Quantum states, maps, measurements and entanglement

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Title: Quantum states, maps, measurements and entanglement
Author: Kuah, Aik-Meng, 1972-
Abstract: 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 .
URI: http : / /hdl .handle .net /2152 /3136
Date: 2008-08-28


Quantum states, maps, measurements and entanglement. Doctoral dissertation, The University of Texas at Austin. Available electronically from http : / /hdl .handle .net /2152 /3136 .

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