Quantum Entanglement And Separability

Quantum entangled states are playing as the key resources in the quantum information processing and computation. Nevertheless, the mathematical structure of the quantum entanglement is still not fully understood yet. We have investigated the properties of entanglement by criterion of separability (non-entanglement) and equivalence of quantum states under local unitary transformations.The separability problem for pure states is quite well understood. However, for mixed states, the results are far from being satisfied not only theoretically but also experimentally. Till now, there are many necessary conditions for separability (for instance, the well known PPT (positive partial transpose) criterion), but very few sufficient conditions. We have derived several sufficient conditions of separability for some special quantum mixed PPT states on multiple quantum spaces. The main results of this part in our thesis include:Firstly, the separability problem for a class of PPT mixed states in 2 × 3 × N system is solved completely. A canonical form for PPT states with rank N and a sufficient condition for separability on 2 × 3 × N are investigated, we obtain that all quantum states with positive partial transposes and rank smaller than N are separable. Comparing to the case of 2 × 2 × N, the separability and entanglement of mixed states in 2 × 3 × N is more complicated. We improve and simplify the methods in the case of 2 × 2 × N, especially in proving Δ = 0. By making an elementary row transformation on a special matrix and using the properties of rank, we get Δ = 0. Then we derive the other unknown elements and get the canonical form. The proof of the main result is in two steps: When N ≥ 6, the results are obtained from two lemmas and the results of bipartite systems; when N < 6, we discuss the problem step by step for different values of N and ranks. The main idea is, according to the...