Quantum Aspects of Cryptography: From Qutrit Symmetric Informationally Complete Projective Operator Valued Measure Key Encryption to Randomnes Quality Control

All digital information is in its most basic form represented through bits; units of information that can exist in two distinct states. Examples are two different states of an electrical switch or two different voltage or current levels allowed in a circuit. Today's most advanced technique for the representation and manipulation of bits is by means of micro-particles, ex. atoms, photons, etc. However, in this regime, the rules are determined by quantum mechanics and hence the basic units of information are called quantum bits or qubits. In this research area, computer science and physics are brought together to open a new way of computing, transmitting and processing information. In quantum cryptography for example, quantum key distribution (QKD) protocols make use of the quantum properties of micro-particles in order to exchange a secret key between two parties over a public channel. The impossibility to measure a quantum particle without disturbing its quantum state guarantees the safety of the key. Concretely, this means that a third party trying to eavesdrop the key must in some way measure the quantum stat of the transmitted particles and consequently introduce detectable anomalies. Quantum cryptographic networds typically make use of symmetric encryption devices with frequent key change. In this particular type of networks, the fresh key is constantly generated by QKD devices and the resulting functionality is comparable to existing classical secure networks but features a higher level of security. Quantum cryptography is already a reality in today's society: The DARPA {54} quantum network, a 10-node quantum cryptographic network, has been running since 2004 in Massachusetts, USA. Another computer network protected by quantum cryptography was implemented in Vienna, Austria and demonstrated during a scientific conference in October 2008. The latter network is called SECOQC {103} and interconnects six different types of QKD links across distant locations. Recently, in October 2010, a metropolitan network also combining six different QKD systems was developed and demonstrated in Tokyo, Japan {74}. Four companies worldwide are currently offering quantum cryptographic systems (Id Quantique, MagiQ Technologies, SmartQuantum, Quintessence Labs). Furthermore, in direct relation to cryptography, several quantum random number generators (QRNGs) have been developed (Quantis {104}, PicoQuant {73}, etc.) and have also found their niche in the industry. Both aspects of (quantum) cryptography; QKD and QRNGs, attract our particular attention in this dissertation. Our first topic of interest is quantum key distribution. When looking at existing QKD protocols, several qubit based protocols have already been developed in the past, among others the one established at the Quantum Lab, NUS in Singapore {55}. This protocol describes two users establishing a secret key by measuring the correlations between entangled qubits thanks to a Symmetric Informationally Complete Positive Operator Valued Measure (SIC POVM). This method minimizes the number of measurements to be made in order to obtain a given accuracy in the estimation of the density matrix of the signal. Such an optimal measurement is particularly important during the reconciliation step of a quantum cryptographic protocol. The signal-to noise ratio is estimated throughout the reconciliation and in typical QKD schemes, this is very data consuming. In comparison to other qubit QKD schemes, the Singapore protocol was shown to be less data consuming when considering the complete procedure, i.e. data exchange and post processing (including reconciliation). Besides, it has been shown tha