Research On Several Problems Of Quantum Entanglement

Quantum information theory,a combination of information theory and quantum mechanics,has greatly developed in the past thirty years.It has tremendous potential applications and provides us with special perspectives on both information theory and quantum physics.In the development of quantum information theory,quantum entanglement,a fundamental concept noticed by Einstein and Schrodinger in the early stage of quantum mechanics but overlooked in its development,has attracted more and more attention.It is considered as the most important resource in quantum information theory and also contributes to many other physics such as condensed matter physics,thus it is now an important concept in physics.Though useful, entanglement proves to be difficult to understand.After many years development, now we still lack a complete perspective on this concept.However,many interesting and important findings have been built till now.In this thesis,we investigate and discuss three aspects of quantum entanglement. Firstly,entanglement describes the nonclassical correlations in quantum system which brings out the violation of Bell type inequalities.These violations have been illustrated in lower dimensional photon systems,here we devise a 4-dimensional photon projective measurement scheme to realize high dimensional violation experiments.Secondly,it's important to separate entangled states from separable states.We make discussions on the partial transposition criterion of such separation. Last,it's most important to give proper entanglement measures,thus we discuss some measures for both bipartite and multipartite entanglement.Our results are listed as follows:Ⅰ.We make an analysis on the partial transposition of bipartite system's density matrix.Partial transposition is an important tool for the discrimination of entangled states and separable states,for 2×2 and 2×3 systems,it provides a necessary and sufficient condition for this discrimination.However,for higher dimensional systems,it cannot give a complete discrimination.Here, through Monto Carlo methods,we found a general result on the number of negative eigenvalues of the partial transposition of N×N system.We also discussed a problem posed by K.Audenaert et.al with Schur theorem in matrix analysis.Although our discussion cannot give a complete solution on the problem,it provides a new perspective.Ⅱ.The test of violations of Bell inequalities provides a experimental way of obtaining the entanglement information in a composite system.As Bell inequalities refer to the correlations among local measurements,the realization of these measurements is necessary.Here,we devise a 4-dimensional projective scheme for photon's time-polarization modes.With this scheme,we can realize the 4×4-dimensional Bell inequality violation test.Also,we propose a quantum key distribution channel.Ⅲ.The quantification of entanglement in an arbitrary state lies in the heart of quantum information theory.For bipartite system pure states,entanglement of entropy provides a calculable and well-motivated entanglement measure. For 2×2 system,this quantity can even be calculated for mixed states, which is a function of Wootters' concurrence.Here,we make an analysis on a recent entanglement monotone on 2×2 system.Via Mintert's results,we find that it is smaller than the square of concurrence,which can also be generalzied to higher-dimensional systems.Ⅳ.The character and quantification of multipartite entanglement is much harder than bipartite entanglement.Usually multipartite systems can be entangled in many different ways,thus we need more than one parameters to describe multipartite entanglement.Here,we analyzed the permutation variance of SLOCC(stochastic local operations and classical communication)invariants of four-qubit system,and suggest an permutation-invariant entanglement monotone for genuine 4-qubit entanglement.With a comparison to known results,we show that this monotone does give measure on the genuine 4-qubit entanglement in cluster-class states.