both the ratio of the prior probabilities and the input state structures. The bound presented in this work is independent of those of former works, and sometimes it can provide tighter bound of the total success probability, as expected.In the last part, we concentrate on the discrimination of quantum operations. Unlike the case of quantum states, where nonorthogonal states cannot be perfectly identified, we find that in principle, unitary transformations can be perfectly identified only with local methods despite of their nonlocal features. This observation indicates that sometimes, quantum operations show more classical features than quantum states, hence many interesting features about quantum states cannot be naively transplanted to the case of quantum operations.