Quantum states, maps, measurements and entanglement

The structure of the set of density matrices, its linear transformations, generalized linear measurements, and entanglement are studied. The set of density matrices is shown to be a convex and stratified set with simplex and group symmetries. Generalized measurements for density matrices are shown to be reducible to one unitary transformation and one von Neumann measurement carried out with an ancillary system of fixed size. Linear maps of density matrices are considered and the volume of the set of maps is derived. Positive but not completely positive maps are studied in consideration of obtaining a test for entanglement in density matrices. Using the Jamiolkowski representation and Schmidt decomposition of the map eigen matrices, several properties of these maps are shown. An algebraic approach to constructing these positive but not completely positive maps is partially formulated. The positivity of the linear map describing the evolution of an open system and its dependence on the initialized to a zero-discord state, the evolution is shown to be given by a completely positive map. In quantum process tomography, the results obtained from a open system that is initially prepared using von Neumann measurements is shown to be described by a bi-linear map, not a linear map. A method for quantum process tomography is derived for qubit bi-linear maps. The difference between preparing states for an experiment by measurement and by stochastic process is analyzed, and it is shown that the two different methods will give fundamentally different outcomes.