Geometry of mixed states and degeneracy structure of geometric phases for multi-level quantum systems. A unitary group approach

We analyze the geometric aspects of unitary evolution of general states for a multilevel quantum system by exploiting the structure of coadjoint orbits in the unitary group Lie algebra. Using the same method in the case of SU(3) we study the effect of degeneracies on geometric phases for three-level systems. This is shown to lead to a highly nontrivial generalization of the result for two-level systems in which degeneracy results in a "monopole" structure in parameter space. The rich structures that arise are related to the geometry of adjoint orbits in SU(3). The limiting case of a two-level degeneracy in a three-level system is shown to lead to the known monopole structure.Comment: Latex, 27 p

States of physical systems in classical and quantum mechanics

We discuss the descriptions of states of physical systems in classical and quantum mechanics. We show that while it is possible to evolve a terminology common to both, the differences in the underlying mathematical structures lead to significant points of departure between the two descriptions both at mathematical and conceptual levels. We analyse the state spaces associated with physical systems described by two and three dimensional complex Hilbert spaces in considerable detail to illustrate how subtle this question can in general be. We highlight the role the Bargmann invariants play in the passage from the Hilbert space to the ray space, the space of states in quantum mechanics, and also in the context of Wigner's theorem on symmetries in quantum mechanics where they originally appeared

CP^n, or, entanglement illustrated

We show that many topological and geometrical properties of complex projective space can be understood just by looking at a suitably constructed picture. The idea is to view CP^n as a set of flat tori parametrized by the positive octant of a round sphere. We pay particular attention to submanifolds of constant entanglement in CP^3 and give a few new results concerning them.Comment: 28 pages, 9 figure

Squeezing of a coupled state of two spinors

The notion of spin squeezing involves reduction in the uncertainty of a component of the spin vector below a certain limit. This aspect has been studied earlier for pure and mixed states of definite spin. In this paper, this study has been extended to coupled spin states which do not possess sharp spin value. A general squeezing criterion has been obtained by requiring that a direct product state for two spinors is not squeezed. The squeezing aspect of entangled states is studied in relation to their spin- spin correlations.Comment: Typeset in LaTeX 2e using the style iopart, packages iopams,times,amssymb,graphicx; 17 pages, 5 eps figure file

Symmetries and conservation laws in classical and quantum mechanics. 1. Classical mechanics

We describe the connection between continuous symmetries and conservation laws in classical mechanics. This is done at successively more sophisticated levels, bringing out important features at each level: the Newtonian, the Euler-Lagrange, and the Hamiltonian phase-space forms of mechanics. The role of the Action Principle is emphasised, and many examples are given