量子糾纏態與量子資訊處理

全文

(1)量子糾纏態與量子資訊處理 Entanglement and Quantum Information Processing. 研究生 : 李哲明. Student : Che-Ming Li. 指導教授 : 褚德三. Advisor : Der-San Chuu. 國立交通大學電子物理研究所博士論文 A Thesis Submitted to Institute of Electrophysics College of Science National Chiao Tung University in partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in. Electrophysics. February 2008. Hsinchu, Taiwan, Republic of China. 中華民國九十七年二月.

(2) Entanglement and Quantum Information Processing Che-Ming Li March 3, 2008.

(3) 量子糾纏態與量子資訊處理研究生 : 李哲明. 指導教授 : 褚德三. 國立交通大學電子物理研究所. 摘要本論文提出量子糾纏與量子資訊處理的理論與實驗研究成果。我們引進且利用系統性的方法分析量子多體糾纏態的關連結構並偵測隱含於多體多維物理系統中的量子關連，我們亦提出了嶄新的量子方案以實現量子糾纏態的製造、純化、量子糾錯及量子搜尋演算法；這些理論方法與結果有助於了解量子力學的基本特徵並開發量子訊息領域中的相關應用。在實驗工作方面，我們發展並開發了用於實現單向量子計算的二光子四量子位元糾纏源。利用高亮度糾纏源所產生的二光子量子態在偏振與空間自由度上的糾纏特性，我們實現了高效率的量子搜尋演算法，此實驗結果顯示二光子超糾纏態可作為快速且精確的光學量子計算之基礎。 . . i .

(4) ENTANGLEMENT AND QUANTUM INFORMATION PROCESSING Student: Che-Ming Li. Advisor: Der-San Chuu Department of Electrophysics National Chiao-Tung University. Abstract The present thesis shows the result of theoretical and experimental study on the physics of entanglement and quantum information processing. We propose a systematic approach to analyze the correlation structures of multipartite entanglement and detect genuine quantum correlations inherent in multipartite multi-level systems. In addition, we introduce novel proposals for entanglement generation, entanglement purification, quantum error corrections, and quantum search algorithm. These theoretical methods and results are both significant for studying the fundamental feature of quantum mechanics and for exploiting the field of quantum information. The experimental work has developed and exploited a source of two-photon four quantum-bit entanglement to realize one-way quantum computing. With the bright source which produces a two-photon state entangled both in polarization and spatial modes, we implemented a highly efficient quantum search algorithm. The experimental result demonstrates that such hyperentangled states could serve as a building block of rapid and precise optical quantum computation.. ii.

(5) 誌謝我感謝指導老師褚德三教授在我的博士生研究階段的指導與支持，得以將此論文完成。於褚教授在物理與研究工作的鼓勵及引領下，增進了專業的知識，並培養了寶貴的科學研究基本態度；我也要感謝褚教授對本論所提出的意見與評論，使本論文趨於完整、充實。我感謝在海德堡大學物理研究所的指導老師 Pan Jian-Wei 教授，引領我進入實驗光學量子資訊的領域，讓我學習到重要的實驗技術與知識，於此獲得實驗與理論相印證之可貴的經驗，亦因此打開了科學研究之視野，也增進了研究的深度與廣度。我要感謝在大學階段就帶領我進入量子資訊領域的黃吉川教授。黃教授深遠的科學研究目標，我因而幸運受益在此研究道路上持續前行，也因受黃教授的科學研究態度的影響，讓我在重要的科研初步就一直保有熱情至今日。我也要感謝王國雄教授在碩士班階段的指導，在研究主題上的協助與相關行政事物的熱情幫忙，我因而可以進入下一個重要的研究階段，除此之外，我也學習到許多做事的態度與待人的道理。我對於曾經一同在研究工作上努力的夥伴們表達我由衷的謝意。我感謝謝金源教授在許多研究課題上的指導，藉由每次深入的討論，培養出重要的解決問題之獨到技巧與方法，也藉此更正了許多錯誤的觀念，進而增進科學研究的興趣與正確的態度；我感謝陳岳男教授引領我討論許多重要的物理研究主題，與在研究工作上的關心、協助與鼓勵，我因此能持續探索有趣的科學問題；我感謝徐立義教授在許多研究課題上對我的啟發，讓我在相互討論中成長；我感謝蘇正耀教授在其所開的課程中教導我量子資訊的基本知識，進而對重要主題有深入的了解與啟發；我感謝 Chen Kai 博士在海德堡的實驗研究裡幫助我釐清許多重要問題，也因重要的實驗瓶頸的突破，使我獲益許多，更是一位絕佳的實驗夥伴；我感謝 Zhang Qiang 博士在海德堡的實驗室耐心地教導我基礎的光學實驗技術，也藉由在實驗問題的討論裡，更正了許多量子力學中的重要觀念；我感謝 Alexander Goebel 熱心地以啟發性的方法帶領我在基礎光學實驗室學習基本的實驗技巧；我也要感謝參與多體糾纏與量子計算實驗的夥伴：Chen Yu-Ao 博士，Chen Shuai 博士，Alois Mair 博士， Yuan Zhen-Sheng 博士，協助我克服許多實驗瓶頸與問題。我要感謝 Tobias Brandes 教授在糾纏態偵測上給予的重要意見，我也要感謝林秀豪教授與羅志偉教授在研究條件量測引致糾纏問題的時期給予的慷慨協助與討論。我要謝謝研究室裡的夥伴林高進博士、邱裕煌、廖英彥博士、院繼祖、陳光胤、王律堯博士、趙國勝博士的勉勵與照顧，重要地，我要感謝簡賸瑞教授一路上的協助與幫忙，從大學部、碩士班以至博士班的友情支持。最後，我感激我親愛的父親、母親、哥哥、大嫂、姊姊與姊夫，以及敬愛的義父義母，沒有您們的支持我無法完成這系列的研究工作。也謝謝許多教過我的老師與幫助過我的朋友們。 . iii .

(6) Contents. 1 Introduction. 1. 1.1 Entanglement, EPR paradox, and Bell’s theorem. . . . . . . . . . . . . . .. 1. 1.2 Entanglement and quantum information processing . . . . . . . . . . . . .. 3. 1.3 Entanglement detections, purifications, and quantum error corrections . . .. 6. 1.4 Experimental generations of quantum entanglement . . . . . . . . . . . . .. 8. 1.5 Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 9. 2 Entanglement and correlation conditions. 12. 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2 Correlation condition and entanglement detection . . . . . . . . . . . . . . 14 2.3 Quantum correlations imbedded in entangled states . . . . . . . . . . . . . 21 2.3.1. Correlation structures of many-qubit GHZ states . . . . . . . . . . 21. 2.3.2. Correlation conditions for bipartite arbitrary-dimensional Bell states 24. 2.4 Correlators embedded in Bell inequalities . . . . . . . . . . . . . . . . . . . 29 2.4.1. Bell inequalities for many qubits . . . . . . . . . . . . . . . . . . . . 29. 2.4.2. Bell inequalities for two qudits . . . . . . . . . . . . . . . . . . . . . 31. 2.5 Correlators imbedded in entanglement witness operators . . . . . . . . . . 39 2.5.1. Detections of genuine many-qubit GHZ states . . . . . . . . . . . . 40. 2.5.2. Inequalities based on the geometry of spin vectors . . . . . . . . . . 41. 2.5.3. Detecting entangled qudits with two local measurement settings . . 46. 2.5.4. Witnesses composed of the kernels of Bell inequalities for qudits . . 53. iv.

