Group Covariance and Optimal Information Processing

Abstract

In quantum information processing, we need to decrease the amount of error as much as possible. To achieve this requirement, we mathematically formulate the error as a function, and seek the information processing to minimize (optimize) it. This chapter gives optimal scheme including optimal measurement under several setting with the group covariance. This approach can be applied to so many topics in quantum information, the quantum state estimation, the estimation of unknown unitary action, and the approximate quantum state cloning. The problem to optimize the estimation procedure cannot be exactly solved with the finite resource setting, in general, however, it can be exactly solved only when the problem has group symmetric property. Hence, we can say that group symmetric property is essential for state estimation. This idea can be applied to the estimation of unknown unitary action. This problem is much more deeply related to group representation because this problem can be investigated by using Fourier transform in the sense of group representation theory. Further, we deal with the approximate quantum state cloning in a similar way because we can say the same thing for approximate quantum state cloning. That is, all of solved examples of approximate quantum state cloning are based on the group symmetry.