Master Degree / Yüksek Lisans Tezleri
Permanent URI for this collectionhttps://hdl.handle.net/11147/3008
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Master Thesis Quantum Walks: Entanglement Between Spatial Degrees of Freedom and Interference in Multi-Photon Walks(Izmir Institute of Technology, 2020) Çakır, Özgür; Çakır, Özgür; 04.05. Department of Pyhsics; 04. Faculty of Science; 01. Izmir Institute of TechnologyQuantum walks can be described as quantum analogues of classical random walks. In quantum walks, the direction of the walker is dictated by the quantum state of a coin in a coherent fashion. Unlike classical random walk with a fair coin, quantum walk has non-Markovian property. First, we studied 2-D quantum walk analytically and numerically with one-walker and two entangled coins to investigate the transfer of the entanglement in initial coins state to spatial degrees of freedom. The coins are Hadamard Coin, Fourier Coin, among which the Fourier coin generates entanglement, thus increase entanglement between spatial degrees of freedom. Here we calculated the amount of entanglement using negativity. In the second part we studied average photon number correlations for 1-D quantum walk with many body bosonic walkers, like different light sources, to investigate quantum interference effects and we showed the second-order intensity correlations function in terms of the probability amplitudes of the 1-D quantum walk with Hadamard coin. We compared the resulting correlations for various initial many photon states.Master Thesis Apollonius Representation and Complex Geometry of Entangled Qubit States(Izmir Institute of Technology, 2018) Parlakgörür, Tuğçe; Pashaev, Oktay; Pashaev, Oktay; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of TechnologyIn present thesis, a representation of one qubit state by points in complex plane is proposed, such that the computational basis corresponds to two fixed points at a finite distance in the plane. These points represent common symmetric states for the set of quantum states on Apollonius circles. It is shown that, the Shannon entropy of one qubit state depends on ratio of probabilities and is a constant along Apollonius circles. For two qubit state and for three qubit state in Apollonius representation, the concurrence for entanglement and the Cayley hyperdeterminant for tritanglement correspondingly, are constant along Apollonius circles. Similar results are obtained also for n- tangle hyperdeterminant with even number of qubit states. It turns out that, for arbitrary multiple qubit state in Apollonius representation, fidelity between symmetric qubit states is also constant along Apollonius circles. According to these, the Apollonius circles are interpreted as integral curves for entanglement characteristics. For generic two qubit state in Apollonius representation, we formulated the reflection principle relating concurrence of the state, with fidelity between symmetric states. The Möbius transformations, corresponding to universal quantum gates are derived and Apollonius representation for multi-qubit states is generated by circuits of quantum gates. The bipolar and the Cassini representations for qubit states are introduced, and their relations with qubit coherent states are established. We proposed the differential geometry for qubit states in Apollonius representation, defined by the metric on a surface in conformal coordinates, as square of the concurrence. The surfaces of the concurrence, as surfaces of revolution in Euclidean and Minkowski (Pseudo-Euclidean) spaces are constructed. It is shown that, curves on these surfaces with constant Gaussian curvature becomes Cassini curves. The hydrodynamic interpretation of integral curves for concurrence as a flow in the plane is given and the spin operators in multiqubit |PP...P states are discussed.