(7) CONTENTS 2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3 Correlation conditions in the stabilizer formalism. 57. 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.2 Stabilizer formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.3 Entanglement witnesses for stabilizer states. . . . . . . . . . . . . . . . . . 63. 3.4 Correlator-beased Bell inequalities for many-qubit graph states . . . . . . . 64 4 Entanglement detection via the condition of quantum correlation. 66. 4.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.2 Generalized GHZ states . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.3 Four-qubit singlet state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.4 Three-qubit W state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 5 Phase-dependent criterion for many-qudit entanglement. 78. 5.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 5.2 Basic idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 5.3 Many-qudit Bell inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . 81 5.4 Entanglement witnesses for many-qudit entangled states . . . . . . . . . . 84 5.5 Conclusion and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 6 Entanglement purification. 89. 6.1 Background and motivation . . . . . . . . . . . . . . . . . . . . . . . . . . 89 6.2 Basic idea of entanglement purification . . . . . . . . . . . . . . . . . . . . 90 6.3 Entanglement purification with a two-map protocol . . . . . . . . . . . . . 95 6.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101. 7 Quantum error-correcting codes and entanglement purification. 103. 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 v.

(8) CONTENTS 7.2 The 5-EPR-pair single-error-correcting code . . . . . . . . . . . . . . . . . 105 7.3 Analytical technique for simplification of the encoder-decoder circuit for a perfect five-qubit error correction . . . . . . . . . . . . . . . . . . . . . . . 110 7.3.1. Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110. 7.3.2. A systematic scenario example . . . . . . . . . . . . . . . . . . . . . 114. 7.4 The encoder-decoder circuit for a perfect five-qubit error correction . . . . 121 7.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125. 8 Generation of many-qubit entanglement via conditional measurements on cavity photons. 127. 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 8.2 Bell states generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 8.3 Multi-qubit W state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 8.4 Quantum teleportation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 9 Quantum search algorithm. 137. 9.1 Quantum search problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 9.2 Quantum searching with certainty . . . . . . . . . . . . . . . . . . . . . . . 138 9.3 An improved phase error tolerance in quantum search algorithm . . . . . . 146 9.4 On a family of quantum search algorithms robust against phase imperfections152 9.5 Hamiltonian and measuring time for analog quantum search . . . . . . . . 158 10 Experimental generation of hyperentangled photons and experimental realization of one-way quantum computing. 168. 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 10.2 Photon source for polarization entanglement . . . . . . . . . . . . . . . . . 169 10.3 Experimental generation of two-photon four-qubit hyperentaled states . . . 171. vi.

(9) CONTENTS 10.4 Experimental demonstration of quantum search algorithm with an one-way quantum computer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 10.4.1 One-way quantum computation . . . . . . . . . . . . . . . . . . . . 175 10.4.2 Experimental realization of one-way quantum search . . . . . . . . 178 10.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 11 Summary and Outlook. 180. 11.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 11.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 A Tightness of Bell inequalities. 188. B Entanglement witnesses of statbilizer states. 190. C Entanglement witnesses of entangled qudits. 191. vii.

(10) List of Figures 6.1 The standard purification LOCC operations including the local controlledNOT operation, single qubit measurement, and local unitary operation in each party. Note that the classical communication is not shown in this figure. 91 6.2 The variations of the yield and the comparing purity (in the inserted diagram) at ten times of the recurrence method. . . . . . . . . . . . . . . . . . 97 6.3 The variations of the yield and the comparing purity (in the inserted diagram) at five times of the recurrence method. . . . . . . . . . . . . . . . . 98 ′ ′ 6.4 The variations of the improved yields Y5,T M 1 and Y5,Ox and the comparing ′ ′ ratio (Y5,T M 1 /Y5,Ox ) (in the inserted diagram). . . . . . . . . . . . . . . . . 100. 7.1 The 1-EPP with notations used in the context. Alice performs U1 and m and then sends her classical result (vA ) to Bob. Bob performs U2 and m, and then combines his own result (vB ) and Alice’s to control a final (i). operation U3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 7.2 The three quantum gate arrays performed in the stage of row operations: (a) for M1 → M′1 ; (b) for M′1 → M′′1 ; and (c) for M′′1 → 1. . . . . . . . . . 118 7.3 The gate array for the transformation M1 → 1. The basic unitary operations are performed in the order from left to right, while if they are performed from right to left, then the inverse transformation M1 → 1 is accomplished. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120. viii.

(11) LIST OF FIGURES 7.4 The perfect five-qubit error correction. (a) The initial tensor product state is encoded to an entangled state |φE i . (b) After suffering from the single(i). qubit error, the state Er |φE i is then decoded, resulting in the final tensor (i). product state(U3 |φi) |a′ b′ c′ d′ i . Here, P = HQ, P+ = QH. (c) The encoder circuit from (a) is rewritten in terms of the gate primitives of an ion-trap quantum computer. . . . . . . . . . . . . . . . . . . . . . . . . . . 122 8.1 (a) The quantum devices with three dot-like quantum wells embedded in a microcavity which is constructed by a ZnTe medium and two Au mirrors. This device can be prepared by the MBE, the e-beam lithography, and the conventional semiconductor processing. (b) Initial state preparation for W state generation. (c) Evolution of the QWs and cavity field for a specific time period. (d) Detection of cavity field for determining the number of the cavity photon. Procedures (b)-(d) are repeated until finishing the entanglement generation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 8.2 The variations of fidelity FN,n and the purification yield YN,n (in the inserted diagram) for cases n = 3(), 6(∇), and 9(△), and for two different kinds of initial states: ρ = p1 + (1 − np) |L0 i hL0 | (dash) and |ψ0 i (solid), in √ √ which the evolution time of each case, τ3 = π/( 10γ), τ6 = π/( 22γ), and √ τ9 = π/(4 2γ) has been set. . . . . . . . . . . . . . . . . . . . . . . . . . . 132 9.1 Variations of φ(θ) (solid) and f (θ) (broken), for α + u = 0, β0 = 10−4 , and β = 10−4 (1), 10−2 (2), 0.5 (3) and 0.7 (4), respectively. The cross marks denote the special case of Høyer [166], while the entire cirles correspond to the optimal choices of φop and θop for α + u = 0, β0 = 10−4 and β = 0.7. The solid straight line 1 corresponds the case φ = θ, while the solid curve 2 is only approximately close to the former. . . . . . . . . . . . . . . . . . 144. ix.

(12) LIST OF FIGURES 9.2 Variations of φ(θ) (solid) and f (θ) (broken), for α + u = 0.1, β0 = 0.1, and β = 10−4 (1), 10−2 (2), 0.5 (3) and 0.7 (4), respectively. The cross marks denote the special case of Høye [166]. The solid curves 1 and 2 are very close, and both of them are only approximately close to the line φ = θ. . . 145 9.3 Variations of exact vaule of Pmax (n)(cross marks), 16β 2 sin2 ( θ2 )/(δ 2 +16β 2 sin2 ( θ2 )) (solid), and 4β 2 /(δ 2 + 4β 2) (dash) for θ = π, δ = 0.01 where β = sin−1 (2−n/2 ).150 9.4 Variations of exact vaule of Pmax (n)(cross marks), 16β 2 sin2 ( θ2 )/(δ 2 +16β 2 sin2 ( θ2 )) (solid), and 4β 2/(δ 2 +4β 2) (dash) for θ = π, δ = 0.001 where β = sin−1 (2−n/2 ).151 9.5 The variation of p¯(β) for cases of Bae-Kwon(solid), Farhi-Gutmann(solid), and Fenner(broken) at the specific measuring times, t1,BK = t1,F G = π/(2Eo ) and t1,F = (π − 2β)/(2Eo). . . . . . . . . . . . . . . . . . . . . . . 164 9.6 Variations of t1 (φ − u) (broken) and Eo (φ − u) (solid), for β = 0.085 (1), β = 0.031 (2), and β = 0.0055 (3). . . . . . . . . . . . . . . . . . . . . . . . 166 10.1 Polarization photon source with two-crystal geometry BBO crystals . . . . 170 10.2 Polarization photons emitted from the first BBO crystal . . . . . . . . . . 171 10.3 Polarization photons emitted from the second BBO crystal . . . . . . . . . 172 10.4 Schematic of experimental setup. . . . . . . . . . . . . . . . . . . . . . . . 173 10.5 Quantum circuit for realization of quantum search algorithm. . . . . . . . . 176 10.6 Box cluster state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 10.7 Quantum circuit involved an action of oracle for quantum search. . . . . . 177 10.8 Quantum circuit composed of four local operations for the step 3 in one way realization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 10.9 A successful identification probability of (96.1±0.2)% is achieved deterministically with feed-forward, while it is (24.9 ± 0.4)% without feed-forward. This depicts that our source of cluster state is ideally suited for such a sort of algorithm’s implementation.. . . . . . . . . . . . . . . . . . . . . . . . . 179. x.

