Kraus operator formalism using a reduced Hamiltonian approach on controlled-unitary operations

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Quantum systems are extremely fragile, and as such, are difficult to isolate from interactions with their environment. Such "open" quantum systems that interact with their environment do not follow the unitary dynamics of "closed" quantum systems. The dynamics of open quantum systems are described using a set of operators called Kraus operators. The goal of this thesis is to formulate a Kraus operator representation for an open quantum system for which unitary dynamics can no longer be used. The strategy used was to extend the single qubit "open" system to a "larger two-qubit closed system", which can evolve using unitary dynamics. The constructed two-qubit system was evolved using controlled unitary operations since such operations have only diagonal block matrix elements. As such, only subspaces can be considered using a technique called the reduced Hamiltonian technique, wherein the qubit of interest (called the target qubit) evolves in two independent subspaces of the control qubit. In each subspace, the target undergoes different unitary evolutions, and parameters of the system are then solved for so that a desired controlled unitary operation is implemented on the combined two-qubit system. The reduced Hamiltonian technique used here allows solving for the system parameters. Once the parameters are solved for, Kraus operators can be derived for the evolution of the target qubit by tracing out the control. Furthermore, since an arbitrary unitary matrix can be written as a product of controlled unitary matrices, the Kraus operator representation for arbitrary open system dynamics can be written. Typically, the Kraus operator formulation takes a separable uncoupled two-qubit system as its initial state. A significant finding of this thesis is that, even though when deriving system parameters, a separable uncoupled state was considered as the initial state, the method can be applied to writing the formulation for states that are not separable. This is significant.