EXPERIMENTAL ENTANGLEMENT WITNESS FAMILY MEASUREMENT AND THEORETICAL ASPECTS OF QUANTUM TOMOGRAPHY

Experimental Entanglement Witness Family Measurement Experimental Entanglement Witness Family Measurement Experimental Entanglement Witness Family Measurement and blahakldasldjl;aklaa;klas Dai Jibo Experimental Entanglement Witness Family Measurement Experimental Entanglement Witness Family Measurement Experimental Entanglement Witness Family Measurement Experimental Entanglement Witness Family Measurement Experimental Entanglement Witness Family Measurement Experimental Entanglement Witness Family Measurement 2015 Thanks Berge! Experimental Entanglement Witness Family Measurement and blahakldasldjl;aklaa;klas DAI JIBO B. Sc. (Hons.) and B. Eng. (Hons.), National University of Singapore SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DRGREE OF DOCTOR OF PHILOSOPHY Centre for Quantum Technologies 2015 Thanks Berge! DECLARATION DECLARATION DECLARATION I hereby declare that this thesis is my original work and it has been written by me in its entirety. I have duly acknowledged all the sources of information which have been used in the thesis. a This thesis has also not been submitted for any degree in any university previously. Dai Jibo 15 April 2015 Thanks Berge! Acknowledgments First and foremost, I would like to thank my supervisor Prof. Berthold-Georg En- glert for accepting me as a Ph. D. student at the Centre for Quantum Technolo- gies (CQT), and tirelessly guiding and supporting me throughout my candidature. Nothing in this thesis would be possible without you firstly providing me with the opportunity to learn and work under your supervision. I am deeply grateful for your trust in me that is always the source of motivation for me in face of obstacles. Thank you so much for the time you spend on me, for the invaluable guidance on my Ph. D. project, insights in physics, and wisdom of life that you have shared with me, which have helped me tremendously along the way and encouraged me never to give up. You are the role model that I deeply admire and will always try to follow. I’d like to thank Prof. Feng Yuan Ping and Prof. Gong Jiangbin for willingly writing the recommendation letters for me and encouraging me to apply for Ph. D. studies in CQT. Thank you also for teaching me in my undergraduate studies and guiding me in my final year project, for which the good time that I had while studying physics as an undergraduate student is one of the reasons that urged me to pursue further studies in physics eventually. And I want to thank Prof. Kwek Leong Chuan for interviewing me and recommending me into the CQT Ph. D. pro- gram. I also wish to thank Prof. Christian Kurtsiefer and Prof. Gong Jiangbin for taking time out of their busy schedule to serve in my thesis advisory committee, and providing me with help and support when I need them. For the experimental part of the thesis, I am very much indebted to Dr. Leonid Krivitsky at the Institute of Data Storage (DSI), A*STAR. Thank you for your patience in teaching me and for all the wonderful experimental techniques that you shared with me. I am greatly thankful for the time that you spend with me in the II Acknowledgments lab, as well as during our numerous enjoyable discussions. Thank you for never giv- ing up on me, and persistently teaching me new experimental skills. The experience of being a member in your team is truly memorable and enriching. I would also like to thank Asst/Prof. Hui Khoon Ng for guiding me on a sub- stantial part of the thesis. Thank you for the numerous help you provided me with and for the many effective discussions we had. Thank you also for the opportunities that you entrusted on me, to supervise the high school students on science projects. The experience of it is both enjoyable as well as rewarding. Moreover, thank you for critically reading this thesis and giving me many invaluable comments and sug- gestions on how to improve it. Throughout my Ph. D. studies, I also received a lot of help from my colleagues in CQT. A special thanks to Dr. Teo Yong Siah, for being always supportive and helpful when I am faced with difficulties. I wish to extend my sincere appreciation to all my colleagues who helped me in one way or another. While it may not be possible to name all of them, I would like to thank Dr. Shang Jiangwei, Dr. Zhu Huangjun, Li Xikun, Max Seah Yi-Lin, Tan Wei Hou, Dr. Han Rui and so on. I would also like to thank Dr. Dmitry Kalashnikov in DSI for sharing with me his experimental expertise. I’d like to especially acknowledge Len Yink Loong, who worked together with me on most of the project in this thesis. Thank you for spending time with me on the experiment in the dark lab. Thank you for helping me with loads of checking and calculations. Thank you for many of the lunch-time talks and discussions that are both enlightening and thought-provoking. I would like to acknowledge the financial support from Centre for Quantum Tech- nologies, a Research Centre of Excellence funded by the Ministry of Education and the National Research Foundation of Singapore. I am grateful to all the administra- tive staff at CQT for providing numerous timely help and a favorable environment at CQT where I can learn and work comfortably. Last but not least, I wish to thank again all the professors, my colleagues, my friends and my family, for your help given to me, care and love shone on me have always been the source of inspiration for me. Thank you. Abstract and Summary Quantum state tomography is a central and recurring theme in quantum information science and quantum computation. In a typical scenario, a source emits a certain desired state which carries the information, or is required for the computational task. Quantum state tomography is needed for the verification and identification of the state emitted by the source. In the first part of the thesis, we focus on the efficient detection of entanglement, a key resource in many quantum information processing tasks. We report an experiment in which one determines, with least tomographic effort, whether an unknown two-photon polarization state is entangled or separable. The method measures whole families of optimal entanglement witnesses at once. We introduce adaptive