(13) Chapter 1 Introduction 1.1. Entanglement, EPR paradox, and Bell’s theorem. Entanglement is one of fundamental pillars in the field of quantum information [1–4]. The remarkable properties of entanglement go essentially beyond the classical correlation constrained by two plausible assumptions, namely locality and realism (local realism) [5, 6]. The assumption of realism states that physical properties of objects have definite values which exist independently of their observation, and the one of locality says that in a causally disconnected manner a measurement of a system does not influence the result of measurement of another system at spacelike separation. Local realism is the essence of the view of Einstein, Podolsky, and Rosen (EPR) [7] on elements of reality. EPR considered that any element of reality must be described by any complete physical theory, and by local realism that was sufficient for the reality of a physical quantity, they showed that quantum mechanics is incomplete. The criterion of EPR is applied to a composite quantum system comprised of two distant particles with a wave function of the form [8]: Ψ = δ(x2 − x1 − x0 )δ(p1 + p2 ), where δ denotes a modified delta function, that is normalizable and possesses an arbitrary high-narrow peak, and x0 is a large distance that is much larger than the range of interaction between particles 1 and 2. From the description of the wave function Ψ, one knows that the total momentum of the system is 1.

(14) CHAPTER 1. INTRODUCTION close to zero and the relative distance of the particles is close to x0 . If one measures x2 , one then can predict with certainty the value of x1 without having any actual influence on particle 1. Then, according to the criterion of EPR, x1 corresponds to an element of physical reality. Furthermore, if one measures p2 , one can predict with certainty the value of p1 without having any actual influence on particle 1. Therefore, according to the criterion of EPR, p1 corresponds to an element of physical reality. However, Heisenberg uncertainty principle precludes one from knowing position and momentum simultaneously. Thus EPR considered that quantum mechanics was an incomplete theory. After EPR’s article, Bohr published a response [9] where he gave the principle of complementarity and argued that the two particles in the situation considered by EPR are always parts of one quantum system and the measurement performed on the first system determines the possible predictions that can be made for the second particle. In addition to Bohr’s reply, Schrödinger [10] claimed that, since the composed system is describe by a single wave function, the two remote particles can influence each other nonlocally. In 1951 Bohm [11] introduced spin-entangled systems and gave a simpler example of the dilemma of EPR. The model of Bohm has become the most studied one for the so-called EPR paradox. The EPR paradox remained a philosophical discussion until Bell [5] in 1964 introduce quantitative criteria for the existence of any local-realistic theory. Bell derived correlation inequalities to show that there is an upper limit to the correlation predicted by local-realistic theories whereas the upper bound can be violated by correlations imbedded in entangled states. The inequalities advocated by Bell are experimental testable. Experiments with entangled pairs have confirmed correlations predicted by quantum mechanics and then show Einstein locality are incompatible with quantum correlations as the proof given in Bell’s theorem [12]. By the inspiration of Bell’s theorem, the so-called Bell inequalities [5, 13–16] for two-level systems have been proposed to experimentally invalidate the point of view of EPR and to show that quantum mechanics is not locally realistic. Furthermore, while 2.

(15) CHAPTER 1. INTRODUCTION entanglement for quantum two-level systems (qubits) is still under intensive study, entangled quantum multi-level systems (qudits) attract much attention for their nonlocal characters [17–21] and advantages in quantum information processing [22–24]. It has been shown that entangled qudit pair can maximally violate the Clauser-Horne-Shimony-Holt (CHSH) inequality [13] and the corresponding violation continues to survive in the limit of infinite dimension [25]. Using the method of linear programming to give necessary and sufficient conditions [26], numerical calculations have demonstrated that contradiction between local realism and quantum mechanics increases with the dimension. Latter, this contradiction has been confirm analytically in [17, 27]. Collins et al. [17] have reformulated Bell inequalities to construct a large family of multi-level inequalities in terms of a novel constraint for local-realistic theories called Collins-Gisin-Linden-Massar-Popescu (CGLMP) inequality. Recently, Son, Lee, and Kim (SLK) [18] presented generic Bell inequalities and their variants for arbitrary high-dimensional systems through the generalized Greenberger, Horne and Zeilinger (GHZ) nonlocality [28].. 1.2. Entanglement and quantum information processing. For the aspect of quantum information processing, the nonlocal features of quantum correlations enable people to perform high-security and novel quantum communication [29, 30]. Moreover, it promotes a novel model of universal quantum computation [31–33]. Quantum communication could be consider as the first application of quantum mechanics, that is based on entanglement, no-cloning theorem, and quantum superposition. Quantum communication involves transmissions of quantum states form one place to another. In 1984, the first quantum-cryptography protocol has been proposed by Bennett and Brassard [34]. The essence of their scheme is the fact that unknown quantum states cannot be cloned. In 1991, the first application of quantum non-locality is introduced by Ekert [29]. In the protocol of Ekert, maximally entangled pairs are utilized for transmission of quantum 3.

(16) CHAPTER 1. INTRODUCTION key and the corresponding security is guaranteed by the distinct features of entanglement cooperating with Bell’s theorem. These novel encryption schemes provide a fundamental improvements compared to conventional ones. In 1993, quantum teleportation was exposed by Bennett et al. [30] in a momentous article entitled ”Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels” . With share maximally entangled pairs together with two classical bits of communication as specified in their protocol, unknown quantum states can be transferred from one place to another without any intermediate location. Quantum teleportation is also central to a number of quantum computation protocols [35, 36]. In addition to the above quantum protocols, super dense coding [37] and quantum secret sharing [38] are based on resources of entangled states for quantum information processing. The former illustrates that two bits of information can be transmitted from sender to receiver by sending only a single qubit. Quantum secret sharing does not only give a procedure required for the goal of secret sharing but also provides a way to detect the presence of an eavesdropper. Many-qubit entanglement are necessary for performing some specific types of protocol of quantum secret sharing. Experimental demonstrations of single-qubit teleportation have been implemented with different physical systems [39–43] . Recently, teleportation of two-qubit composed systems has been experimentally realized with photonic qubits successfully [44]. As for quantum secret sharing, in Re. [45] four-party secret sharing with four-photon singlet states has been experimentally preformed. On the other hand, in order to achieve the aim of long-distance entanglement-based communication, up to now experiments have demonstrated over distances of up to 144 km using polarization-entangled photons via free-space links through the atmosphere. In Re. [46], the violation of CHSH-type [13] Bell inequality shows the distinct features of entanglement observed 144 km apart and then the Ekert protocol for quantum key distribution has been demonstrated successfully. However, to dissolve the problems about limitation communication distance further, quantum relays [47] and quantum memories [48, 49], i.e., quantum repeaters, are needed. 4.

