Two Important Problems on Quantum Coherence
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Two Important Problems on Quantum Coherence Thesis submitted in partial fulfillment of the requirements for the degree of B. Tech and Master of Science by Research in Computer Science and Engineering by Udit Kamal Sharma 201202094 [email protected] Center for Security, Theory and Algorithmic Research (CSTAR) International Institute of Information Technology Hyderabad - 500 032, INDIA December 2017 Copyright © Udit Kamal Sharma, 2017 All Rights Reserved International Institute of Information Technology Hyderabad, India CERTIFICATE OF AUTHORSHIP I, Udit Kamal Sharma, declare that the thesis, titled “Two Important Problems on Quantum Coher- ence”, and the work presented herein are my own. I confirm that this work was done wholly or mainly while in candidature for a research degree at IIIT-Hyderabad. Date Signature of the Candidate International Institute of Information Technology Hyderabad, India CERTIFICATE It is certified that the work contained in this thesis, titled “Two Important Problems on Quantum Coherence ” by Udit Kamal Sharma, has been carried out under my supervision and is not submitted elsewhere for a degree. Date Adviser: Dr. Indranil Chakrabarty Acknowledgments I would like to thank my supervisor Dr. Indranil Chakrabarty, for his immense guidance, patience and helpful suggestions, without which, my journey in the uncharted waters of research would have been impossible. I would like to thank him for first introducing me to this field of quantum information theory and motivating me thereafter to pursue research in this direction. I would like to thank all the faculty and non-teaching staff at CSTAR for providing me an excellent convivial environment to pursue my research. I would like to express my gratitude to the members of my research group - Sourav Chatterjee, Palash Pandya, Maharshi Ray, Gaurav Singh, Manish Shukla, Aditya Jain, Luv Agarwal and Dhrumil Patel. The weekly group discussions were extremely helpful in clarifying my doubts during my incipient days of being a neophyte in this field and further provided me insight every time I found myself in a fix. Furthermore, I would also like to thank Prof. G.P. Kar and his amazing research group at ISI Kolkata for the enlightening and untiring discussions in the summer of 2015. I am grateful to all my wing-mates, especially Aabhas Majumdar, Yogesh Maheshwari, Mihir Wad- wekar, Vrushank Vyas and Gaurav Mishra, for having lots of discussions on a plethora of topics and for motivating me throughout, even in the moments when I felt most dejected. Finally, I would like to thank my family, for their unconditional and unending love, support and encouragement. v Abstract Quantum information theory has emerged as one of the new frontiers of science and technology over the last few decades. Unlike classical information theory that deals with the classical bit, quantum information theory is centred around the concept of a qubit. In contrast to the classical bit that can either be a 0 or 1, the qubit exists in a superposition of both states at the same time. The importance of quantum resources is evident from the fact that are certain information processing tasks like telepor- tation which cannot be accomplished with only classical resources and even though there are certain tasks like secret sharing which have classical counterparts, their outputs are greatly enhance when the classical resources are combined with quantum resources. While quantum superposition and entan- glement are well-established quantum resources, recently, significant research has been carried out to explore the possibility of quantum coherence as a resource for information processing applications. This thesis is divided into four chapters. While the first two chapters deal with the fundamentals of quantum information theory and quantum coherence, the last two chapters primarily deal with the two research problems on quantum coherence. The first chapter serves as a revision of the important introductory concepts of quantum information theory, starting with the significance of the famous Stern-Gerlach experiment and the rise of ran- domness to mathematical formalism and postulates of quantum mechanics. Then, we explore the connection between randomness and lack of information and further explore how classical informa- tion is different from quantum information alongwith a brief overview of its corresponding measures. Finally, the chapter ends with a layout of widely-used quantum resources alongwith its utility in some of the famous information processing tasks. The second chapter mainly deals with a brief synopsis on quantum coherence. At first, it discusses the formulation of resource theory of quantum coherence, which is followed by an introduction to the widely accepted framework of quantifying coherence and some of the popularly used coherence quantifiers. Finally, the chapter concludes with the application of coherence in