MASARYK UNIVERSITY FACULTY}w¡¢£¤¥¦§¨ OF I !"#$%&'()+,-./012345<yA|NFORMATICS Quantum Entanglement MASTER’S THESIS Boris Ranto Brno, 2013 Declaration Hereby I declare, that this paper is my original authorial work, which I have worked out by my own. All sources, references and literature used or excerpted during elaboration of this work are properly cited and listed in complete reference to the due source. Boris Ranto Advisor: Colin Wilmott, M.Sc., Ph.D., prof. RNDr. Jozef Gruska, DrSc. ii Acknowledgement I would like to thank Colin Wilmott and Jozef Gruska for all the sup- port and advice with which they provided me throughout the thesis. iii Abstract The core element of all the phenomena presented in this thesis is quantum entanglement. In order to present and outline the phenom- ena related to the quantum entanglement, the mathematical frame- work of quantum mechanics is presented at the beginning of this thesis. Afterwards, the idea of quantum entanglement along with the uniqueness of this resource and brief application of this resource are outlined. The study of entanglement of two, three and n parties with respect to the tasks of entanglement classification, entanglement de- tection and entanglement distillation follows in order to explore the structure of entanglement. iv Keywords Quantum mechanics, multipartite entanglement, entanglement the- ory, classification of entangled states, detection of entanglement, dis- tillation of entanglement v Contents 1 Introduction & Preliminaries ..................1 1.1 History ............................2 1.2 Hilbert spaces ........................5 1.3 Pure quantum states ....................9 1.4 Products & Operators .................... 11 1.5 Measurements ........................ 15 1.6 Postulates of quantum mechanics ............. 18 2 Introduction to Entanglement ................. 20 2.1 Entanglement of pure states ................ 21 2.2 Schmidt decomposition .................. 22 2.3 Entanglement of mixed states ............... 25 2.4 Bell inequalities ....................... 29 2.5 Properties of entanglement ................ 31 3 Bipartite entanglement ..................... 37 3.1 Classification ......................... 38 3.2 Detection ........................... 40 3.3 Distillation .......................... 49 4 Tripartite entanglement ..................... 53 4.1 Classification ......................... 54 4.2 Detection ........................... 60 4.3 Distillation .......................... 63 5 Multipartite entanglement ................... 66 5.1 Classification ......................... 68 5.2 Detection ........................... 72 5.3 Distillation .......................... 75 6 Conclusion ............................. 77 vi 1 Introduction & Preliminaries This first chapter of thesis covers the very basic principles of quan- tum mechanics and will serve as an introduction to the topics that will be covered in this thesis. The aim of this chapter is to motiv- ate the study of quantum mechanics for use in quantum information processing and provide reasonable background for understanding of the phenomena that is covered by this thesis. The second chapter of this thesis will serve as an introduction to quantum entanglement that is one of the most important and inter- esting phenomena of quantum mechanics. The chapter also provides a discussion that suggests that thanks to quantum entanglement, the theory of quantum mechanics may not be superseded by a more standard and intuitive physical theory. At the end of the chapter a very basic entanglement manipulation protocols will be presented in order to demonstrate the power as well as a deficiency of quantum entanglement. Afterwards, in three consecutive chapters, three very important tasks concerning the entanglement will be discussed for quantum systems of consecutively increasing complexity. These three tasks in- clude the classification of entanglement, detection of entanglement and distillation of entanglement. Each of these tasks provides a dif- ferent level of insight to the theory of quantum mechanics and in par- ticular quantum entanglement. The task of entanglement classifica- tion provides an insight to the very basic structure of quantum entan- glement. The remaining two tasks further refine the understanding of the structure of entanglement with concern to the applications of quantum entanglement that are more practical in their nature. Fur- thermore, the task of entanglement detection focuses on locally pre- pared quantum systems while the task of entanglement distillation is of a greater importance for preparation of quantum systems amongst remote parties. However, before any quantum formalism will be presented in this thesis, a brief history of the field of quantum mechanics will be discussed in order to provide important insights to the theory of quantum mechanics and further motivate the study of this intriguing theory. 