(17) CHAPTER 1. INTRODUCTION Entanglement is also a resource for universal quantum computing.. One-way quan-. tum computation [31–33] is performed with a certain multipartite entangled source, a cluster state [31], and local measurements on the constituents, and then it is also called measurement-based quantum computation. Before the model of one-way quantum computation is introduced, a quantum computer including the mathematical model and the corresponding experimental realizations is designed for logic circuit [50] of universal quantum gates [51] that require highly controlled interactions between selected qubits. It has been proven that one-way quantum computer can simulate any quantum logic circuit [32]. Three experiments have created four-photon cluster states and then demonstrated quantum one-way computation by performing quantum search algorithm [52–54]. Quantum mechanical algorithms involves utilization of quantum effects and have become very popular in the field of computation science because they can speed up a computation over classical algorithms. Famous examples include Deutsch-Josza algorithm [55], the factorizing algorithm discovered by Shor [56], and the quantum search algorithm well-develpoed by Grover [57, 58]. If there is an unsorted database containing N items, and out of which only one marked item satisfies a given condition, then using Grover’s algorithm one will √ find the object in O( N) quantum mechanical steps instead of O(N) classical steps. It has been shown that Grover’s original algorithm is optimal [59–61]. Through four-photon cluster states, Deutsch-Josza quantum algorithm [55], that is a quantum method to identify whether a given function is constant or balanced, has also been experimentally realized in the one-way approach [62]. Besides, the compiled version of Shor’s quantum factoring algorithm has been demonstrated by using photonic qubits [63]. In addition to qubits for quantum information protocols discussed above, qudits are very useful for several different kinds of quantum communication tasks. It has been shown that quantum key distribution with higher alphabets is more secure than that based on qubits [64–66]. The coin-flipping and the Byzantine agreement problems can be solved by making use of qutrits (quantum three-level systems) [67]. Entangled qutrits can be used to solve two-party communication complexity problem [68]. N-party N-level supersinglet 5.

(18) CHAPTER 1. INTRODUCTION states can help to solve the problems which have no solution using classical method: N-strangers, secret sharing, and liar detection problems [69].. 1.3. Entanglement detections, purifications, and quantum error corrections. Quantum communication protocols for tasks such as quantum key distributions [29], quantum teleportation [30], and quantum super dense coding [37] rely on the transmission of maximally entangled qubit pairs over quantum channels between a sender (Alice) and a receiver (Bob). The quantum channel, however, is always noisy due to the interaction with the environment and even possibly the measurement controlled by an eavesdropper. Therefore, the pairs shared by Alice and Bob are no longer of the desired pure ones to begin with a quantum processing. The resource in the noisy channel then can be viewed as a mixed state, or equivalently, an ensemble of pure states with definite random probabilities. The fidelity of the pure states in the ensemble are random so should be unknown to Alice and Bob. Accordingly, first, Alice and Bob need to find efficient experimental methods to detect whether a experimental output is indeed entangled. Then, they could take an action of entanglement purification to regain, at least asymptotically, the desired maximally entangled pure state if the mixed state is distillable. This aim can be achieved by Alice and Bob, using consecutive local operations and classical communications (LOCC). Above processes are also necessary for many-party protocols of quantum communication, e.g., experimental achievement of open-destination teleportation [43]. In addition to the importance of entanglement detections for quantum communication, detecting genuine many-qubit correlations of multipartite entanglement is also crucial for performing faithful measurement-based quantum computation [31–33]. Since Bell inequalities can be consider as a means to feature quantum correlations in the corresponding violations, it is natural to think that Bell inequalities are useful for entanglement detections. However, there are two difficulties in utilizing Bell inequalities 6.

(19) CHAPTER 1. INTRODUCTION for entanglement detections. One involves experimental difficulty and another is about limits of their intrinsic utilities. First, for detecting N-qubit entangled states, the number of local measurement settings (see the definition in the second chapter) required increases with exponentially with N [14–16]. Second, Bell inequalities cannot always detect an entangled state with some specific characters of quantum correlation, e.g., detections of genuine multipartite entanglement [70]. To resolve these problems, entanglement witnesses are introduced to detect entanglement [71]. Entanglement witness operators rely on an use of the whole or partial knowledge of an entangled state to be created, which are designed for distinguishing entangled states from separable ones. Furthermore, entanglement witnesses can be designed for detecting genuine multipartite entanglement [70], and some witnesses for truly multipartite entanglement require fewer local measurement settings [72, 73] when used in experiments. The first entanglement purification protocol (the IBM protocol) was developed by Bennett et al. [74, 75] in achieving a faithful quantum teleportation. Soon later, an improved protocol entitled “Quantum Privacy Amplification” (QPA, or the Oxford protocol) was addressed by Deutsch et al. [76] in consideration of the security of a quantum cryptography over noisy channels. Both the IBM and Oxford protocols are capable of purifying a desired maximally entangled pure state from every distillable mixed state whose components are not learned by Alice and Bob initially. It is worth noting that Bennett et al. [74] have presented the equivalence between the entanglement purification protocol based on one-way classical communication, that is different from the IBM protocol with two-way classical communication, and the fivequbit quantum error-correcting code [77, 78]. Quantum states can be encoded into qubits through quantum error-correcting codes [79, 80]. With an introduction of redundancy, the encoded data can tolerate little errors which are due to decoherence in some individual qubits. Then, quantum error-correcting codes play a crucial role in scalable quantum computation and communication to preserve the gain in computational time and in security. 7.

(20) CHAPTER 1. INTRODUCTION The experimental purification of entangled qubits of IBM protocol has been demonstrated by using entangled-photon source [81]. In addition to the IBM and Oxford protocols, further extensions cover purifications of many-qubit W [82] and graph states [83] and multi-level GHZ states [84].. 1.4. Experimental generations of quantum entanglement. For the aspect of generating entanglement in real physical systems, many different architectures and schemes have been proposed. An entanglement can be generated in atom[85–87] and ion-trap systems [88], superconducting charge [89, 90] and flux [91] qubit systems. However, in order to perform quantum information processing, in addition to entanglement generations there are several criteria for measuring how good physical systems are. To realize quantum computation, the requirements of the physical systems involve scalability, isolation, initialization, measurement, and controllably interactions for universal quantum gates [92]. To achieve quantum communication, the physical systems carrying information are expected to transmit between remote places [93, 94]. According to these reasons, optical quantum systems [36, 93–96] are important candidates for quantum information processing and then become leading approaches over the past few years. Many experimental achievements of tasks of quantum information processing are attained with optical quantum systems. Polarization-entangled photons emitted by the process of spontaneous parametric down-conversion (SPDC) [97, 98] in a nonlinear crystal has been widely utilized to analyze quantum correlation and to experimentally demonstrate quantum computation and quantum communication, e.g., the experimental realizations of the quantum protocol mentioned above [39–41, 43–46, 52–54, 62, 63, 81], entanglement of six photons in graph states [99], and test of non-local realism [100]. These experiments are designed to process information encoded in qubits. 8.

(21) CHAPTER 1. INTRODUCTION Recently, due to the advantages and distinct characteristics of qudits as mentioned above, many researches have paid attention to generations of hyperentanglement. Orbital angular momentum entangled photons generated from the SPDC process have been experimentally realized and provide a resource to study quantum correlation inherent in multilevel systems [20]. Using this kind of entangled source, violation of three-level Bell inequalities has been experimentally confirmed and quantum key distribution with qutrits has also been demonstrated [23]. In addition to the polarization and orbital angular momentum of photons, utilizing accessible degrees of freedom including path modes [54, 101, 102], enery time, time bin [103–105], and every degree of freedom [106], one can produce hyperentanged photon sources. Since a hyperentangled state is in a larger Hilbert space, this feature can be used to perform 100% efficient complete Bell-state analysis with only linear elements [107], to purify entanglement [101], and to realize all versus nothing test of quantum mechanics [102]. An experimental CGLMP test for energy-time entangled qutrits has been reported in Re. [105]. The experimental scheme for deterministic and efficient quantum communication based on hyperentanglement has also been proposed [108]. In particular, experimental realization of one-way quantum computing with two-photon four-qubit hyperentangled states has been reported in Re. [54].. 1.5. Outline of the Thesis. We have proposed several novel ideas and proposals for quantum information processing and experimentally demonstrated one important element of quantum computation during 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 processing. We start with a study into the properties of correlations inherent in multipartite. 9.

(22) CHAPTER 1. INTRODUCTION 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 generation, 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 entanglement 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 inequalities 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 proposed to purify any distillable state to a desired maximally entangled pure state. 10.

