Products of Tensorial Sets of Operators

U. Fano, A. R. P. Rau

José F. Cariñena, Jesús Clemente-Gallardo, Jorge Alberto Jover-Galtier +1 more

Abstract A geometric description of the space of states of a finite-dimensional quantum system and of the Markovian evolution associated with the Kossakowski–Lindblad operator is presented. This geometric setting is based on two composition laws on the space of observables defined by a pair of contravariant tensor fields. The first one is a Poisson tensor field that encodes the commutator product and allows us to develop a Hamiltonian mechanics. The other tensor field is symmetric, encodes the Jordan product and provides the variances and covariances of measures associated with the observables. This tensorial formulation of quantum systems is able to describe, in a natural way, the Markovian dynamical evolution as a vector field on the space of states. Therefore, it is possible to consider dynamical effects on non-linear physical quantities, such as entropies, purity and concurrence. In particular, in this work the tensorial formulation is used to consider the dynamical evolution of the symmetric and skew-symmetric tensors and to read off the corresponding limits as giving rise to a contraction of the initial Jordan and Lie products.

Jack A. Tuszyński, J.M. Dixon

Abstract The reduced matrix elements of products of tensor operators, coupled to form a non‐zero rank, are investigated to emphasize the importance of correctly incorporating all tensorial couplings. It has recently been realized that serious inaccuracies can arise, in some cases, unless the complete many body nature of the full operator and the states involved is properly incorporated. Exact results are presented for one‐ and two‐body operators which include all couplings and compare and contrast them with some approximate methods.

Sylvia Pulmannová

Edgard Elbaz