The Local Unitary Equivalence And LOCC Distinguishability Of Quantum States

Quantum information theory is a cross field of Mathematics, Quantum mechanic-s, Classical information theory and Computer science. Quantum information can help greatly improve computing speed, ensure information security and increase the storage of information in Quantum communication and Quantum computation. Quantum entangle-ment is one of the substantial character of quantum information. The entorpy of quantum state remain unchanged under local unitary transformation. So the classification undei local unitary transformation is an important question. The LOCC distinguish of quan-tum states is also a basic question of quantum information which is conducive to the understanding of quantum non-locality, quantum information extraction and quantum channel capacity et al.The problems of the thesis researching are the local unitary equivalence of pure states in multiple quantum system, the one-way LOCC indistinguishability of orthogonal quantum states, the LOCC indistinguishability of product states. The main results are showed in the following:1. The local unitary equivalence of pure states. First, we transfer the local unitary equivalence question of pure states into the local unitary equivalence question of HOSVD states. Then we find the necessary condition of HOSVD states to be locally unitary equivalent:the matrices Mi,kψ,mkm and Mi,kψm are simultaneously unitary equivalence. Next, we define the canonical forms of quantum states, then transform the local unitary equivalence question of multipartite pure states into the unitary equivalence of canonical under direct group. Finally, we give an effective method for judging the local unitary equivalence of pure states.2. The one-way LOCC indistinguishability of orthogonal quantum states. We use the necessary and sufficient condition of maximally entangled states can be distinguished by one-way LOCC. Then we construct[d/2]+2 maximally entangled states in Cd×Cd(d≥ 4) which cannot be distinguished by one-way LOCC. Actually, we can optimize our before result get 3(?) —1 maximally entangled states which cannot be distinguished by LOCC. Finally, we find 4 maximally entangled states in Cd×Cd(d≥ 4) which cannot be distinguished by one-way LOCC.3. The LOCC indistinguishability of product states. First, we find 6d — 9 product states in Cd× Cd which cannot be distinguished by LOCC, where d is odd. The set is an subset of the states which are constructed in the paper [1]. Our result give an answer to the problem which is raised in this paper. Then we generalized our methods to arbitrary quantum system Cm×Cn, and find 3(m+n) — 9 product states which cannot be distinguished by LOCC. Finally, we give a set only contains 5 orthogonal product states in Cm×Cn(m≥3,n≥3) which are proved to be LOCC indistinguished.