Quantum communication protocols invoke one of the most fundamentallaws of quantum mechanics, namely the superposition principle whichleads to the no-cloning theorem. During the last three decades, quantumcryptography have gone from prospective theories to practical implementationsscalable for real communication. Scientist from all over the world havecontributed to this major progress, starting from Stephen Wiesner, CharlesH. Bennett and Gilles Brassard who all developed the theory of QuantumKey Distribution (QKD). QKD lets two users share a key through a quantumchannel (free space or fiber link) under unconditionally secure circumstances.They can use this key to encode a message which they thereaftershare through a public channel (internet, telephone,...). Research developmentshave gone from the ordinary 2-User Quantum Key Distribution oververy small free space distances to distances over 200 km in optical fiber andQuantum Key Distribution Networks.As great experimental achievements have been made regarding QKDprotocols, a new quantum communication protocol have been developed,namely Quantum Secret Sharing. Quantum Secret Sharing is an extensionof an old cryptography scheme called Secret Sharing. The aim of secretsharing is to split a secret amongst a set of users in such a way that thesecret is only revealed if every user of this set is ready to collaborate andshare their part of the secret with other users.We have developed a 5-User QKD Network through birefringent singlemode fiber in two configurations. One being a Tree configuration and theother being a Star configuration. In both cases, the number of users, thedistances between them and the stability of our setup are all well competitivewith the current worldwide research involving similar work.We have also developed a Single Qubit Quantum Secret Sharing schemewith phase encoding through single mode fiber with 3, 4 and 5 parties. Thelatter is, to the best of our knowledge, the first time a 5-Party Single QubitQuantum Secret Sharing experiment has been realized.

Three-dimensional topological insulators are materials that have a bulk band gap like a traditional insulator, but which hold topologically protected conducting surface states. In this thesis we present a numerical analysis of the surface states of topological insulator nanowires in the tight-binding approximation. We carry out the calculations at zero temperature under the presence of coaxial and perpendicular magnetic fields using Dirac Hamiltonians to model the surface. The results are obtained using Kwant, a Python package first developed in 2014 by Groth et al. for the purpose of aiding in the creation of quantum transport simulations in tight-binding models. The main focus is the self-contained and complete study of the behaviour of the conductance in clean and disordered systems, as well as to serve as an introduction to Kwant. We first study the main properties of quantum transport in mesoscopic systems, and present the scattering problem in the tight-binding approximation, which is the one treated in Kwant. We review the main properties of topological insulators, as well as the history of their discovery. We then present Kwant in detail, and illustrate its inner workings by considering the example of a clean wire. We study clean wires and show the existence of the perfectly transmitted mode under a coaxial magnetic field, obtain the quantisation of the conductance expected from the Laundauer-Büttiker formalism, and recover Fabry-Pérot oscillations when considering highly doped leads. We discuss how disorder can be introduced in our systems to simulate more realistic models, analyse its effects in the period of the conductance oscillations, and recover the robustness to disorder of the perfectly transmitted mode. Finally, we comment on how this thesis can be expanded to cover a wider range of systems and phenomena. / Tredimensionella topologiska isolatorer är material som har ett bulkbandgap som traditionella isolatorer, men som har topologiskt skyddade ledande yttilstånd. I detta arbete presenterar vi en numerisk analys av yttilstånden hos topologiska isolator nanotrådar i tight-binding approximationen vid nolltemperatur, under närvaron av koaxiala och vinkelräta magnetfält med användning av Dirac-Hamiltonians för att modellera ytan. Resultaten erhålls med hjälp av Kwant, ett Python-paket som först utvecklades 2014 av Groth et al. i syfte att underlätta skapandet av simuleringar för kvanttransport i tight-binding modeller. Huvudfokus ligger på en självständig och komplett studie av beteendet hos konduktansen i rena och oordnade system, samt att fungera som en introduktion till Kwant. Vi studerar först de huvudsakliga egenskaperna hos kvanttransport i mesoskopiska system och presenterar spridningsproblemet i tight-binding approximationen, vilket är det som behandlas i Kwant. Dessutom går vi igenom de viktigaste egenskaperna hos topologiska isolatorer, samt deras upptäckthistoria. Sedan pre- senterar vi Kwant i detalj och illustrerar dess inre funktioner genom att titta på en ren tråd. Vi studerar rena trådar och visar förekomsten av det perfekt överförda läget under ett koaxialt magnetfält, erhåller kvantiseringen av den förväntade konduktansen från Laundauer-Büttiker-formalismen och återfår Fabry-Pérot-oscillationer när vi överväger starkt dopade ledare. Sedan diskuterar vi hur oordning kan införas i våra system för att simulera mer realistiska modeller, analysera dess effekter under tiden för oscillationer vid konduktans och återfå robustheten mot oordning av det perfekt överförda läget. Slutligen kommenterar vi hur detta arbete kan utvidgas för att täcka ett bredare spektrum av system och fenomen.

