Analiza splątania układów kwantowych dla ścisłych rozwiązań podstawienia Bethego
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Date
2013-01-09
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Entanglement analysis of quantum subsystems for the exact Bethe Ansatz solutions
Abstract
Głównym obiektem badań jest jednowymiarowy izotropowy pierścień magnetyczny o (N={4,5,6,7}). W każdym z węzłów znajduje się spin ½ który może oddziaływać jedynie ze spinami z sąsiednich węzłów. Jak dobrze wiadomo rozpatrywany model jest ściśle rozwiązywalny a rozwiązania możemy otrzymać poprzez tzw. podstawienie Bethego. Formalne podobieństwo (w sensie struktury multiliniowej) do komputerów kwantowych powoduje iż magnetyk ten staje się niezwykle ciekawym obiektem z punktu widzenia kwantowego przetwarzania informacji. W niniejszej rozprawie podano dwa sposoby wyodrębnienia podukładów (węzły, struny). Analizując splątanie węzłów posłużono się dwiema znanymi z literatury miarami: współbieżność (concurrence) oraz negatywność (negativity). Otrzymane wyniki wskazują iż splątanie węzłów jest raczej sporadyczne. Analiza splątania strun wymagała rozwinięcia nowych narzędzi. Wyniki wskazują na istnienie znacznego splątania pomiędzy strunami.
In this work we have considered one-dimensional isotropic magnetic rings with (N={4,5,6,7}) nodes each with the spin ½ with isotropic exchange interactions occurring only between nearest neighbours. It is well known that the energy eigenproblem for such model can be solved exactly and the solutions are given by the Bethe Ansatz. Observe, that there is strong formal similarity (in sense of multilinear structure) between the Hilbert space of the Magnet and the space of quantum computer states. The main purpose of this thesis was to analyse the entanglement between differently chosen subsystems of system in each of its eigenstates. The most obvious choice was to study entanglement between two nodes. In this case we have tested two well-known measures: negativity and concurrence. Obtained results show that the presence of entanglement between nodes is rather rare. A more subtle way of choosing subsystems is based on the notion of strings. In this thesis, we have developed new tools allowing us to detect and measure quantum entatnglement between strings. We have observed, in each of cases under our investigation, a very strong entanglement between these objects.
In this work we have considered one-dimensional isotropic magnetic rings with (N={4,5,6,7}) nodes each with the spin ½ with isotropic exchange interactions occurring only between nearest neighbours. It is well known that the energy eigenproblem for such model can be solved exactly and the solutions are given by the Bethe Ansatz. Observe, that there is strong formal similarity (in sense of multilinear structure) between the Hilbert space of the Magnet and the space of quantum computer states. The main purpose of this thesis was to analyse the entanglement between differently chosen subsystems of system in each of its eigenstates. The most obvious choice was to study entanglement between two nodes. In this case we have tested two well-known measures: negativity and concurrence. Obtained results show that the presence of entanglement between nodes is rather rare. A more subtle way of choosing subsystems is based on the notion of strings. In this thesis, we have developed new tools allowing us to detect and measure quantum entatnglement between strings. We have observed, in each of cases under our investigation, a very strong entanglement between these objects.
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Keywords
Magnetyk heisenberga, Heisenberg magnet, Podstawienie Bethego, Bethe Ansatz, Splątanie, Entanglement