Research On Correlations Of Quantum States And PT-symmetry Quantum Theory

As an important resource in quantum information processing, quantum en-tanglement is an essential feature in quantum mechanics. Recently, with the development of quantum information theory, some people found that quantum correlations could play an important role on many quantum information processing tasks without entanglement, so it becomes a hot topic in quantum information. Moreover, Bender and the collaborator discovered in1998that not every Hamiltonian is Hermitian and proposed fT-symmetry theory, hereafter, this leads to attracted more and more concern. This dissertation is de-voted to research the characterization of quantum correlations, the structure of quantum channels preserving correlations, and to propose the PT-frames and CPT-frames as well as adiabatic approximation in PT-symmetric theory by use of the methods in operator theory and matrix theory.In Chapter1, we introduce some research background and status on our main con-tents, and list some notations, definitions and theorems.In Chapter2, we study the characterization and quantity of quantum correlations. Firstly, based on the matrix representation of an operator under an orthonormal basis, three sufficient and necessary conditions of a state being classical correlated are estab-lished. Secondly, a sufficient and necessary condition for a convex combination of two classical correlated states to be classical correlated is given. It is proved that the set of all classical correlated states is a perfect, nowhere dense and compact subset of the space of all states. Based on our new characterization of classical correlated states, a quantity Q(p) is associated to a state p. It is shown that a state p is quantum correlated if and only if Q(ρ)>0. Especially, we prove that the Werner state Wλ in a two-qubit system with a single real parameter A is quantum correlated if and only if λ≠0.25. Lastly, a classifica-tion of quantum correlations on tripartite mixed states and corresponding characterization are studied.In Chapter3, we discuss local quantum channels that preserve classical correla-tions. Based on a characterization of classical correlated states in Chapter2, the rela-tionships among classical correlation-preserving local quantum channels, commutativity-preserving local quantum channels and commutativity-preserving quantum channels on each subsystem are obtained. Furthermore, for a two-qubit system, the general form of a classical correlation-preserving local quantum channel is showed. Moreover, it is proved that sufficient and necessary conditions for existence of a quantum channel φ and agener-alized unitary operation ε sending Ai to Bi(1≤i≤k)for two given families{Ai}i=1k,{Bi}i=1k of matrices, respectively. As an application, a sufficient and necessary condition for ex-istence of a unitary duality quantum computer with given input-output states is obtained.In Chapter4, we investigate abstract PT-symmetric theory. PT-frames and CPT-frames on a Hilbert space are introduced, and it is proved that the spectrum and point spectrum of a PT-symmetric closely defined linear operator are all symmetric with respect to the real axis. For a linear operator H on Cd, H has unbroken PT-symmetry if and only if it has d different eigenvalues and the corresponding eigenstates are eigenstates of PT. Given a CPT-frame, a new positive definite inner product is induced and called CPT-inner product. The relationship between the CPT-adjoint and the Dirac adjoint of a closely defined linear operator is discussed, and a PT-symmetric operator having a CPT-frame is CPT-Hermitian if and only if it is symmetric, in that case, it is similar to a Hermitian operator. At last, existence of an operator C consisting of a CPT-frame is explored.In Chapter5, we discuss time evolution and adiabatic approximation in PT-symmetric quantum mechanics. By use of the CPT-frames on a Hilbert space and the induced pos-itive definite inner product in Chapter4, the case when the Hamiltonian is dependent of the time t is considered. For PT-symmetric Hamiltonian H(t) and its a C(t)PT-frame, the evolving equation described by H(t) is given. A quantitative sufficient condition and error estimations of the adiabatic approximation for the PT-symmetric Hamiltonian are derived.