We have proposed several novel ideas and proposals for quantum information processing and experimentally demonstrated one important element of quantum computation dur-ing the time of my Ph. D. studies. Our research mainly concentrates on entanglement detection, on entanglement generation, on entanglement purification, on quantum error corrections, on quantum search algorithm, and on the experimental creation of four-qubit hyperentangled states and realization of one-way quantum computation. We investigate into several key subjects involved in almost the whole process of quantum information pro-cessing. We start with a study into the properties of correlations inherent in multipartite
entangled states and then provide a new insight into entanglement detections including Bell inequalities, entanglement witness operators, and the connections between them. We improve the purification protocols of entanglement and then design a new efficient one.
Furthermore, we give an analytic and systematic way to construct quantum circuits for both entanglement purification and quantum error corrections. For entanglement genera-tion, we propose a scheme for generating a many-qubit entangled state with translational symmetry. We also analyze the quantum search algorithm in detail and experimentally perform a quantum search by one-way realization successfully. A summary is given as follows.
Chapter 2 Quantum correlations imbedded in many-qubit and two-qudit entangled states are described by novel criteria of correlation for dependent systems. Correlation structures of Bell inequalities and entanglement witness operators are in terms of correlation criteria proposed. Several robust and efficient Bell inequalities and en-tanglement witnesses are also introduced.
Chapter 3 We apply the correlation criteria to the stabilizer formalism and discuss the entanglement of stabilizer states in a new point of view. Entanglement witnesses for stabilizer-entangled states that required only two local measurement settings when used in experiments are given.
Chapter 4 Entanglement witnesses for detecting several different kinds of many-qubit entangled states that are useful for quantum information processing are proposed.
Chapter 5 General correlation criteria for many-qudit entanglement are introduced.
We reveal the essential elements of the GHZ paradoxes and the generic Bell inequal-ities for many qudits are comprised of the criteria introduced. Several witnesses for multipartite entangled qudits are proposed.
Chapter 6 Standard entanglement purification protocols based on hybrid maps are pro-posed to purify any distillable state to a desired maximally entangled pure state.
Chapter 7 An analytical method to simplify the encoder-decoder circuit for a perfect five-qubit quantum error correcting code that is converted from its equivalent one-way entanglement purification protocol is introduced.
Chapter 8 We study how dot-like single quantum well excitons, which are coupled to single-mode cavity photon, evolve into maximally entangled state as a series of conditional measurements are taken on the cavity field state.
Chapter 9 Detailed analyses of the constructions of quantum search algorithm are pre-sented in this chapter. We focus on the accuracy and noise tolerance of the quantum algorithm.
Chapter 10 We experimentally develop a two-photon cluster state source entangled both in polarization and spatial modes. We also utilize the created hyperentangled qubit source to give a experimental demonstration of one-way quantum computation. A quantum search task is performed in an one-way realization.
Chapter 11 We summarize the main results in the thesis and give an outlook.
Entanglement and correlation conditions
2.1 Introduction
Bell inequalities are results about local realism, and then violations of which by entan-gled states can be considered as a means to feature the distinct properties of quantum correlations. In this situation, three main questions arise: (i) Is there a necessary condi-tion of quantum correlacondi-tion associated with some entangled state in the kernels of Bell inequalities? While Bell inequalities are based on the local realistic theories, we wonder whether their kernels can provide conditions of correlation for entangled states. (ii) What is the connection between the correlation structures of Bell inequalities for qubits and the ones for qudits? Can it be utilized to analyze the correlation properties of both entangled qudits and many-qubit entanglement? (iii) What is the connection between the correla-tion structures of Bell inequalities and entanglement witnesses? Can the kernels of Bell inequalities be used to construct entanglement witnesses for qudits?
The goal of this chapter is threefold. First, we introduce necessary conditions of correlation for many-qubit and two-qudit entanglement. Second, we reveal that the Bell inequalities for many qubits introduced by Clauser-Horne-Shimony-Holt (CHSH) [13],
Mermin [14], and Seevinck-Svetlichny [15], and the Collins-Gisin-Linden-Massar-Popescu (CGLMP) [17] and Son-Lee-Kim (SLK) [18] inequalities for bipartite arbitrarily high-dimensional systems are composed of the correlation conditions proposed. The general correlation functions of the CHSH inequality proposed by Fu [109] are also shown to consist of conditions of correlation. Bell inequalities based on correlation criteria for qudits are introduced. In addition, we show that the Durkin-Simon inequalities [110] for many-qubit entanglement can be rephrased in terms of correlation criteria. Third, we use the criteria to construct the first entanglement witness operator for detecting a two-qudit Bell state. In particular, this witness needs only two local measurement settings (see below) when used in experiments and is very robust against noise, independent of the number of levels. Further, two novel and robust witnesses for qudits are proposed.
The conditions of correlations for Bell inequalities are also utilized to construct witness operators for qudits. In short, the condition presented is common among Bell inequalities for qudits and many qubits. The constructions introduced show connections between Bell inequalities and entanglement witnesses.
This chapter is organized as follows. We start in Sec. 2.2 by revisiting the scenario of a many-party Bell-type experiment for identifying the correlations between outcomes of measurements. Then we present the basic idea of the condition of correlation and intro-duce the dependence criterion for qubit and two-qudit correlations. Since many-qubit GHZ and two-qudit Bell states are very useful for quantum information processing and under intensive study in entanglement physics, in Sec. 2.3 we proposed different kinds of correlation conditions to analyze their correlation characters. In Sec. 2.4, we show the criteria of correlations introduced in Sec. 2.3 are the kernels of the Bell inequalities that have been presented. We also introduce Bell inequalities based on the conditions of quan-tum correlations for qudits. In Sec. 2.5 we give a novel entanglement witness operator for detecting states close to a two-qudit Bell state. We also consider entanglement detections of two given multilevel entangled states. In addition, we give witness operators for N-qubit GHZ states and analyze the structure of the inequalities beased on the geometry of
spin vectors by the conditions proposed. Then a conclusion follows. Finally, in Appendix A we give a proof to show the tightness of the Bell inequalities for qudits proposed in Sec.
2.4.