(23) CHAPTER 1. INTRODUCTION 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 oneway 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 presented 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.. 11.

(24) Chapter 2 Entanglement and correlation conditions 2.1. Introduction. Bell inequalities are results about local realism, and then violations of which by entangled 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 condition of quantum correlation 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 correlation 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],. 12.

(25) CHAPTER 2. ENTANGLEMENT AND CORRELATION CONDITIONS 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 highdimensional 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 twoqudit 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 introduce the dependence criterion for many-qubit and two-qudit correlations. Since manyqubit 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 quantum 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 Nqubit GHZ states and analyze the structure of the inequalities beased on the geometry of 13.

(26) CHAPTER 2. ENTANGLEMENT AND CORRELATION CONDITIONS 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.. 2.2. Correlation condition and entanglement detection. In a N-party Bell-type experiment, measurements on each spatially-separated particle are assumed to be performed with two distinct results (d distinct results for d-level Bell-type experiments) from two different observables. In each run of the experiment, each party chooses one observable for a simultaneous measurement on the particle in parallel. Let us denote the local measurement setting by M = (V1 , V2 , ..., VN ), where Vi represents the observable chosen by the ith party. After measurements, a set of results, (v1 , v2 , ..., vN ), where vi ∈ {0, 1} (vi ∈ {0, 1, ..., d − 1} for d-level Bell-type experiments), is acquired. If sufficient runs of such measurements have been made under the chosen local measurement setting, the correlation between experimental outcomes can be revealed through analytical analyses of experimental records. In analogy, experiments for bipartite multilevel systems work in the same way as mentioned above. For quantum mechanical representation, we introduce an operator of the form Vî =. 1 X. (−1)vi vî ,. (2.1). vi =0. where vî = |vi iVi Vi hvi | and {|vi iVi } is a complete set of orthonormal basis vectors for the N ˆ observable Vî . Each N-product operator of the form Vˆ ± = ± N i=1 Vi can be represented. explicitly by. Vˆ ± = Cˆ0± + Cˆ1± ,. (2.2). 14.

(27) CHAPTER 2. ENTANGLEMENT AND CORRELATION CONDITIONS where ˆm − 1 ˆm ) ⊗ 0 ˆ m¯ , Cˆ1+ = (1 ˆm − 0 ˆm ) ⊗ 1 ˆ m¯ , Cˆ0+ = (0. (2.3). ˆm − 1 ˆm ) ⊗ 1 ˆ m¯ , Cˆ1− = (1 ˆm − 0 ˆm ) ⊗ 0 ˆ m¯ , Cˆ0− = (0. (2.4). and ˆm = 0 ˆm = 1. m X O. v1 ,...,vm i=1 m X O. vî for vî for. v1 ,...,vm i=1. ˆ m¯ = 0. X. N O. m X. i=1 m X. ˆ m¯ = 1. X. vm+1 ,...,vN i=m+1. . vi = 1,. i=1. vî for. vm+1 ,...,vN i=m+1 N O. . vi = 0,. N X. . vi = 0,. i=m+1. vî for. N X. . vi = 1,. (2.5). i=m+1. . and = denotes equality modulus two. Expectation values of Vˆ ± for some physical states, denoted by hVˆ ± i, are typically called N-point correlation functions. Here we will give a new insight into hVˆ ± i via their elements Cˆ0± and Cˆ1± . Determining the expectation values of Cˆ0± and Cˆ1± can provide information about correlation between the subsystems composed of the first m qubits and the rest of the system. Theorem 1. If measured outcomes show that expectation values of operators satisfy hCˆ0± i > 0 and hCˆ1± i > 0, or, hCˆ0± i < 0 and hCˆ1± i < 0, the outcomes of measurements performed on the subsystem of the first m qubits are correlated with the ones performed on the subsystem of the last N − m qubits [111]. Proof. If the subsystems are independent, we have the following relations ˆ m i − h1 ˆ m i)h0 ˆ m¯ i, hCˆ1+ i = (h1 ˆ m i − h0 ˆ m i)h1 ˆ m¯ i, hCˆ0+ i = (h0. (2.6). ˆ m i − h1 ˆ m i)h1 ˆ m¯ i, hCˆ1− i = (h1 ˆ m i − h0 ˆ m i)h0 ˆ m¯ i. hCˆ0− i = (h0. (2.7). and. 15.

(28) CHAPTER 2. ENTANGLEMENT AND CORRELATION CONDITIONS ˆ m i + h1 ˆ m i = 1, h0 ˆ m¯ i ≥ 0, and h1 ˆ m¯ i ≥ 0 for any physical systems, it turns out Since h0 that hCˆ0+ ihCˆ1+ i ≤ 0 and hCˆ0− ihCˆ1− i ≤ 0. Thus a contradiction reveals the dependency of one subsystem on another one. Then hVˆ ± i is not just a N-point correlation function but a general one composed of ± hCˆ0,1 i that gives nc conditions of dependence for correlations between any two subsystems. with m qubits and N − m ones, where nc =. ⌊N/2⌋. X k=1. f (N, k). N! , k!(N − k)!. (2.8). f (N, k) = 2−δ[k,⌊N/2⌋] for even N, δ[·] denotes Kronecker delta symbol, and f (N, k) = 1 for odd one. Take N = 3 for example. A correlation function hVˆ1 Vˆ2 Vˆ3 i involves three conditions, i.e., nc = 3, to describe correlations between subsystems including the following classifications, {[1, 2, 3]}: [1|2, 3], [2|1, 3], and [3|1, 2], where [i|j, k] denotes the correlation between the ith qubit and the subsystem composed of the j th and kth ones. ¯ to represent nc differFor N qubits, we use the denotation {[1, 2, ..., N]} or {[m, m]} ± ent kinds of partitions for correlation, and we sometimes use the notations Cˆ0[m, ¯ and m] ± Cˆ1[m, ¯ emphasizing the correlations between two specific subsystems denoted by m and m]. ¯ respectively. m ± By the same idea of constructing Cˆ0,1 for qubits, we introduce the following sets of. operators for two-qudit correlations: (q) ˆ ⊗ U(k), ˆ Cˆk = [kˆ − T (k)]. (2.9). for k = 0, 1, ..., d − 1 and q = 1, ..., γd , where T and U are injective maps such that ˆ 7→ kˆ′ , U(k) ˆ 7→ kˆ′′ , and k ′ 6= k, and each set {T (k)} ˆ composed of T (k)’s ˆ is numbered T (k) ˆ and hence the sets of operators by q. Take d = 3 for example, we have two sets of {T (k)}. 16.

(29) CHAPTER 2. ENTANGLEMENT AND CORRELATION CONDITIONS (q). {Cˆk } could be (1) (1) (1) {Cˆ0 = (ˆ0 − ˆ1)ˆ0, Cˆ1 = (ˆ1 − ˆ2)ˆ1, Cˆ2 = (ˆ2 − ˆ0)ˆ2}, (2) (2) (2) {Cˆ0 = (ˆ0 − ˆ2)ˆ0, Cˆ1 = (ˆ1 − ˆ0)ˆ1, Cˆ2 = (ˆ2 − ˆ1)ˆ2}.. ˆ = kˆ is used in this example. For d = 2, we get the sets of operators for qubits where U(k) (1) (1) ˆ = k, ˆ or Cˆ (1) = Cˆ − and Cˆ (1) = Cˆ − introduced above: Cˆ0 = Cˆ0+ and Cˆ1 = Cˆ1+ for U(k) 0 0 1 1. ˆ = kˆ′′ and k ′′ 6= k. Then it is clear that the number of sets {Cˆ (q) } depends on the for U(k) k (q) ˆ number of {T (k)}. For general d, we have γd sets of {Cˆk }, where γ2 = 1, γ3 = 2, γ4 = 9,. γ5 = 44, γ6 = 285, and. γd = (d − 2). d−1 Y v=3. v + (d − 1)γd−2 ,. (2.10). for d ≥ 7. The correlation between outcomes of measurements performed on two remote qudits can be revealed by the help of the following theorem. (q) Theorem 2. If measured outcomes show each expectation value hCˆk i in the qth set (q). {hCˆk i} is positive or each one is negative, the outcomes of measurements performed on the first qudit are correlated with the ones performed on the second qudit [112]. (q) Proof. If the subsystems are independent, one can recast hCˆk i as. (q) ˆ − hT (k)i)hU( ˆ ˆ hCˆk i = (hki k)i,. Since. (2.11). P ˆ P ˆ ˆ ˆ (q) h ki = k k hT (k)i = 1 and hU(k)i ≥ 0, hCk i > 0 for all k’s is impossible for. independent subsystems. Then a contradiction indicates the dependency of the first qudit on the second one.. With the above two theorems, one can feature a many-qubit or two-qudit entangled state in sets of correlation conditions proposed under different local measurement settings. These conditions can be considered as necessary ones for the entangled state under. 17.