the fields of quantum thermodynamics, quantum algorithms, interference phenomena and phase discrimination. The third chapter addresses the first research topic of the thesis : broadcasting of quantum coher- ence. As quantum coherence has recently emerged as a key candidate for use as a resource in various vi vii quantum information processing tasks, hence, it is of utmost importance to explore the possibility of creating a greater number of coherent states from an existing coherent pair. This process is known as broadcasting of coherence. This chapter starts with a brief discussion on the famous no-cloning theorem, followed by an overview of approximate cloning transformations. After that, the motivation behind broadcasting of coherence is explored along with an introduction to the definitions of optimal and non-optimal broadcasting. In both these definitions, the most general two-qubit state is taken as the input to the cloner, while the most incoherent states are used as blank states of the cloner. This chapter leads to two important results. Firstly, it has been proved that while optimal broadcasting is not possible (for both local and non-local cloners), non-optimal broadcasting cannot be ruled out. Secondly, in case of non-optimal broadcasting, the coherence introduced in the output states of the cloner will always be lesser than the initial coherence of the input state. Finally, three classes of mixed states, namely, the statistical mixture of the most coherent state (MCS) and the most incoher- ent state (MIS), the Werner-like states (WLS) and the Bell-diagonal states (BDS), are taken up for obtaining their respective ranges of non-optimal broadcasting in terms of their corresponding input state parameters. The fourth and final chapter deals with the second research topic of the thesis : some peculiar prop- erties of robustness of coherence (ROC). Here, it has been shown that robustness of coherence, in contrast to many popular quantitative measures of quantum coherence derived from the resource the- oretic framework of coherence, may be sub-additive for a specific class of multi-partite quantum states. Furthermore, this chapter also highlights how the sub-additivity is affected by admixture with other classes of states for which ROC is super-additive. Moreover, it has been shown that pairs of quantum states may have different orderings with respect to relative entropy of coherence, l1-norm of coherence and ROC and the difference in ordering for coherence measures chosen pairwise has been numerically studied. Contents Chapter Page 1 Introduction to Quantum Information Theory ::::::::::::::::::::::: 1 1.1 Stern Gerlach experiment . 1 1.2 Rise of randomness . 3 1.3 Mathematical formalism of quantum mechanics . 6 1.3.1 Hilbert space . 6 1.3.2 Linear operators . 8 1.3.3 Dirac notation . 9 1.3.4 Outer product notation . 9 1.3.5 Tensor products . 9 1.4 Postulates of quantum mechanics . 10 1.5 Preliminaries of quantum information theory . 12 1.5.1 Density matrix representation of a quantum state . 12 1.5.2 Bloch sphere representaion of a general state in C2 . 15 1.5.3 Quantum measurements . 17 1.6 Quantifying information . 18 1.6.1 Classical information . 19 1.6.2 Quantum information . 21 1.7 Quantum information processing . 23 1.7.1 Resources . 23 1.7.2 Applications . 25 2 Quantum Coherence : A Brief Outlook :::::::::::::::::::::::::: 28 2.1 Resource Theory of quantum coherence . 28 2.1.1 Constraints and operations . 29 2.1.2 Coherence as a resource . 31 2.1.3 Coherence in distributed scenarios . 32 2.2 Framework for quantifying quantum coherence . 33 2.2.1 Distance-based quantifiers of coherence . 34 2.2.2 Distillable coherence and coherence cost . 37 2.2.3 Convex roof quantifiers of coherence . 38 2.2.4 Robustness of coherence . 38 2.2.5 Measuring Quantum Coherence with Entanglement . 39 2.3 Applications of quantum coherence . 40 2.3.1 Quantum thermodynamics . 41 viii CONTENTS ix 2.3.2 Quantum algorithms . 43 2.3.3 Interference phenomena . 45 2.3.4 Quantum phase discrimination . 45 3 Broadcasting of Quantum Coherence via Cloning ::::::::::::::::::::: 47 3.1 No-cloning theorem . 47 3.2 Approximate cloning transformations beyond the No-cloning theorem . 49 3.2.1 State independent cloning transformations . 49 3.2.1.1 Local state independent cloning machine . 50 3.2.1.2 Non local state independent cloning machine . 50 3.3 Broadcasting of entanglement and correlations . 50 3.4 Broadcasting of coherence . 52 3.4.1 Motivation behind broadcasting of coherence . 52 3.4.2 Optimal broadcasting of coherence . 53 3.4.2.1 Local optimal broadcasting . 53 3.4.2.2 Non-local optimal broadcasting . 54 3.4.3 Impossibility of optimal broadcasting . 55 3.4.4 Non-optimal broadcasting of coherence . 58 3.4.4.1 Local non-optimal broadcasting . 58 3.4.4.2 Non-local non-optimal broadcasting . 58 3.4.5 Theorems in non-optimal broadcasting . 58 3.5 Non-optimal broadcasting for particular mixed states . 62 3.5.1 Mixture of MIS and MCS . 62 3.5.2 Werner-like states (WLS) .