Tensor-product Decompositions Of Unitary Operators And Applications

Quantum computing and quantum information are a discipline of information processing tasks based on quantum mechanical systems.Entanglement is a unique resource of quantum mechanics and plays a key role in most applications of quan-tum computing and quantum mechanics.The operator theory has a high degree of unity and comprehensiveness,which can reveal the common attributes and interre-lationships of many problems in mathematics and physics.In this thesis,we study tensor product decomposition of unitary operators in a tensor product space and application.Some sufficient and necessary conditions for the unitary operator for eiH to be a tensor product of two unitary operators are given when H is a self-adjoint operator.Some sufficient and necessary conditions for the elvolution operator of a composite quantum system can be written as a tensor product of ones on subsys-tems are established.When the Hamiltonian of a composite system has a special structure,it is proved that the initial state of the system is separable(classical cor-related)if and only if the state at any time is separable(classical correlated).This article is divided into three chapters,the specific content is as follows:The first,chapter introduces the relevant historical background,research status and research significance of this paper,and reviews the basic concepts and definitions of quantum mechanics and quantum information and operator theory used in this paper.The second chapter firstly studies the necessary and sufficient conditions of the separability of the exponential operator eH when H is separable.Then it is discussed that the separabilit.y of exponential operator eH when H is a diagonal matrix.Fi-nally,for the general self-adjoint operator,a necessary and sufficient condition the exponential operator for eH to be separable is obtained by using diagonalization of a self-adjoint operator.The third chapter firstly studies the necessary and sufficient conditions for the unitary operator eiH generated by the self-adjoint operator H to be decomposed into the tensor products of two unitary operators.Secondly,it is discussed first that the separability of the unitary operator eiH when H is an arbitrary self-adjoint operator,and several related lemmas are proved as auxiliary.Finally,when the Hamiltonian H in the composite system has a special structure,it is proved that the initial state of the system is separable if and only if it is separable at any time.