(30) CHAPTER 2. ENTANGLEMENT AND CORRELATION CONDITIONS (q) ± study. We call the expectation values hCˆk i and hCˆ0,1 i correlators due to their utilities for. correlations. We give three concrete examples to illustrate how correlators work for analyzations of the correlation structures of given states and the basic idea of entanglement detections based on correlators: (a) A two-qubit pure entangled state in the following representation:. |φi = sin(ξ) |00i + cos(ξ) |11i ,. (2.12). for 0 < ξ < π/4, where |v1 v2 i = |v1 i ⊗ |v2 i and |vi i is the eigenstate of Pauli-operator σz with eigenvalue (−1)vi , can be described by correlators that correspond to the operators Cˆ0Z = Cˆ0+ = (ˆ0 − ˆ1)ˆ0 and Cˆ1Z = Cˆ1+ = (ˆ1 − ˆ0)ˆ1. By a direct calculation, one obtains the correlators hCˆ0Z i = sin2 (ξ) and hCˆ1Z i = cos2 (ξ) for the state |φi, which reveals the correlation properties when observed in the local measurement setting Mz = (Z, Z) where Z = σz . The state |φi can also be shown in another representation, e.g., |φi = a(|00iX + |11iX ) + b(|01iX + |10iX ),. (2.13). where a = [cos(θ) + sin(θ)]/2, b = [cos(θ) − sin(θ)]/2, and |vi iX is an eigenstate of Paulioperator σx with an eigenvalue (−1)vi . This representation provides the information of probability distribution for {|v1 v2 iX } when measured in the setting Mx = (X, X) where X = σx . From which, one can construct correlators, and the characters of correlation can be described by hCˆ0X i = hCˆ0X i = sin(2ξ)/2 where Cˆ0X = Cˆ0+ = (ˆ0 − ˆ1)ˆ0 and Cˆ0X = Cˆ1+ = (ˆ1 − ˆ0)ˆ1. (b) The probability distribution for |φi when measured with the setting Mz is the same as the one of the following mixture of product states: ρφ = sin2 (ξ) |00i h00| + cos2 (ξ) |11i h11| .. (2.14). Then we have the correlators hCˆ0Z iρφ = sin2 (ξ) and hCˆ1Z iρφ = cos2 (ξ) and know outcomes 18.

(31) CHAPTER 2. ENTANGLEMENT AND CORRELATION CONDITIONS of measurements for these particles are dependent. When the state ρφ is represented in the basis {|v1 v2 iX }, the probability for observing an element in {|v1 v2 iXX hv1 v2 |} of ρφ is 1/4, which implies that these two particles are independent. This fact can be shown by the correlators hCˆ0X i = hCˆ1X i = 0. From the above examples, one has. P. l=X,Z. P1. ˆ k=0 hCkl i >. P. l=X,Z. P1. ˆ. k=0 hCkl iρφ .. From. which, it is worth noting that determining a sum of correlators associated with two different local measurement settings can help us to distinguish the entangled state |φi from the separable state ρφ . This idea and approach can be applied to detections of truly many-qubit entanglement and bipartite entangled qudits. For any many-qubit system composed of two independent parts, outcomes of measurements should satisfy

(32)

(33)

(34)

(35) X

(36)

(37) ± ˆ ˆ ˆ ˆ ¯ i − h1 ˆ m¯ i)| ≤ 1 h C i

(38) ¯

(39) = |(h0m i − h1m i)(h0m k[m,m]

(40)

(41). (2.15). k. for any measurement settings chosen. Whereas, for some specific entangled states, one P ± can feature properties of entanglement to be created in | k hCˆk[m, ¯ i| = 1 for several local m] measurement settings chosen and consider which as necessary conditions for the entangled. ¯ associated with any state. Furthermore, we could give all conditions of correlations [m, m] two subsystems of the many-qubit entangled state under study. Thus we can use these conditions of genuine many-qubit entanglement to rule out biseparable correlations. For two independent qudits observed under any measurement settings, a sum of correlators should follow the criteria

(42)

(43)

(44) X

(45) X X

(46) (q)

(47) ˆ − hT (k)i)| ˆ 2 |(hki |hU(kˆ′ )i|2

(48) hCˆk i

(49) ≤

(50)

(51) ′ k. k. k. ≤ 1.. Then entanglement conditions |. P. ˆ (q) k hCk i|. (2.16). = 1 for all local measurement settings con-. sidered can be very useful to detect entangled qudit pairs. Using the idea introduced above can promote constructions of many-qubit and two-qudit entanglement witness operators that require only two local measurement settings. Even though the conditions 19.

(52) CHAPTER 2. ENTANGLEMENT AND CORRELATION CONDITIONS |. P. ˆ± k hCk[m,m] ¯ i|. = 1 and |. P. ˆ (q) k hCk i|. = 1 cannot be satisfied by all entangled states con-. sidered, the above approach still can be applied to entanglement detections if more local measurement settings are used. See the case discussed in the second subsection of Sec. 2.5. (c) The state vector of a two-qubit singlet state is represented by 1 |ψi = √ (|01i − |10i). 2. (2.17). √ √ (2) (1) (2) (1) If Vˆ1 ∈ {Vˆ1 = Z, Vˆ1 = X} and Vˆ2 ∈ {Vˆ2 = −(Z + X)/ 2, Vˆ2 = (Z − X)/ 2}, we have four different local measurement settings M = (Vˆ1 , Vˆ2 ) to give four sets of correlators. (rt) (rt) The operators of correlators are as follows: Cˆ0 = Cˆ0+ = (ˆ0 − ˆ1)ˆ0, Cˆ1 = Cˆ1+ = (ˆ1 −. ˆ0)ˆ1 for (rt) ∈ {(11), (21), (22)} and Cˆ0(12) = Cˆ0− = (ˆ0 − ˆ1)ˆ1, Cˆ1(12) = Cˆ1− = (ˆ1 − ˆ0)ˆ0 , (r) (t) where the superscripts (rt) mean an observable Vˆ1 and another one Vˆ2 are chosen (rt) for measurements. The correlators can be easily calculated, and then we have hCˆ0 i = √ (rt) hCˆ1 i = 1/2 2. When collecting all of the correlator operators proposed above, one gets. B =. 2 X 1 X. (rt) Cˆk. r,t=1 k=0. (1) (1) (2) (2) (2) (1) (1) (2) = Vˆ1 Vˆ2 + Vˆ2 Vˆ2 + Vˆ1 Vˆ2 − Vˆ1 Vˆ2 .. (2.18). Local-realistic theories predict that B ≤ 2, which is called the CHSH inequality [13], whereas the entangled state |ψi predicted by quantum mechanics provides a violation by √ P ˆ (rt) i = 2 2. It is remarkable that the kernel of the CHSH inequality [13] is comrt hC. (rt) (rt) posed of necessary conditions of the state |ψi in terms of the correlators (hCˆ0 i, hCˆ1 i).. In what follows, we will use correlators to analyze the most studied many-qubit and two-qudit entangled states: the N-qubit GHZ state [113] and the two-qudit Bell state. The correlators proposed are necessary for states to be the entanglement under study and play important roles in identifying quantum correlations including ruling out biseparable correlations and ones predicted by local-realistic theories.. 20.

