Bell tests in physical systems Seung-Woo Lee St. Hugh's College, Oxford A thesis submitted to the Mathematical and Physical Sciences Division for the degree of Doctor of Philosophy in the University of Oxford Trinity Term, 2009 Atomic and Laser Physics, University of Oxford Bell tests in physical systems Seung-Woo Lee, St. Hugh's College, Oxford Trinity Term, 2009 Abstract Quantum non-locality and entanglement in realistic physical systems have been of great interest due to their importance, both for gaining a better understanding of quantum physical principles and for applications in quantum information process- ing. Both quantum non-locality and entanglement can be e®ectively detected by testing Bell inequalities. Thus, ¯nding Bell inequalities applicable to realistic phys- ical systems has been an important issue in recent years. However, there have been several conceptual di±culties in the generalisation of Bell inequalities from bipartite 2-dimensional cases to more complex cases, which give rise to many fundamental questions about the nature of quantum non-locality and entanglement. In this the- sis, we contribute to answering several fundamental questions by formulating new types of Bell inequalities and also by proposing a practical entanglement detection scheme that is applicable to any physical system. To start with, we formulate a generalised structure of Bell inequalities for bipar- tite arbitrary dimensional systems. The generalised structure can be represented either by correlation functions or by joint probabilities. We show that all previously known Bell inequalities can be written in the form of the generalised structure. Moreover, the generalised structure allows us to construct new Bell inequalities in a convenient way. Subsequently, based on this generalised structure, we derive a Bell inequality that ful¯lls two desirable properties for the study of high-dimensional quantum non-locality. The ¯rst property is the maximal violation of Bell inequalities by maximal entanglement which agrees with the intuition of \maximal violation of local i ii realism by maximal entanglement". The second property is that the Bell inequality written in correlation space should exactly represent a boundary between quantum mechanics and local realism. In contrast to any previously known Bell inequality, the derived Bell inequality is shown to satisfy both conditions. We apply this Bell inequality to continuous variable systems and demonstrate maximal violation by the maximally entangled state associated with position and momentum. We then formulate a generalised Bell inequality in terms of arbitrary quasi- probability functions in phase space formalism. This includes previous types of Bell inequalities formulated using the Q and Wigner functions as limiting cases. We show that the non-locality of a quantum system is not directly related to the negativity of its quasi-probability distribution beyond the previously known fact for the case of the Wigner function. We also show that the Bell inequality formulated using the Q-function permits the lowest detector e±ciencies out of the quasi-probability distributions considered. Finally, we present a general approach for witnessing entanglement in phase space by signi¯cantly ine±cient detectors. Its implementation does not require any additional process for correcting errors in contrast to previous proposals. Moreover, it allows detection of entanglement without full a priori knowledge of detection e±ciency. We show that entanglement in single photon entangled and two-mode squeezed vacuum states is detectable by means of tomography with detector e±- ciency as low as 40%. This approach enhances the possibility of witnessing entan- glement in various physical systems using current detection technologies. Contents Abstract i Chapter 1. Introduction 1 Chapter 2. Basic Concepts 9 2.1 High-dimensional quantum systems . 9 2.1.1 Quantum states in high-dimensional Hilbert space . 9 2.1.2 High-dimensional physical systems . 12 2.2 Bell's theorem . 14 2.2.1 Bell's theorem and Bell inequalities . 14 2.2.2 Bell test experiments . 18 2.2.3 Loopholes in Bell tests . 20 2.2.4 Polytope representation of Bell's theorem . 23 2.3 Quantum entanglement . 24 2.3.1 Entanglement . 24 2.3.2 Genuine entanglement . 26 2.3.3 Entanglement Witness . 27 2.4 Phase space representations . 29 2.4.1 Generalised quasi-probability functions . 29 2.4.2 Bell inequalities in phase space . 30 iii iv Contents Chapter 3. Generalised structure of Bell inequalities for arbitrary- dimensional systems 33 3.1 Introduction . 33 3.2 Generalised arbitrary dimensional Bell inequality . 35 3.3 Violation by Quantum Mechanics . 40 3.4 Tightness of Bell inequalities . 46 3.5 Remarks . 