Research On Related Problems Of Quantum Error Correction

In quantum information processing and computation,owning to the existence of noise,quantum codes passing though the quantum channel may change,thus wrong information may be produced.It is necessary to correct the errors in practice.In order to achieve this goal,the quantum codes which contain quantum information are divided into different classes.The relevant conceptions such as quantum error correcting code,strongly-preserved code,weakly-preserve code,and noiseless code are proposed.In this paper,we study the noiseless code and the error-correcting code space which are closely related to quantum error correcting.Firstly,according to the definition of noiseless code,we study the relationship between the noiseless code and the strongly(or weakly)-preserved code and obtain some results.Second-ly,in order to correct the error information produced by the quantum channel as much as possible,we introduce the concept of maximally error correcting code s-pace(MECCS)after discussing the properties of error correcting code spaces,study the properties of MECCS and solve out the dimensions of the MECCSs of two spe-cific quantum channels.This thesis is divided into three chapters and arranged as follows:In Chapter 1,we introduce some research backgrounds and status on our main contents,and review some basic definitions of quantum mechanics and quantum information which will be used in this paper.In Chapter 2,we give some properties of noiseless quantum and explore re-lationships of strongly preserved code,weakly preserved code and noiseless codes.When the image of a code C under a quantum channel ? is contained in the fixed-point set of ?,we prove that C is a noiseless code of ? if and only if it is a weakly preserved code of ?.For an arbitrary quantum channel ? and a quantum code C,we also prove that C is a noiseless code of ? if and only if it is a noiseless code of any convex combination of ? and ?2.Lastly,we present a weakly preserved code and a noiseless code of a quantum channel of the quantum system C2(?)C2.In Chapter 3,we introduce first the concept of a maximally error-correcting code space of a quantum channel and prove that any error-correcting code space of a quantum channel must be contained in an maximally error-correcting code space of that channel.We also prove that every quantum channel has always an maximally error-correcting code space.Moreover,we have given a sufficient and necessary condition for the direct sum of two orthogonal error-correcting code space of the same channel to be an error-correcting code space of that channel.Lastly,By using the rank-k numerical range by Choi,we have obtained a characterization for the case where C(k).