Study On Purification Of Entanglement And Local Distinguishability Of Orthogonal Quantum States

This thesis will probe the property of non-locality from the following fields: inseparability criterion of quantum states, purification or distillation of entanglement, local distinguishability of orthogonal quantum states, and the relations among information entropy, distillable entanglement and local distinguishability of orthogonal states. The main contents of this thesis are shown below:1. A necessary and sufficient condition of the separability of a mixed state in bipartite systems is presented by decomposing a higher dimension Hilbert space into many 2 2 subspaces. This condition comes down to the problem of finding the solution of a set of quadratic equations. The solution of this set of quadratic equations is a set of orthonormal vectors, which makes the equations more easily solved. By same means we present a lower bound on entanglement of formation of states in 2 n systems. This lower bound can be calculated easily and has distinct physical meaning.2. We discuss the distillation of entanglement from finite copies of a mixed state. Two conceptions of quasiseparable state and distillable subspace are introduced. First a necessary condition of distillability from finite copies is given, then a necessary and sufficient condition of distillability from finite copies and the most efficient distillation protocol are given. These conclusions can be generalized to multi-partite mixed states. It is shown that the distillation of entanglement (from finite copies or infinite copies of a mixed state) is to project out a subspace, the projection of all copies of the mixed state in this subspace is a pure entangled state.3. We consider the orthogonality and the distinguishability of a set of arbitrary states in multi-partite systems. It is shown that if a set of orthogonal states are distinguishable by local operations and classical communication (LOCC), the product vectors in every orthogonal states should be orthogonal to the other orthogonal states. Employing this conclusion we also prove a specially simple criterion: if the sum ofthe Schmidt numbers of a set of bipartite states in a quantum system is bigger than the dimensions of Hilbert space of the system, the states are not LOCC distinguishable. We also present a sufficient and necessary condition of local distinguishability of a set of complete orthogonal prodect states. These results can improve the known conclusion in local distinguishability, and may be also useful in understanding the essence of non-locality and discussing the distillation of entanglement.4. The relations of information entropy, distillation of entanglement and distinguishability of orthogonal states are discussed. In terms of information an new interpretation for the distillation of entanglement and the distinguishability of orthogonal states is given. Using a special protocol we give a sufficient and necessary condition for the local distinguishability of states, and gain the maximal yield of the distillable entanglement. It is shown that information entropy, the local distinguishability of states and the distillable entanglement are closely and generally related. This means that non-locality and the uncertainty in a quantum stystem are related.