Detection And Quantification Of Quantum Entanglement

After a fundamental nonclassical aspect of entanglement was recognized by Ein-stein, Podolsky, and Rosen in 1935, Werner proposed the accurate definitions of sep-arability and entanglement in 1989. In recent years, with the rapid progress in quan-tum information science, it has become more and more clear that entanglement is a valuable quantum resource and acts as an important role in many other physical phe-nomenon such as quantum phase transition and efficient simulation of many body systems. Therefore, the detection and quantification of entanglement became fun-damental problems in quantum information theory. On the one hand, several suffi-cient conditions for detection of entanglement have been found. On the other hand, a considerable amount of effort on quantification of entanglement has also been made. However, entanglement is not yet fully understood. Despite a great deal of effort in the past years, it is a challenging task and remains an open question to decide whether a given state is entangled or not and if it is entangled how to quantify it.In this dissertation, we mainly discuss the detection and quantification of entan-glement in discrete variable systems and the entanglement detection in continuous variable systems. The thesis focuses on the following three parts.I. As mentioned above, many sufficient conditions for detection of entanglement have been found, such as the Peres-Horodecki positive partial transpose (PPT) crite-rion, the computable cross norm or realignment (CCNR) criterion, local uncertainty relations (LURs), covariance matrix criterion, the criterion based on Bloch represen-tations. PPT criterion is a necessary and sufficient condition for 2 x 2 and 2×3 systems, but only necessary for higher dimensional cases. We propose several nec-essary conditions for separable states in discrete variable systems. It is believed that these conditions complements PPT criterion since it can detect many bound entangled states which PPT criterion cannot detect. In Section 2.2, we optimize the nonlinear entanglement witnesses (NEWs) based on local orthogonal observables, which are in-troduced by O. Giihne et al.. The optimized NEWs are always strictly stronger than the original NEWs. Section 2.3 introduces a powerful and computationally simple separability criterion based on the extensions of the realignment map. The criterion is strictly stronger than the CCNR criterion and the criterion based on Bloch represen-tations. Furthermore, we also propose a tighter LUR criterion in Section 2.4. Using arbitrarily chosen operators{Ak} and{Bk} of subsystems A and B, the tighter LUR criterion can detect more entangled states than the original LUR criterion.II. A considerable amount of effort on quantification of entanglement in discrete variable systems has been made. For instance, Wootters has analytically derived a perfect measure of two qubits, which is so-called concurrence. Furthermore, gener-alized concurrence in bipartite higher dimensional cases, such as I-concurrence, has been pointed out as well. Unfortunately, theⅠ-concurrence of mixed states is given as a convex roof for all possible ensemble realization. Therefore, it is generally difficult to be calculated. Recently, lower and upper bounds on I-concurrence have attracted much interest, which are relatively easier thanⅠ-concurrence itself to get. In Section 3.2, we obtaine a lower bound on theⅠ-concurrence in bipartite systems. We also obtaine observable upper bounds for the squared concurrence in Section 3.3, which are the dual inequalities of the observable lower bounds introduced by Mintert and Buchleitner. In Section 3.4, we define a separability measure for separable states.III. There are a lot of conditions for detection of entanglement in continuous variable systems, and many of them are corollaries of (or equivalent to) PPT criterion. Therefore, the corollaries and equivalences of PPT criterion cannot be used to detect bound entangled states. In Section 4, We propose two sufficient conditions for detec-tion of entanglement in continuous variable systems. All of them are strictly stronger than the condition proposed by Duan et al. Thus, they are also necessary and suf-ficient condition for separability of two-mode Gaussian states. Furthermore, one of our sufficient conditions can be used to detect bound entangled Gaussian states, which cannot be detected by the positive partial transpose criterion.