Standard Quantum Error Correction And Operator Quantum Error Correction

The primary task of realizing effective quantum computing is to overcome deco-herence,to monitor and correct errors in t,he calculation process in a timely manner is main aim of quant,um error correction.This thesis is devoted to discussion on some problems related to standard quantum error correction and operator quantum error correction.The full text is divided into three chapters.The first chapter introduces the research background and present situation of the topics discussed in this paper and introduces some basic concepts and existing results used later.In the second chapter,we introduce the standard quantum error correction.Firstly,the definition of standard quantum error correction,maximally error cor-rection code space and the power of error correction code are given.At the same time,equivalent characterizations of error correction are given.Secondly,generated channels and generating matrices are introduced.Some examples are given to illus-trate the possible relationship between two quantum channels.We prove that two quantum channels can be generated from one another if and only if the generating matrix is invertible.Quantum channel preserver is given.And the necessary and sufficient condition for quantum channel preserver is that,it is a unitary matrix.Also,a relation between an error correction code and the related error correction channel of a generated channel and those of the original channel is obtained.Third-y,equivalent channels are defined and a relation between an error correction code,the related error correction channel,maximally error correction code space and the power of error correction code of an equivalent channel and those of the original channel is explored and illustrated by some examples.Lastly,the application of unitary equivalent channel related to diagonal channel is given.In the third chapter,the definition of operator quantum error correction and equivalent characterizations of it are given.Then,a relation between error correction subsystem code of equivalent channel and that of the original channel is explored.The necessary and sufficient condition for a subsystem to become an error correction subsystem of a quantum channel is given.And it is proved that when the condensed matrix of a diagonal quantum channel has a full rank,it must have no operator quantum error correction code.