Quantum Fisher Information And Quantum Metrology

Quantum mechanics, together with the theory of relativity, has become the most impor-tant corner stone of modern physics since it was founded in the 1920s. In the meanwhile, it also provides a solid theoretic foundation for the rapid development of high technologies. Besides, quantum mechanics has brought a massive impact on our philosophy. Local real-ism has been stripped away, and the features of quantum correlations such as entanglement and discord have been studied and the benefits of which are reaped by the state of art high tech giants. Disciplines like Quantum computation, quantum communication and quantum information emerges rapidly in the last 30 years. Meanwhile, quantum metrology, which covers quantum mechanics, statistical inference and precision measurements, is another new emerging cross discipline in quantum technology. It has drawn plenty of attention in the research community and showed its great potential both in theoretic and in practical appli-cations. In this paper, we will study some aspects of quantum metrology theory, especially a vital concept called quantum Fisher information.In the first chapter, we briefly retrospect the history of quantum mechanics and give light on some of its latest progress. Then we introduce the background of the birth of quantum metrology and its potential applications. We report the current research interest and trend in quantum metrology.In the second chapter, we introduce the definition and calculations of quantum Fisher information. Firstly, the origin of Fisher information is explained. In the following,we review the Cramer-Rao theorem, which is a bedrock of the statistical inference theory. After that, we introduce two methods to generalize the concept of Fisher information to the domain of quantum field. The first method is a straightforward generalization by using the concept of POVM(positive operator valued measurement). The second method takes advantage of the uncertainty relation and the error-propagation formula. In the final part of this chapter, we gives a classical way to compute the quantum Fisher information by decomposing the initial density operator.In the third chapter, we discuss the phase estimation problem in optical interferometry. At the beginning of this chapter, we briefly introduce the Mach-Zehnder interferometer and give the mathematical representation of the interferometer in the language of SU(2) group. Then we focus on the evolution of the entangled coherent state inside the interferometer. We considered the effect of photon loss on the entangled coherent state, calculate its quan-tum Fisher information in the photon loss scenario. We also provide a method to calculate quantum Fisher information for a unitary phase parametrization process.In the fourth chapter, we focus on the theory of unitary parametrization process and give the quantum Fisher information for this process. By introducing the concept of a Hermitian operator as the generator of the parametrization process, we find the connection between quantum Fisher information and the spectral of the initial state along with the generator operator. Following that,we mainly consider a category of unitary parametrization governed by a su(2) type Hamiltonian and find the maximal quantum Fisher information. We then apply the theoretical results to several practical examples.The summary is given in the fifth chapter.