The Separability Of Density Matrices Of Quantum Mixed States And ◇-products Of Positive Operators

We investigate separability of density matrices of quantum mixed states and the◇-products of positive operators. We devote to present some necessary conditions for separability of density matrices of quantum mixed states and study some algebraic properties of the◇-products-of positive operators. This paper is organized as follows.In Chapter 1, we introduce mathematical background of quantum mechanics and Dirac notations from linear algebra and operator theory that are needed in the sequel. The main contents including some definitions and basic properties about Hilbert space, out product, tense product, density operator, trace, the separability and entanglement of quantum state, quantum effect and sequential product.In Chapter 2, we study the separability of a density matrix of a quantum mixed state. Firstly, we introduce some well known necessary separability criteria. Then we discuss the properties ofλ-matrices by using the generators of the special unitary group SU(n) and obtain a representation of the density matrix of a bipartite quantum system. Finally, we give a necessary condition for separability of a density matrix of a mixed state in a bipartite quantum system.In Chapter 3, we study some algebraic properties of the◇-products of positive operators. Firstly, we introduce the◇-products by considering two physically relevant numerical quantities(namely, the concurrence and fidelity), which play an important role in quantum computation and quantum information. Then by com paring a◇-product with a sequential product, we give some properties of◇-products and some counterexamples. Finally, we obtain a characterization of transformations that preserve concurrence or fidelity between different Hilbert spaces.