Beyond the simplest case of bipartite qubits, the composite Hilbert space of multipartite systems is largely unexplored. In order to explore such systems, it is important to derive analytic expressions for parameters which characterize the system's state space. Two such parameters are the degree of genuine multipartite entanglement and the degree of mixedness of the system's state. We explore these two parameters for an N-qubit system whose density matrix has an X form. We derive the class of states that has the maximum amount of genuine multipartite entanglement for a given amount of mixedness. We compare our results with the existing results for the N = 2 case. The critical amount of mixedness above which no N-qubit X-state possesses genuine multipartite entanglement is derived. It is found that as N increases, states with higher mixedness can still be entangled.
In this thesis, the complex Clifford algebra and its representation are introduced as a foundation of our geometrical description of quantum states. An inner product and a metric are defined on the matrix representation in a natural way, so that it has the structure of a Hilbert space and a Riemannian manifold. In particular, the set of D x D pure density operators which de-scribe n-qubit pure states, where D = 2\\(^n\\) and n is a natural number, is characterised as the set of D x D positive operators whose associated quadratic forms satisfy the Fierz identities and the normalisation condition. Then this set is, in fact, nothing but the complex projective space ‚Äövëvá‚Äövëv¥\\(^{D-1}\\). Further-more, the fibre bundle S\\(^{2D-1}\\) ‚Äövúv¿ ‚Äövëvá‚Äövëv¥\\(^{D-1}\\) is derived from the construction of the space of D x D pure density operators, and its bundle projection provides the natural correspondence between the two formulations of quantum mechanics, namely, the state vector formulation and the density operator formulation. The single qubit state case and the two-qubit pure state case are ex-plored intensively as examples. For the single qubit case, both pure and mixed states are discussed explicitly in terms of the Clifford algebra description along with the `mixedness' of a single qubit state, and the Riemannian structure of the space of single qubit pure density operators is examined. For the two-qubit case, the Clifford algebra description is discussed explicitly in relation to the Fierz identities, and, by employing the reduced density operators, it is shown that the concurrence of a two-qubit pure density operator coincides with the 'mixedness' of the reduced density operators. In addition, from this viewpoint, the \EPR paradox\" is examined as a more geometrically precise illustrated example."
We investigate the dynamical relations among entanglement, mixedness, and nonlocality, quantified by concurrence
C, purity
P, and maximum Bell function
B, respectively, in a system of two qubits in a common structured reservoir. To this aim we introduce the
C-
P-
B parameter space and analyze the time evolution of the point representative of the system state in such a space. The dynamical interplay among entanglement, mixedness, and nonlocality strongly depends on the initial state of the system. For a two-excitation Bell state the representative point draws a multibranch curve in the
C-
P-
B space and we show that a closed relation among these quantifiers does not hold. By extending the known relation between
C and
B for pure states, we give an expression among the three quantifiers for mixed states. In this equation we introduce a quantity, vanishing for pure states, which in general does not have a closed form in terms of
C,
P and
B. Finally, we demonstrate that for an initial one-excitation Bell state, a closed
C-
P-
B relation instead exists and the system evolves, remaining always a maximally entangled mixed state.