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Nasser Metwally, Aly Mohamed (2002): Entangled qubit pairs. Dissertation, LMU München: Faculty of Physics
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Abstract

The aim of this dissertation is to study the kinematics of 2-qubit pairs. A 2-qubit state is specified by the Bloch vectors for each qubit and a 3 × 3 dyadic, representing the expection values of the joint observables. In what follows the main results are listed. • The behaviour of a 2-qubit state under a local transformation is investi-gated, where we obtain a set of 9 quantities, constructed from the Bloch vectors and the cross dyadic, which are invariant under local transforma-tion. We establish a generic form for a 2-qubit state, which helps us decide if given two 2-qubit states belong to the same family or not. Using this form, which depends on the characteristic values of the cross dyadic, one can distinguish between all two-qubit families, where we obtain six classes of families of locally equivalent states. Those families classify all 2-qubit states, four of them consist of two subclasses each. • A simple criterion is obtained to check if a 2-qubit state is separable or not. This criterion does not depend on the eigenvalues of the partial trans-position of the state in question but only on three numbers, those are the coefficients of the eigenvalue equation. These numbers are written as a function of the Bloch vectors and the cross dyadic. Also using the charac-teristic values of the cross dyadic, one can decide if the CHSH inequality is obeyed or violated. • The properties of Lewenstein-Sanpera decompositions are employed to split a given state into its optimal separable and pure parts. The degree of separability is an important part in this decomposition, where a state is more useful for quantum information purposes, the smaller its degree of separability. The optimal degree of separability is obtained analytically for some important cases as a function of the Bloch vectors and the cross dyadic of the state in question: For the simple Werner state, where our result coincides with that obtained numerically. Also for the self-transposeed states, the generalized for Werner states and rank-2 states. • The technique of the rank-2 case is employed to study the case of rank-3 and the full rank, where we obtain all possible decompositions. Among them, we choose the one which has the largest splitting parameter. In this situation a numerical procedure could help in obtaining the optimal de-composition. Given a density matrix, one obtains its eigenvalues and corre-sponding eigenvectors. Using the rank-2 method to obtain all the possible decompositions, then choose the one corresponding to the largest degree of separability. If this decomposition is the optimal one, a technique is used to check the optimality. If it is not optimal, an optimization procedure is performed. • There is an important relation between the degree of separability S and what is the so called concurrence C which appears in the definition of the entanglement of formation. In general, the sum of the degree of separabil-ity and the concurrence is less than one, but equal to one for states with vanishing Bloch vectors. On the other hand both of them run from 0 to 1. • A criterion is introduced to decide if a given decomposition is the optimal one or not. Two inequalities are obeyed if the decomposition in question is the optimal one and are violated if it is not. • The effect of the unitary operators and BCNOT operations are described on the dynamical variables. An alternative presentation of the IBM and Oxford purification protocols is obtained by using theses variables. One could introduce the degree of separability as a purification parameter, where the purified state has a smaller degree of separability than the initial one. Employing the properties of the characteristic values of the cross dyadic, makes the Oxford protocol much faster.