Abstract (eng)
Entanglement is a fascinating curiosity of quantum physics that distinguishes it considerably from classical concepts. On the one hand it implicates surprising philosophical aspects such as the incompatibility of local realistic theories with quantum physics, on the other hand it can be successfully implied in quantum information and quantum communication tasks to improve protocols with respect to classical procedures.
It is still an open mathematical problem to determine whether a quantum state is entangled or not; there is no operational procedure for a general state on an arbitrary dimensional Hilbert space. For pure states and lower dimensional bipartite systems, e.g., for two qubits, the problem is solved, since computable necessary and sufficient conditions for separability (i.e. for being not entangled) were found. Moreover, if one seeks to quantify the entanglement of a quantum state, this can be conveniently done for a system of two qubits. For higher dimensional and/or multipartite systems much has been accomplished in the context of entanglement detection and quantification, but the problem cannot be seen as solved at all, and much has still to be investigated. The aim of this thesis is to present new methods to detect and quantify entanglement for systems beyond two qubits. States of these systems are, as usual, described by density operators that are usually put into matrix notation. For high dimensional and/or multipartite systems, these density matrices can become, however, quite unhandy. A mathematical tool to express density operators in a compact and simpler way is provided by the Bloch vector decomposition. In this notation we decompose the density operator into a complete and orthogonal basis of operators of the operator space. For qubits this notation is well known and usually one uses the Pauli spin-1/2 operators as the operator basis. For qudits, i.e. states of arbitrary dimensional systems, however, there is no unique generalization of the Pauli operator basis. We therefore present three possible choices of operator bases: the generalized Gell-Mann operator basis, the polarization operator basis, and the Weyl operator basis. We furthermore provide a method to find the decomposition of any operator (not necessarily density operators) into one of the three operator bases via the decomposition of so-called standard operators. As an application example of the method we consider the maximally entangled two-qudit state and decompose it according to each basis.
In the context of entanglement detection, entanglement witnesses are an important and frequently used tool. They are observables and detect the entanglement of a state if they reveal a negative expectation value for it, and if at the same time it is known that the expectation value has to be non-negative for all separable states. This latter condition is, however, in general not straightforward to show. A simplification in this connection is the usage of Bloch decompositions, which offer new relations and help to prove the condition in many cases.
A particular class of entanglement witnesses are geometric entanglement witnesses. These allow a direct Euclidean geometrical representation of operator space. We show the importance of these witnesses by giving shift methods that can detect entangled and in particular bound entangled states, and help to determine the set of separable states for convex subsets of states. Examples for a three-parameter family of two-qutrit states are given.
Geometric entanglement witnesses can also be applied for geometrical quantification of entanglement, in particular for determining the Hilbert-Schmidt measure of entanglement. We give examples for the isotropic two-qudit state and two-parameter families of two-qubit and two-qutrit states.
Finally, we address the problem of entanglement detection in multi-qubit systems. In this context we provide a general construction of entanglement witnesses using static structure factors. In solid state physics, structure factors describe dynamical properties of solids in scattering experiments. The structural witnesses can detect many genuinely multipartite entangled states, such as Dicke states and Dicke-like states with changed phases of the constituting terms. Moreover, they contain two-point correlations only and are apt for experimental application for various physical systems.