Kvantsammanflätning har varit ett populärt ämne bland fysiker i snart 100 år då det tydligt belyser hur annorlunda kvantmekanikens värld är jämfört med den klassiska verklighet vi lever i. Med tiden har kvantsammanflätning blivit mer och mer välförstått och teknologier ämnade att utnyttja det har de senaste årtionden kommit allt närmare till industriell använding. Kvantdatorer är fortfarande i forskningsstadiet men idag excisterar det en kvantdator som kan lösa vissa problem betydligt mycket snabbare än en klassisk dator. På grund av algorithmer som Shors faktoriseringsalgoritm och Grovers sökalgoritm så riskerar dagens krypteringsprotokoll för kommunikation att bli otillräckliga. Som svar på detta har en fysikalisk icke-hackbar krypterings metodik tagits fram i form av QKD. Det baseras på att generara krypteringsnycklar från slumptal och att dessa distribueras tack vare kvantsammanflätning. För att lyckas med detta så krävs generering av sammanflätade kvanttillstånd, kvantbitar, samt singel-fotonsdetektorer. I den här masteruppsatsen har en kvantprick karaktäriserats och används för att generera sammanflätade kvantbitar i QNP-gruppens lab på KTH samt för att skicka enstaka fotoner via Stockholms fibernät till Ericsson i Kista där de detekteras av singel foton detectorer. Multifoton sannolikheten har uppmäts till 0.049 för exciton fotoner samt 0.169 för biexciton fotoner i labbet medan ett värde på 0.176 har uppmäts för exciton fotoner detekterade hos Ericsson, vilket är betydligt lägre än singel emission gränsen 0.5 (dvs foton källan sänder ut singel fotoner). Synkronisering av data är avgörande för att få QKD att fungera varpå en post process-tidssynkroniserings metod baserad på biexciton-exciton kaskad-sönderfall har implementerats i lab. / Quantum entanglement has been a popular topic amongst physicists for almost 100 years as it clearly illuminates the extreme difference between the quantum mechanical world and our classical reality. Over time, the quantum physical property of entanglement became more and more well understood and technologies utilizing entanglement are coming closer to reach industry. Quantum computers are still in the research stage but there already exists a quantum computer capable of solving tailored problems significantly faster than a classical computer. Due to algorithms like Shor’s factorization algorithm and Grover’s search algorithm the current cryptography schemes used to ensure secure communication risk rendering obsolete. A response to this was the invention of the theoretically un­hackable Quantum key Distribution (QKD) scheme, based on generating and distributing random cryptography keys by using quantum entanglement. To achieve this, the generation of entangled photons, or qubits, as well as detection of single photons is required. In this thesis a Quantum Dot (QD) is characterized and used to generate quantum entangled states in the Quantum Nano Photonics (QNP)­group lab at KTH as well as sending single photons via the metropolitan fiber network in Stockholm to Ericsson in Kista, where they are detected using single photon detectors. A multi­photon emission probability of 0.049 was measured for the exciton emission and 0.169 for the biexciton emission in the KTH lab as well as a probability of 0.176 was measured for the exciton photons sent to Kista which is significantly lower than the single emitter limit of 0.5 (i.e. the source is emitting pure single photons). Synchronization of data is of high importance in order to implement a working QKD scheme, therefore a post process temporal synchronization method based on the biexciton­exciton cascaded decay is implemented in the lab.

Qubits (quantum bits) are what runs quantum computers, like a bit in classical computers. Quantum gates are used to operate on qubits in order to change their states. As such they are what ”programmes” a quantum computer. An unfortunate side effect of quantum physics is that coupling a quantum system (like our qubits) to an outside environment will lead to a certain loss of information. Reducing this decoherence effect is thus vital for the function of a quantum computer. Geometric quantum computation is a method for creating error robust quantum gates by using so called geometric phases which are solely reliant on the geometry of the evolution of the system. The purpose of this project has been to develop physical schemes of geometric entangling two-qubit gates along the Schmidt sphere, a geometric construct appearing in two-qubit systems. Essentially the overall aim has been to develop new schemes for implementing robust entangling quantum gates solely by means of interactions intrinsic to the computational systems. In order to create this gate four mutually orthogonal states were defined which together spanned the two-qubit state space. Two of the states were given time dependent variables containing a total of two angles,which were used to parameterize the Schmidt sphere. By designing an evolution for these angles that traced out a cyclical evolution along geodesic lines a quantum gate with exclusively geometric phases could be created. This gate was dubbed the ”Schmidt gate” and could be shown to be entangling by analyzing a change in the concurrence of a two qubit system. Two Hamiltonians were also defined which when acted upon the predefined system of states would give rise to the aforementioned evolution on the Schmidt sphere. The project was successful in creating an entangling quantum gate which could be shown by looking at difference in the concurrence of the input and output state of a two-qubit system passing through the gate.