The Basic Theory And Properties Of Density Operator In Quantum Computation

Density operator is an important concept in quantum computation.It can not only describe the pure state quantum system,but also describe the mixed state quantum system.Moreover,the reduced density operator derived on the basis of the density operator can simultaneously describe the state of subsystems in the unknown quantum system and composite quantum system.Further speaking,density operator and reduced density operator are widely used in the fields of quantum noise,quantum error correction,system measurement,and quantum algorithm as the basic quantities in quantum computing.Given a composite quantum(physical)system,the density state operator and the reduced density operator are first used to describe the quantum state in the system.Second,the nature of other physical quantities in quantum system is analyzed based on the representation of the quantum state,and the properties of the density operator and the reduced density operator.Finally,the properties of the quantum system applied to real life.Obviously,the density operators and the reduced density operators play an important basic role in this process.At the same time,it also stimulates the improvement of density operators and reduced density operators and their properties in order to provide more detailed and thorough analysis and application of the properties of quantum(physical)systems.The main findings of this article are as follows:(1)The singular value decomposition form,the power form,and the Jordan standa-rd form of the density matrix in pure and mixed states are given.Based on the representation of the density matrix Jordan standard form,the condition that the density matrix is not a unitary matrix is given.At the same time,it is proved that in any two-body qubit system,the track of the reduced density matrix of the subsystem is less than 1.It also proves that the three-body quantum bit system has the coherence.In addition,the density operator has been expanded as a mathematical theory for distinguishing quantum states.(2)The representation of the reduced density operator of the n-1 body subsystem in any n-body system is proved.The basic properties of the subsystems ?~A,?~B and ?~A(?)?~B in the composite quantum system are obtained.Based on their basic properties,the relationships between eigenvalues and eigenvectors of ?~A,?~B and ?~A(?)?~B are proved.Finally,the properties of the reduced density operator is used to prove the purification of the composite quantum(physical)system.