1 1. INTRODUCTION &PRELIMINARIES 1.1 History The field of quantum mechanics started as early as the middle of the nineteenth century when physicist Gustav Kirchhoff showed that the emitted energy of a blackbody depends solely on the temperature and the frequency of the emitted energy. The blackbody is a concept in physics and is known as both a perfect absorber and a perfect emitter in the sense that it absorbs or emits all the energy. Kirch- hoff demonstrated that the emitted energy is dependent on these two quantities however, he was not successful in correctly formulating the blackbody energy emission rate. Hence, he set a challenge to all physicists to find the correct formulae for the emitted blackbody en- ergy. Over the coming years there were several attempts to tackle the blackbody problem, many of which turned out to be incomplete, un- til the problem was finally solved by Max Planck in 1900. Amazingly, as soon as Planck discovered the problem it was only a matter of hours before he derived the correct formulae for blackbody emission. While his deduction was triggered by an ingenious conjecture, he remained unsatisfied with his work and sought out an exact reason that would justify in theoretical terms the reason for his conjecture. In his workings, Planck made important assumption that energy is not continuous but rather discrete quantity. In particular, Planck stated that energy consists of quanta of energy. Initially, Planck’s explanation was not taken very seriously within physics. However, there was a physicist that became interested in the Planck’s theory about the discreteness of energy and he went by name Albert Einstein. Einstein used Planck’s idea to derive his sem- inal work on discrete properties of light, and in 1905, Einstein made very important realisation – if the light behaves as if it was made out of a quanta then perhaps it is also emitted and absorbed in that way. Einstein used this realisation to describe the photoelectric effect. This quanta of the light – the light particle – is now known as photon. The main problem with his theory was that the theory of classical electro- dynamics postulated that the light is a wave. It was counterintuitive to think that something can be both a particle and a wave. This led to the phenomenon that is now known as the wave-particle duality of light. 2 1. INTRODUCTION &PRELIMINARIES While Einstein’s explanation of the photoelectric effect provided for testable predictions, it took ten years to design an experiment that could show that his predictions were correct and hence, in the mean- time his quantisation did not receive widespread support. However, Einstein found another application for quantisation within the the- ory of specific heats1. This use of the quantisation was soon suppor- ted by experiments and brought a broader audience to the quant- isation debate. This eventually resulted in the Solvay Congress in 1911 where similar ideas were mooted. Here, as the second youngest physicist at the conference, Einstein gave a talk on the specific heats and offered his insights on electromagnetic radiation. The next stop for quantisation was the model of atom. Niels Bohr, in his doctoral studies, noticed inconsistencies in one of the theor- ies trying to describe the model of atom – Rutherford’s model. In Rutherford’s model, the electrons circle around a dense core. Bohr’s intuition led him to the idea of quantisation of the model. Bohr sug- gested that electrons circle the dense core in orbits and that for an electron to switch the orbits it must overcome a quanta of energy. Fol- lowing the Bohr’s result, Einstein successfully connected the Bohr’s model with blackbody radiation. In the same paper, Einstein also postulated that the transition between the orbits can not be determ- ined precisely. In particular, he stated that the transition can only be determined probabilistically. Bohr’s model of atom allowed for a much more general quantum theory. Unfortunately, this task was not as simple as one might wish. While some parts of the model could be described classically, some could not be described that way. Continuous work on the theories with conjunction with Bohr’s model resulted in the concepts known as the old quantum theory. The old quantum theory was simply a collection of results built around the Bohr’s model of atom. The col- lection of the results was quite successful in explaining many exper- iments but suffered from inconsistencies and incompleteness. There was not a single failure as such rather a collection of little missing doubts. Moreover, the number of these doubts was overwhelming. To overcome these obstacles and