Existence And Construction Of Error-correcting Code Spaces Of Diagonal Quantum Channels

In the processing of the quantum information,the environment always coupling of the primary system.It is hard to avoid the error which caused by noise.So,how to control the noise effectively is the most important content in the research of the quantum information.And the quantum error correcting code space is one of the most effective method.This method has been widely used in the quantum compu-tation and the quantum communication.In this paper,we propose the definition and the properties of diagonal quantum channels,and the necessary and sufficien-t condition for ? to be a diagonal quantum channel though the Kraus operators.Then we propose the definition of the condensed vector set,represented as V,and the condensed matrix,represented as W,which of diagonal quantum channels.And prove that,for a certain diagonal quantum channel ?,in spite of the corresponding condensed matrix is not unique,but the rank of the condensed matrix is unique.At the same time,a necessary and sufficient condition for ? being uncorrectable is presented,? is uncorrectable if and only if the rank of W is full.On the basis,we proved that ? is correctable on the whole space if and only if the rank of W is one.Finally we use a concrete example to present a way to find the code space of ?.And,we also present some simple applications that diagonal quantum channel can be used in the quantum error detection and the correction.The construction of chapters and the concrete contents of this paper are as follows:Chapter 1:We introduced some research background and status on our main contents,and some basic concepts.Chapter 2:In order to research the diagonal quantum channel,firstly,we give some properties of the quantum channel.Then we give the definition of diagonal quantum channel.Finally,we present the necessary and sufficient condition for E to be a diagonal quantum channel.Chapter 3:In order to research the error correcting code space of the diagonal quantum channel,first of all,we give the definition of the ? can be corrected.Secondly,we present the definition of the compression vector and the compression matrix.On this basis,we give the relationship between different compression matrix that different Kraus operators of a same diagonal quantum channel corresponded.Finally,a necessary and sufficient condition for ? being uncorrectable is presented,? is uncorrectable if and only if the rank of W is full.On the basis,we proved that? is correctable on the whole space if and only if the rank of W is one.At the same time we use a concrete example to present a way to find the code space of ?.Chapter 4:In the part,we give some equivalent conditions of the quantum error detection.And we give some simple applications using diagonal quantum channel contact with error detection.