48 Chapter 4. Maximal violation of tight Bell inequalities for maximal entanglement 51 4.1 Introduction . 51 4.2 Optimal Bell inequalities . 53 4.3 Extension to continuous variable systems . 60 4.4 Conclusions . 63 Chapter 5. Testing quantum non-locality by generalised quasi-probability functions 65 5.1 Introduction . 65 5.2 Generalised Bell inequalities of quasi-probability functions . 68 5.3 Testing Quantum non-locality . 70 5.4 Violation by single photon entangled states . 74 5.5 Violation by two-mode squeezed states . 76 5.6 Discussion and Conclusions . 79 Chapter 6. Witnessing entanglement in phase space using ine±cient detectors 83 6.1 Introduction . 83 6.2 Observable associated with e±ciency . 85 Contents v 6.3 Entanglement witness in phase space . 86 6.4 Testing single photon entangled states . 88 6.5 Testing two-mode squeezed states . 90 6.6 Testing with a priori estimated e±ciency . 91 6.7 Conclusions . 93 Chapter 7. Conclusion 95 Bibliography 99 Chapter 1 Introduction Bell's theorem [1] has been called \the most profound discovery of science" [2] and also \one of the greatest discoveries of modern science" [3]. These words highlight the importance of the quantum features i.e. quantum non-locality and entanglement that were then found based on Bell's theorem. Quantum non-locality and entangle- ment show striking properties that can not be understood in the context of classical physics, and thus shift the paradigm of understanding fundamental principles in physical systems. Moreover, these quantum features are a promising resource for quantum information processing that is a revolutionary technology superior to clas- sical counterpart in various ways [4]. In these senses observing quantum non-locality and entanglement in physical systems can be seen as a surprising discovery. Historically, the concepts of quantum non-locality and entanglement appeared for the ¯rst time in the famous Einstein-Podolski-Rosen (EPR) paradox [5]1. This paradox was formulated with the theory of local realism in mind to prove the incom- pleteness of quantum mechanics. Later Bell showed in his theorem that local realism leads to constraints on correlations between measurement results carried out on two separated systems [1]. These constraints, known as Bell inequalities, provide a pos- sible method for testing the validity of quantum mechanics against local realism. It 1The concept of entanglement was also introduced by E. SchrÄodinger[6, 7]. 1 2 Introduction was theoretically shown that Bell inequalities are violated by the quantitative pre- dictions of quantum mechanics in the case of entangled states [1, 8]2. Subsequently, experiments have con¯rmed the validity of quantum mechanics by demonstrating the violation of Bell inequalities [9, 10]. In general, quantum non-locality and entan- glement manifest themselves in such a counterintuitive way, precisely through the violation of the local realism [9, 10] and by permitting stronger correlations beyond the level allowed in classical physics [11]. Quantum non-locality and entanglement in simple models such as bipartite or tripartite 2-dimensional systems are currently well understood [11]. However, little is known about them in realistic physical systems composed of many particles with many degrees of freedom, which we will call `complex systems' throughout this thesis. In fact most physical systems in nature are complex systems, and thus generalisations of Bell's theorem to complex systems have been regarded as one of the most important challenges in quantum mechanics [12, 13, 14, 15, 16, 17, 18, 19, 20, 11, 21, 22]. There are several motivations for studying Bell's theorem in complex systems. Firstly, generalising Bell's theorem to complex systems would provide a way to ob- serve quantum features in the macroscopic world. Macroscopic systems are complex systems that are generally governed by classical physics, and thus observing quan- tum properties in those systems seems to be di±cult. However, recent studies on e.g. the relation of entanglement with macroscopic observables [21] o®er possibili- ties to investigate quantum properties in macroscopic systems [23]. For example, Bell type inequalities constructed with macroscopic observables would allow one to detect quantum non-locality and entanglement in macroscopic physical systems. Secondly, Bell inequalities can help us to investigate quantum phenomena arising in realistic physical systems. For example, studying the role of entanglement in a 2Note that not all entangled states violate Bell inequalities. 3 quantum phase transition is one of the most interesting issues in recent relevant research [24]. In fact, quantum phase transition can be regarded as a change of the dominant degrees of freedom of a system. Thus Bell inequalities de¯ned in both degrees of freedom