The Mathematical Aspects of Quantum Maps

Dariusz Chruściński

Some mathematical aspects of quantum evolution described by local in time master equations are discussed. We analyze the basic characterization of quantum evolution in terms of the corresponding dynamical map. Simple criteria which enable us to discriminate between Markovian and non-Markovian evolution are provided.

Arul Murugan Lakshminarayanan

Sudhavathani Simon, S. P. Rajagopalan, R. Simon

Demosthenes Ellinas, Christos Konstandakis

A toy model relaxation of the unitarity of a quantum search algorithm is introduced and the resulting parametric quantum search is investigated mathematically via a one-parameter family of completely positive trace preserving maps by introducing the oracle algebra (an SU(2) isomorphic matrix algebra). The parametric search is shown to be conditionally as fast as the original algorithm. Geometrically, it is related to spherical geodesics which exhibit an azimuthal symmetry, while its search map and density matrix enjoy geometric and structural symmetries.

Paul A. Benioff

Some aspects of the physical nature of language are discussed. In particular, physical models of language must exist that are efficiently implementable. The existence requirement is essential because without physical models no communication or thinking would be possible. Efficient implementability for creating and reading language expressions is discussed and illustrated with a quantum mechanical model. The reason for interest in language is that language expressions can have meaning, either as an informal language or as a formal language associated with a mathematical or physical theory. It is noted that any universally applicable physical theory, or coherent theory of physics and mathematics together, includes in its domain physical models of expressons for both the informal language used to discuss the theory and expressions of the theory itself. It follows that there must be formulas in the formal theory that express some of their own physical properties. Inclusion of intelligent system in the theory domain means that the theory, e.g., quantum mechanics, must describe in some sense its own validation. Maps of language expressions into physical states are discussed as are conditions under which such a map is a Godel map. The possibility that language is also mathematical is very briefly discussed.