Local Distinguishability Of Orthogonal Quantum States

Quantum Information Science is a new and developing interdiscipline, which includes Quantum Mechanics, Mathematics, Informatics, Computer Science and so on. Its flourishing development promotes other relative disciplines in recent years. In Quantum Information, there is a series of basic issues completely different from classical informatics, one of which is about local distinguishability of a set of orthogonal quantum states.As we know, under micro quantum conditions, the solution space of Schrodinger equation belongs to Hilbert space characterized by the Superposition Principle. The Principle makes it possible that quantum states describing different quantum systems are not orthogonal, which accordingly brings a series of problems violating our established cognition such as local indistinguishability and quantum non-cloning. However, under macro classical conditions, as extremely noisy environment around macroscopic systems, systems are continually measured by the surrounding and can't keep themselves in coherent superposition states. Thus, the states describing these systems are naturally orthogonal and perfectly distinguishable. In fact, we can't determine in which quantum state a multiparticle quantum system is when its state belongs to a set of orthogonal states but only local operations and classical communication (LOCC) are allowed. This is local distinguishability of orthogonal quantum states, which is a basic problem in quantum information and quantum mechanics. By intensive study of this problem, we can get a better understanding about quantum channel capacity, quantum information extraction, quantum nonlocality and so on.In this graduation thesis, we study this problem in terms of Schmidt ranks and get some useful conclusions as follows:I. We associate this problem with the generator of group SU(N), from which we get a necessary condition of local distinguishability. According to this condition, we can systematically analyze how to locally distinguish a set of multipartite orthogonal states.II. Instead of LOCC, we consider local projective measurement and classical communication (LPCC). When allowed operations are confined with LPCC, we get the necessary condition that if a set multipartite orthogonal quantum states can be distinguished by LPCC, then the sum of Schmidt ranks of these states should be less than the total dimension of the Hilbert space in which the states belong to. But, this necessary condition can't be generalized to LOCC condition.III. Local indistinguishability of a subspace means all orthogonal bases in this subspace are locally indistinguishable. Watrous et al. first proved that in a Hilbert space n (?) n (n≥3), the orthogonal complement of the space spanned by the maximally entangled state is locally indistinguishable. In this thesis, we take advantage of a new method and obtain more general conclusions on this topic. We proved that in a m (?) n(m, n≥3) Hilbert space, the orthogonal complement of the space spanned by a Schmidt rank-3 maximally entangled state is locally indistinguishable. We further demonstrate that, when both parties' dimensions are greater than three, the orthogonal complement of the space spanned by any entangled state whose Schmidt rank is greater than three has no locally distinguishable orthonormal bases. Therefore, Watrous's subspace is just a sub-class in our results.IV. In II, we confine allowed operation with LPCC and get a necessary condition of local distinguishability in terms of Schmidt ranks. Now, we confined the number of orthogonal states and allowed operation is LOCC. When a quantum system belongs to one of a set of 2n-l orthogonal states in a Hilbert space 2(?) n, we get a sufficient and necessary condition to determine local distinguishability, that is, the sum of Schmidt ranks of these states should be less than 2n, which means at most one of the 2n-l states is entangled.