The Generalized Concurrence And Application

In this thsis, we mainly give the separable conditions that a linear combination of twovectors is separable by Generalized Concurrence. Generalized Concurrence is a measure ofentanglement. Wootters gave an entanglement measure named Concurrence of two-qubit state,then people promoted this measure to high-dimensional two-body quantum systems. We call itGeneralized Concurrence. But Generalized Concurrence is only defined on the pure quantumstates. In this thsis, we study separable situations of mixed quantum states whose rank are 2, 3,n.In chapter 1, the preliminaries of this paper are given. These are quantum state, someoperator definitions, spectral decomposition, Schmidt decomposition and knowledge of traces,PPT criterion, the range and the support of density operatorρ.In chapter 2, firstly, we give the definition of Iα. It is a necessary amount which candefine Generalized Concurrence. From this, Generalized Concurrence is defined. Necessaryand sufficient condition when the two-body quantum pure states on separable is given. And thespecific values of Generalized Concurrence of the two-body maximum entanglement states isalso given. These appear in the article as properties of Generalized Concurrence.Generalized Concurrence is an important tool in studying the two-body quantum purestates. In chapter 2, we have known that the quantum state is separable if and only if its Gen-eralized Concurrence is equal to zero. In chapter 3 using that, I proved necessary and suffi-cient condition when a linear combination of two separable vectors is separable. For two states, , there exists a complex number , such that is separable if and only if for any complex number k, is separable. The processof the proof also gives the concrete form of two vectors when there is non-zero complex num-ber that makes this linear combination of this two vectors are separable. Two separable states , and is the Schmidt decomposition of .If there exists a complex number , such that is separable, then |E2? can bewritten in one of the following formsor This makes the second expression vector further clarified. In addition, the other condition isgiven. That is a linear combination of an inseparable vector with any vector. Two vectors , if is inseparable, then there are not more than twocomplex numberλsuch that is separable. And, then there are not more than onecomplex numberλsuch that is separable when is separable.In short, this chapter discusses all cases of combinations of the two two-body quantum linearstates.In Chapter 4, the problem when the two-body mixed quantum state of rank 2 is separableis solved. There are two cases when the two-body mixed quantum state of rank 2 is separable. is separable. Then one case isthat and are separable. The other case is that and are inseparable, and thereare two pure real or pure imaginary numbers making separable. So, necessaryand sufficient condition when the two-body mixed quantum state of rank 2 is separable is given.On this basis, we further study the two-body mixed quantum states of rank n(n≥3). Somespecific examples show that the conditions are not necessary are also given.It is convenient using Generalized Concurrence to determine whether the two-body quan-tum pure states are separable. So we solve all the problem whether the two-body mixed quantumstate of rank 2 is separable. But, in the paper I can not solve all the problem when the rank isn(n≥3). Whether the use of Generalized Concurrence is able to fully determine that two-bodyquantum states of rank n(n≥3) is separable? Can the situation whether the linear combinationof three pure quantum states is separable be all discussed? These problems are being furtherinvestigated.