(53) CHAPTER 2. ENTANGLEMENT AND CORRELATION CONDITIONS. 2.3. Quantum correlations imbedded in entangled states. 2.3.1. Correlation structures of many-qubit GHZ states. The three-qubit GHZ state is first discussed in the GHZ argument [113] which provides important insights into tripartite entanglement. Entanglement embedded in a three-qubit GHZ state has been shown very useful to investigate both noncontextual variables and Bell-EPR theorems [114]. In addition to the three-qubit GHZ state, the generalized Nqubit GHZ sates have attracted much attentions. Many Bell inequalities for many qubits [14–16] have been shown to be violated by N-qubit GHZ states. Furthermore, six-atom [85] and six-photon [99] GHZ states have been demonstrated experimentally. In this subsection, we utilize three different types of correlators to specify the features of N-qubit correlation of a N-qubit GHZ state which is of the state vector: 1 |Φi = √ (|0i⊗N + |1i⊗N ). 2. (2.19). The classification of correlator depends on which kind a local measurement setting is chosen and how many settings are used in Bell-type experiments. These correlators will be utilized to subsequent investigations on entanglement detections. Specification 1. Firstly, we introduce alternative dichotomous observable for qubits by Y +X Y −X Vˆk ∈ {X, Y }, VˆN ∈ { √ , √ }, 2 2. (2.20). for k = 1, 2, ..., N − 1, where Y denotes the Pauli-operator σy . Since each party has two choices to perform measurements, there are 2N possible sets of local measurement settings. Then we give the following operators of correlators Cˆ0l = γl Cˆ0+ , Cˆ1l = γl Cˆ1+ ,. (2.21). for l = 1, 2, ..., 2N , where l is used to number 2N different measurement settings and γl. 21.

(54) CHAPTER 2. ENTANGLEMENT AND CORRELATION CONDITIONS are constants. By a direct calculation, we have the correlators γl (−1)n2 (n2 +1)/2 √ hCˆ0l i = hCˆ1l i = , 2 2. (2.22). √ where n2 denotes the number of Vˆk = Y and VˆN = (Y − X)/ 2 chosen in a setting numbered l. If we assign γl = (−1)n2 (n2 +1)/2 , then each correlator has the same sign and √ becomes hCˆ0l i = hCˆ1l i = 1/2 2. Specification 2. The observable of each particle designed for measurement is given by Vˆk ∈ {X, Y },. (2.23). for k = 1, 2, ..., N. Although there are 2N possible settings of local measurements, we focus only on settings in which there are even number of Y ’s chosen for measurements. For odd N and 2n Y ’s where n is odd, we introduce correlator operators of the form Cˆ0l = Cˆ0− , Cˆ1l = Cˆ1− ,. (2.24). for l = 1, 2, ..., 2N −1 − 1. For even N and 2n Y ’s where n is even, the operators of correlators are given by Cˆ0l = Cˆ0+ , Cˆ1l = Cˆ1+ ,. (2.25). for l = 1, 2, ..., 2N −1. For a N-qubit GHZ state, the correlators proposed are all the same: 1 hCˆ0l i = hCˆ1l i = . 2. (2.26). Specification 3. In this specification, we use only two local measurement settings to. 22.

(55) CHAPTER 2. ENTANGLEMENT AND CORRELATION CONDITIONS feature correlation structure in correlators:. Mz = {Z, Z, ..., Z}, Mx = {X, X, ..., X}.. (2.27). When a N-qubit GHZ state is measured under the setting Mz , correlations between some subsystem composed of m qubits and the rest can be described by correlators corresponding to the operators: ˆ Cˆ0Z[m,m] ¯ = (0mZ − 1mZ )0mZ ¯ , C1Z[m,m] ¯ = (1mZ − 0mZ )1mZ ¯ , where 0m(m)Z = ¯. N. i∈m(m) ¯. vî for all vi = 0, 1m(m)Z = ¯. N. i∈m(m) ¯. (2.28). vî for all vi = 1, and m and. m ¯ denote the subsystems comprised of m and N − m qubits respectively. For instance, to detect three-qubit GHZ state the correlator operators have the explicit representations: Cˆ0Z[i,pq] = (ˆ0i − ˆ1i )ˆ0p ˆ0q , Cˆ1Z[i,pq] = (ˆ1i − ˆ0i )ˆ1p ˆ1q , where the set of subscripts (ipq) is used to number qubits. For each set of correlator, we have 1 ˆ hCˆ0Z[m,m] ¯ i = hC1Z[m,m] ¯ i = . 2. (2.29). For the sets of correlators constructed under the setting Mx , their constructions are ˆ similar to the ones of (Cˆ0Z[m|m] ¯ , C1Z[m|m] ¯ ) and represented by ˆ+ ˆ ˆ+ Cˆ0X[m|m] ¯ = C0[m|m] ¯ = C1[m|m] ¯ , C1X[m|m] ¯ .. (2.30). + ˆ+ where the operators Cˆ0[m| ¯ and C1[m|m] ¯ are of the forms as Eq. (2.3). Take three-qubit m]. operators of correlators for example, they are of the forms: Cˆ0X[i,pq] = (ˆ0i − ˆ1i )(ˆ0p ˆ0q + ˆ1p ˆ1q ), Cˆ1X[i,pq] = (ˆ1i − ˆ0i )(ˆ0p ˆ1q + ˆ1p ˆ0q ).. 23.

(56) CHAPTER 2. ENTANGLEMENT AND CORRELATION CONDITIONS The correlators corresponding to the above operators can be easily calculated and given by 1 ˆ hCˆ0X[m,m] ¯ i = hC1X[m,m] ¯ i = . 2. 2.3.2. (2.31). Correlation conditions for bipartite arbitrary-dimensional Bell states. We proceed to introduce correlators to study the correlation structure of a bipartite arbitrary-dimensional Bell state: d−1. 1 X |Ψi = √ |vi ⊗ |vi . d v=0. (2.32). The constructions of correlators are based on the basic idea shown in the introduction and mainly in the second theorem. The generalized Bell state plays an important role both in violations of Bell inequalities for two qudits [17, 18] and in quantum communication protocols [24]. Experiments have demonstrated generalized Bell states for d = 3 successfully and even used for further applications [23]. Specification 1. The correlator operators presented in this specification can be formulated by the following general form (r) ′(r) (t) (rt) (r) ′(r) (t) Cˆk (v1 , v1 , v2 ) = (ˆ v1 − vˆ1 ) ⊗ vˆ2 ,. (2.33). (r) where the superscripts, (rt), (r), and (t), mean that the measurement Vˆ1 numbered r (t) and the one Vˆ2 numbered t have been selected from two choices by each party. Four. 24.

(57) CHAPTER 2. ENTANGLEMENT AND CORRELATION CONDITIONS (r). ′(r). (t). designations of (v1 , v1 , v2 ) associated four local measurement settings are given by (1) . ′(1) . (2) (v1 = −k, v1 = 1 − k, v2 = k),. (2) . ′(2) . (1) (v1 = d − k − 1, v1 = −k, v2 = k),. (h) . ′(h) . (h) (v1 = −k, v1 = d − k − 1, v2 = k),. (2.34). . for h = 1, 2 and k = 0, 1, ..., d−1, where = denotes equality modulo d . The set of operators (rt). (q). {Cˆk } is a special case of the general one {Cˆk } discussed in the second theorem, and (rt). for each measurement setting {Cˆk } involves one set of correlation condition rather than (q). γd conditions for {Cˆk }. To evaluate the correlators concretely, we choose specific sets of complete orthonormal (h). (h). basis {|vj iV (h) } for projectors {ˆ vj j.