measurement schemes that greatly speed up the entanglement detection. The witness family measurement enables informationally complete (IC) quantum state tomography if the individual family gives inconclusive results. On average, only about three families need to be measured before the entanglement is detected and the IC state tomography is hardly necessary. However, in a realistic experiment, not only the quantum state to be recon- structed, but additional parameters in the experimental setup are also unknown, for example, the efficiency of the detectors, the total number of copies emitted, etc. Furthermore, the assumption of a closed quantum system is also only an approxi- mation, and there are often the ignored bath degrees of freedom which interact with the system. The second part of the thesis aims at these aspects. For the former aspect, based on the idea of credible regions, we construct joint optimal error re- gions for the system state and the other unknown parameters. By marginalizing over the nuisance parameters, one can obtain a marginal likelihood which only de- pends on the parameter of interest. We illustrate the method and technique with IV Abstract and Summary several examples. Some of them display unusual features in the likelihood function. For the latter aspect, we show how one uses ideas from quantum tomography or state estimation to deduce a reasonable and consistent system-bath state. In typ- ical experimental situations, such a state turns out to be uncorrelated or almost uncorrelated between the system and the bath. a [...]... theoretical aspects: the issue of additional unknown parameters and that of coupling between the system and environment during quantum state tomography Below is a more detailed outline of this thesis In Chapter 2, we present a short review of quantum mechanics and basic ideas in quantum state tomography that are needed to follow this thesis For the short review on quantum mechanics, the polarization of light... 123 7 Conclusion and Outlook Bibliography 125 XI VIII Contents List of Figures 2.1 Geometry of states and entanglement witnesses: The set of separable state ρsep is convex, whereas the set of entangled state ρent is not An entanglement witness W defines a hyperplane in the state space which separates the separable states and a partial set of entangled states An optimal entanglement witness Wopt touches... turns out that most of these desired states possess entanglement, a key resource in many quantum information processing tasks Hence entanglement verification and detection is also of critical importance in quantum information science and quantum computation Entanglement witnesses have been introduced such that if we know the state, then we can choose a suitable witness to detect its entanglement However,... reconstruct the input state and then determine whether there is entanglement in the state or not Hence, in a certain sense, quantum state tomography is a central and recurring theme in quantum information science and quantum computation In a realistic quantum state tomography experiment, not only the quantum state to be reconstructed is unknown, but also some additional parameters in the experimental setup,... of the system This interaction is however often ignored when one describes such quantum state tomography experiments It is then the aim of this thesis to study some of the issues raised above In the first part of the thesis, we focus on state preparation and the efficient detection of entanglement, presenting two experiments that have been performed The second part of the thesis aims at tackling two theoretical. .. algorithm and other quantum computation tasks Entanglement plays a crucial role in quantum information protocols such as teleportation In all these examples, the quantum system is manipulated to perform certain information tasks or computations In a nutshell, quantum information and quantum computation are about finding ways of utilizing the quantum system in theory, and about gaining better and more... states of various purities and de- 5 grees of entanglement depending on the parameters of the VPRs The method can be easily implemented for various experiments that require the generation of states with controllable degrees of entanglement or mixedness As an application of the source described in Chapter 3, we discuss an experiment done on entanglement witnesses in Chapter 4 Besides testing the utility and. .. control and know the complete knowledge of the state, then everything about the system in the future is completely determined However, the development of quantum theory brings challenges to such a point of view: a fundamental feature of quantum theory is that it is probabilistic The complete knowledge of the state of the system now does not enable us to predict the outcomes of all possible measurements... efficiency of the detectors, the dark counts of the detectors, etc Furthermore, quantum systems tend to interact with the environ- 4 1 Introduction ment that is surrounding them, and then decohere The assumption that we have a closed quantum system is just an approximation, though often a good one Such interaction generally develops correlations between the system and environment, often in the form of entanglement, ... The collection of the red lines form the SCR for this set of data for cλ = 0.9, and the blue lines are for cλ = 0.5 5.3 81 Size (the blue curve) and credibility (the red curve) as functions of λ for the primitive prior for the regions in Fig 5.2 The experimenter interested in the SCR of his desired credibility c can determine the required value of λ and check if a given value of z and n0 is in the . Jibo Experimental Entanglement Witness Family Measurement Experimental Entanglement Witness Family Measurement Experimental Entanglement Witness Family Measurement Experimental Entanglement Witness. Witness Family Measurement Experimental Entanglement Witness Family Measurement Experimental Entanglement Witness Family Measurement 2015 Thanks Berge! Experimental Entanglement Witness Family Measurement. Experimental Entanglement Witness Family Measurement Experimental Entanglement Witness Family Measurement Experimental Entanglement Witness Family Measurement and blahakldasldjl;aklaa;klas Dai