Plenary Lecture

BASES IN QUANTUM CHEMISTRY,
QUANTUM COMPUTATION AND QUANTUM INFORMATION

Prof. Maurice R. Kibler
Université de Lyon,
Institut de Physique Nucléaire,
université Lyon 1 and CNRS / IN2P3,
43 Bd du 11 Novembre 1918,
F-69622 Villeurbanne Cedex, France
m.kibler@ipnl.in2p3.fr

Abstract: The interest of symmetry adapted bases (atomic orbitals, molecular orbitals, spin waves, etc.) is well-known in quantum chemistry. In particular, the spherical harmonics (e.g., in atomic spectroscopy) and cubical, tetragonal or trigonal harmonics (e.g., in crystal-field theory and ligand field theory) is quite familiar to the practitioner in theoretical chemistry and chemical physics. The symmetry adapted functions generally span bases for finite-dimensional Hilbert spaces associated with reducible or irreducible representations of a symmetry group. A Hilbert space of finite dimension d can describe a system of qudits (qubits correspond to d=2, qudits to d arbitrary) used in quantum computation and quantum information. It is the object of this paper to construct bases which play an important role in quantum measurements and quantum information theory (quantum state tomography and quantum cryptography). Such bases, as for example mutually unbiased bases (MUBs) and positive operator valued measures (POVMs) bases, can be generated from methods similar to the ones used for building symmetry adapted bases. We develop here an approach that gives a complete solution for the construction of MUBs in the case where the dimension d of the considered Hilbert space is a prime number. We also give the starting point for studying the case where d is the power of a prime number.

Brief Biography of the Speaker:

Professor Maurice R. Kibler is a theoretician interested in applications of quantum mechanics and group theory to theoretical physics, mathematical physics and quantum chemistry. Symmetries and supersymmetries, based on the use of groups (finite groups, Lie groups, graded Lie groups, and quantum groups) as well as algebras (Lie algebras, graded Lie algebras, Hopf algebras, and quantum algebras), play a central role in his researches achieved alone or in the framework of numerous collaborations. These researches can be classified into several categories. (1) In crystal- and ligand-field theory, he established a connection between the point charge electrostatic model and the angular overlap model. Furthermore, he developed a model, through the use of the Wigner-Racah algebra of a chain of groups starting with the group SU(2), for describing the energy levels and the intensities of multiphoton electronic transitions for a partly-filled shell ion (transition metal or lanthanide or actinide ion) in a surrounding with a given symmetry. (2) He contributed to the development of invariance and noninvariance dynamical groups and quantum groups for dynamical systems in: (i) atomic spectroscopy (in connection with the SO(4,2) symmetry of the hydrogen atom, the Hartmann system of quantum chemistry and related systems, a deformed Aufbau Prinzip, and the periodic table of chemical elements); (ii) in molecular and nuclear physics (for the study of vibrational and rotational bands of molecules and deformed and superdeformed nuclei); and (iii) in particle physics (for the study of quantum-deformed dual amplitudes). (3) He also introduced the concept of k-fermions, which are objects interpolating between fermions and bosons, and contributed, via the introduction of a generalized Weyl-Heisenberg algebra, to the development of fractional supersymmetric quantum mechanics. (4) In mathematical physics, he made important contributions to the study of the Wigner-Racah algebra of a chain of finite or compact groups, he developed the notion of nonbijective canonical transformations (in connection with Cayley-Dickson algebras and Hopf algebras), and he introduced the concept of Lie algebras under constraints. (5) Most of his present researches are devoted to fractional supersymmetric quantum mechanics (with the study of shape invariant potentials) and to quantum information theory (with the study of bases useful in quantum information and quantum computation). In particular, he is involved in the understanding of the so-called mutually unbiased bases (used in quantum cryptography and quantum tomography) and positive operator valued measures (used in quantum measurements).

Prof. Kibler is a member of the Theory Group of the Institute of Nuclear Physics of Lyon (a component of the IN2P3/CNRS, France). He teaches mathematics for physicists and engineers at the University Lyon 1 (a component of the University of Lyon) and at the ARCNAM Rhone-Alpes (a component of the CNAM, France). He is the author or co-author of 160 scientific publications in journals, books and proceedings, of several articles in encyclopedias, and of one textbook. He is regularly involved in international conferences and workshops.