(58) E

(59) (h)

(60) vj. (h). and nj. (h). (h). (h). = |vj iV (h) V (h) hvj |} and operators {Vˆj }, where j. j. d−1. (h). Vj. 1X 2πv (h) (h) = exp[i (v + nj )] |vi , d v=0 d j. (2.35). (h) are local parameters that manifest observable Vˆj . For a set of local parameters. given by (1). (1). (2). (2). n1 = 0, n2 = 1/4, n1 = 1/2, n2 = −1/4,. (2.36) (r). (t). the joint probabilities for obtaining a set of measured outcome (v1 , v2 ) for the state |Ψi is [17] (r). (t). hˆ v1 ⊗ vˆ2 i = (r). (t). 1 2d3 sin2 [ πd (v (rt). + n(rt) )]. (r). (t). ,. (2.37) (rt). where v (rt) = v1 + v2 and n(rt) = n1 + n2 . Therefore, the correlators hCˆk i can be calculated analytically, and then we arrive at (rt). hCˆk i =. 1 2 π 2 3π [csc ( ) − csc ( )]. 2d3 4d 4d. (2.38). 25.

(61) CHAPTER 2. ENTANGLEMENT AND CORRELATION CONDITIONS (rt) Since hCˆk i > 0 for all k’s with any finite value of d, we ensure that outcomes of mea-. surements performed on particles of a state |Ψi are dependent under four different local measurement settings. (r). ′(r). (t). Specification 2. We can generalize the designations (v1 , v1 , v2 ) introduced in the above specification to more general cases. The correlator operators are given by (rt) (r) ′(r) (t) Cˆkα = (ˆ v1 − vˆ1 ) ⊗ vˆ2 .. (2.39). We propose the following designation as the kernel of the second specification of correlation: (1) . ′(1) . (2) (v1 = k − α, v1 = k + α + 1, v2 = k),. . (2) . (1) (v1 = k − α − 1, v1′ (2) = k + α, v2 = k),. (h) . ′(h) . (h) (v1 = k + α, v1 = k − α − 1, v2 = k),. (2.40). for h = 1, 2, k = 0, 1, ..., d − 1, and α = 0, ..., ⌊d/2 − 1⌋.

(62)

(63) E D

(64) (h) (h)

(65) (h) v The sets of projectors {ˆ vj =

(66) vj j

(67) } for the first qudit and the second (h) (h) Vj Vj

(68) E

(69) (h) }, where one are defined by two specific sets of complete orthonormal basis {

(70) vj (h) Vj.

(71) E

(72) (h)

(73) v1.

(74) E

(75) (h)

(76) v2. d−1. (h). V1. 1X 2πv (h) (h) = exp[i (v1 + n1 )] |vi , d v=0 d d−1. (h). V2. 2πv 1X (h) (h) exp[i (−v2 + n2 )] |vi , = d v=0 d. (2.41). (h). and the set of local parameters {nj } chosen is the same as the one used in the first specification. We have the correlators (rt). hCˆkα i =. 1 (1 + 4α)π (3 + 4α)π {csc2 [ ] − csc2 [ ]}, 3 2d 4d 4d. (2.42). which are positive for all k’s and α’s considered. For a given α, we have one necessary. 26.

(77) CHAPTER 2. ENTANGLEMENT AND CORRELATION CONDITIONS condition of the generalized Bell state. Thus one can feature the quantum correlations embedded in the bipartite d-level Bell state in 4⌊d/2⌋ sets of correlators and hence have 4⌊d/2⌋ necessary conditions. (r). Specification 3. There is a specific relation between the projector vˆ1. ′(r). and vˆ1. via. the variable α introduced in the second specification. Then it is natural to consider a generalization about sets of correlators containing two variables. For this motivation, we introduce correlators of the form (r) (rt) ˆ ) ⊗ vˆ(t) , Cˆkηµ = (ˆ v1 − v ′(r) 1 2. (2.43). where (r) . (r) . (t) (v1 = η − k, v1′ = µ − k, v2 = k),. (2.44) (h). for k = 0, 1, ..., d − 1, and η and µ are introduced variables. Let the projectors {ˆ vj } are the same as the one introduced in the first specification, refer to Eq. (2.35) and definitions (r). (t). before which. Then the probability for getting a set of result (v1 , v2 ) is (r). (r). hˆ v1 ⊗ vˆ1 i =. 1 sin2 [π(v (rt) + n(rt) )] . d3 sin2 [ πd (v (rt) + n(rt) )]. (2.45). From which we have the correlators: (rt) hCˆkηµ i = hˆ η ⊗ ˆ0i − hˆ µ ⊗ ˆ0i,. (2.46) (rt). for all k’s, and for each local measurement setting chosen, all of the correlators hCˆkηµ i satisfy either of the conditions: (rt). (2.47). (rt). (2.48). hCˆkηµ i > 0, for k = 0, 1, ..., d − 1, hCˆkηµ i < 0, for k = 0, 1, ..., d − 1,. 27.

(78) CHAPTER 2. ENTANGLEMENT AND CORRELATION CONDITIONS i.e., the criteria of dependence. Specification 4. In the previous three specifications, we use four local measurement settings to investigate correlations. Whereas we will give correlators under two measurement settings in this specification. For the first setting, our operators of correlators are of the form: (q) ˆ ⊗ k, ˆ Cˆk = [kˆ − T (k)]. (2.49). (q) for k = 0, 1, ..., d − 1 and q = 1, ..., γd, where kˆ = |ki hk|. The above formulation of {Cˆk }. follows the same definition as the one introduced in the second theorem, and note that ˆ = kˆ to the present specification. we have applied U(k) For the second measurement setting, we choose a specific complete set of orthonormal basis vectors {|vj iFj } where d−1. |vj iFj. 1X 2πv = exp(−i vj ) |vi , d v=0 d. (2.50). to represent correlator operators, and then we give the following operators of correlators (q) ˆ ⊗ vˆ2 , CˆkF = [kˆ − T (k)]. (2.51). . for k = 0, 1, ..., d − 1 and q = 1, ..., γd, where vˆj = |vj iFj Fj hvj | and v2 + k = 0. By a direct calculation, one has the correlators 1 (q) (q) hCˆk i = hCˆkF i = , d. (2.52). for all k’s and q’s, which satisfies the condition of dependence.. 28.

(79) CHAPTER 2. ENTANGLEMENT AND CORRELATION CONDITIONS. 2.4. Correlators embedded in Bell inequalities. Entanglement manifests itself via correlations in different directions of measurements. In the previous section, we feature quantum correlations of the genuine many-qubit GHZ states and the two-qudit Bell state in the correlators under different local measurement settings. These correlators can be considered as necessary conditions of the states under study. In addition to the CHSH inequality discussed in Sec. 2.2, further, we will reveal that four Bell inequalities that have been presented are composed of correlators.. 2.4.1. Bell inequalities for many qubits. Seevinck-Svetlichny inequality In the first specification for the N-qubit GHZ state, we give 2N sets of correlators corresponding 2N local measurement settings to describe the correlation structure of a GHZ state. It is worth noting that each set of correlator (Cˆ0l , Cˆ1l ) provides information about correlations between any two subsystems of the N-qubit state and give nc sets of necessary conditions of a generalized GHZ state. Each correlator has the same value whatever a ˆ partition is chosen, i.e., hCˆ0l[m,m] ¯ i = hC0l[m,m] ¯ i =. 1 √ , 2 2. for l = 1, ..., 2N and for all different. ¯ , which describes the properties of genuine multipartite partitions involved in {[m, m]} entanglement. A N-qubit genuine multipartite entanglement cannot be created without participation of all of the N particles. It is an interesting question whether one can use these correlators associated characters of many-qubit correlation to rule out correlations predicted by local-realistic theories. Using a linear combination of the correlators. CΦ1 =. X l. hCˆ0l i + hCˆ1l i. (2.53). could be a means of the identification of a N-qubit GHZ state, which helps to approach the question mentioned above. Recently, Seevinck and Svetlichny [15] introduce a new. 29.