Research On Dynamical Symmetries And Non-classical Correlations In Quantum Mechanics

The thesis intends to concern about the dynamical symmetries in a quantum sys-tem, and the relation between the subsystems in a composite quantum system. Namely, a series of systems in relativistic quantum mechanics with dynamical symmetries are studied; and the nonclassical correlations in a composite system, which play the most important role in quantum information, are discussed in several aspects. Some related concepts are also investigated, such as the Schwinger representation which is connected with the dynamical symmetry in isotropy harmonics oscillator, and the fidelity in quan-tum information which is found to be related to the bounds of entanglement in our results. The details are as follow.The Dirac Hamiltonian with equal scalar and vector potentials is said to have the spin symmetry. Its total angular momentum can be divided into two conserved parts. The orbital part has been shown to be three of the generators of the SU(3) dynamical symmetry group when the potential takes the harmonic oscillator form in [Ginocchio, Phys. Rev. Lett.95,252501 (2005)]. This type Dirac Hamiltonian with the Coulomb potential is called as the Dirac hydrogen atom with spin symmetry. It is shown to have a SO(4) dynamical symmetry. The germinators of the SO(4) group are the conserved orbital angular momentum and the Runge-Lenz type vector derived in our work. The corresponding Casimir operator leads to the energy spectrum naturally. This type hy-drogen atom is connected to a four-dimensional Dirac system with equal scalar and vector harmonic oscillator potential, by the Kustaanheimo-Stiefel transformation with a constraint.In the two-dimensional Dirac Hamiltonian with equal scalar and vector potentials, the deformed orbital angular momentum L are constructed, which commutes with the Hamiltonian and corresponding to the three-dimensional result. When the potential takes the Coulomb form, the system has an SO(3) symmetry, and similarly the har-monic oscillator potential possesses an SU(2) symmetry. The generators of the sym-metric groups are derived for these two systems separately. The corresponding energy spectra are yielded naturally from the Casimir operators. Their non-relativistic limits can return to the nonrelativistic two-dimensional hydrogen atom and harmonic oscilla-tor respectively.Noticing that the above mentioned Dirac systems are equivalent to scalar particles in the same potentials, we studied the dynamical symmetries of the two-dimensional Klein-Gordon equations with equal scalar and vector potentials. The dynamical sym-metries are considered in the plane and the sphere respectively. The generators of the SO(3) group corresponding to the Coulomb potential, and the SU(2) group correspond-ing to the harmonic oscillator potential are derived. Moreover, the generators in the sphere construct the Higgs algebra. The algebraic solutions of these Klein-Gordon sys-tems are also obtained.We give a introduction of the Higgs algebra method to solve the nonrelativistic harmonic oscillator and the two-dimensional Smorodinsky-Winternitz system. And, we construct the Higgs algebra generators in the 2D Dirac equation with equal scalar and vector Smorodinsky-Wintemitz potentials, which is our first attempt to investigate the dynamical symmetry in a noncentral Dirac system. The scheme to derive the generators of the Higgs algebra in the Dirac system, are directly extended from the one of the radially symmetric system.As a related concept of the dynamical symmetry in the harmonic oscillator, the non-standard Schwinger fermionic representation of the unitary group is studied by using n-fermion operators. One finds that the Schwinger fermionic representation of the U(n) group is not unique when n≥3. In general, based on n-fermion operators, the non-standard Schwinger fermionic representation of the U(n) group can be established in a uniform approach, where all the generators commute with the total number operators. The Schwinger fermionic representation of U(Cnm) group is also discussed.A geometric interpretation for the A-fidelity between two states of a qubit system is presented, which leads to an upper bound of the Bures fidelity. The metrics defined based on the A-fidelity are studied by numerical method. An alternative generalization of the A-fidelity, which has the same geometric picture, to a N-state quantum system is also discussed.The bounds of entanglement in [F. Mintert and A. Buchleitner, Phys. Rev. Lett. 98 (2007) 140505] and [C. Zhang et. al., Phys. Rev. A 78 (2008) 042308] are proved by using two properties of the Bures fidelity. In two-qubit systems, for a given value of concurrence, the states achieving the maximal upper bound, the minimal lower bound or the maximal difference upper-lower bound are determined analytically.As an attempt to understand the entanglement in high-dimensional systems, we present an alternative decomposition of two-qutrit pure states in a form|Ψ>=P1/(?)=(|00>+ |11>+|22>)+P2/(?)(|01>+|12>)+p3eiθ|02>, based on maximally entangled states in the full-and sub-spaces of two qutrits. Similar to the Schmidt decomposition, all two-qutrit pure states can be transformed into the alternative decomposition under local unitary transformations, and the parameter P1 is shown to be an entanglement invariant.In the investigation of the entangled multi-partite systems. We improve the conti-nuity approach given by Zhou [D. L. Zhou, Phys. Rev. Lett.101,180505 (2008)], and obtain the irreducible multi-party correlations in two families of n-qutrit Greenberger-Horne-Zeilinger type states. The pure states in one of the families are shown to have irreducible m-party (2< m< n) correlation in addition to 2-party and n-party correla-tions. We also derive the irreducible multi-party correlations in the n-qutrit maximal slice states, which can be uniquely determined by its (n-1)-qutrit reduced density ma-trices among pure states. This enlightens us to give a discussion about how to character-ize the pure states with irreducible n-party correlation in arbitrarily-high-dimensional systems.The nonclassical relations between the subsystems of a entangled system also are considered in the viewpoint of locality and nonlocality. The quantum probability dis-tribution arising from single-copy von Neumann measurements on an arbitrary two-qubit state is decomposed into the local and nonlocal parts, in the approach of Elitzur, Popescu and Rohrlich [A. Elitzur, S. Popescu, and D. Rohrlich, Phys. Lett. A 162,25 (1992)]. A lower bound of the local weight is proved being connected with the degree of entanglement, concurrence, of the state, such pLmax=1-e(ρ). The local probabil-ity distributions for two families of mixed states are constructed independently, which accord with the lower bound.