This book treats the central physical concepts and mathematical techniques used to investigate the dynamics of open quantum systems. To provide a self-contained presentation, the text begins with a survey of classical probability theory and with an introduction to the foundations of quantum mechanics, with particular emphasis on its statistical interpretation and on the formulation of generalized measurement theory through quantum operations and effects. The fundamentals of density matrix theory, quantum Markov processes, and completely positive dynamical semigroups are developed. The most important master equations used in quantum optics and condensed matter theory are derived and applied to the study of many examples. Special attention is paid to the Markovian and non-Markovian theory of environment induced decoherence, its role in the dynamical description of the measurement process, and to the experimental observation of decohering electromagnetic field states. The book includes the modern formulation of open quantum systems in terms of stochastic processes in Hilbert space. Stochastic wave function methods and Monte Carlo algorithms are designed and applied to important examples from quantum optics and atomic physics. The fundamentals of the treatment of non-Markovian quantum processes in open systems are developed on the basis of various mathematical techniques, such as projection superoperator methods and influence functional techniques. In addition, the book expounds the relativistic theory of quantum measurements and the density matrix theory of relativistic quantum electrodynamics.

Machine learning algorithms learn a desired input-output relation from examples in order to interpret new inputs. This is important for tasks such as image and speech recognition or strategy optimisation, with growing applications in the IT industry. In the last couple of years, researchers investigated if quantum computing can help to improve classical machine learning algorithms. Ideas range from running computationally costly algorithms or their subroutines efficiently on a quantum computer to the translation of stochastic methods into the language of quantum theory. This contribution gives a systematic overview of the emerging field of quantum machine learning. It presents the approaches as well as technical details in an accessible way, and discusses the potential of a future theory of quantum learning.

We present an experimental study of higher-dimensional quantum key distribution protocols based on mutually unbiased bases, implemented by means of photons carrying orbital angular momentum. We perform ($d+1$) mutually unbiased measurements in a classically simulated prepare-and-measure scheme and on a pair of entangled photons for dimensions ranging from $d=2$ to 5. In our analysis, we pay attention to the detection efficiency and photon pair creation probability. As security measures, we determine from experimental data the average error rate, the mutual information shared between the sender and receiver, and the secret key generation rate per photon. We demonstrate that increasing the dimension leads to an increased information capacity as well as higher key generation rates per photon. However, we find that the benefit of increasing the dimension is limited by practical implementation considerations, which in our case results in deleterious effects observed beyond a dimension of $d=4$.

A generalization of the stochastic wave-function method to quantum master equations which are not in Lindblad form is developed. The proposed stochastic unraveling is based on a description of the reduced system in a doubled Hilbert space and it is shown that this method is capable of simulating quantum master equations with negative transition rates. Non-Markovian effects in the reduced systems dynamics can be treated within this approach by employing the time-convolutionless projection operator technique. This ansatz yields a systematic perturbative expansion of the reduced systems dynamics in the coupling strength. Several examples such as the damped Jaynes-Cummings model and the spontaneous decay of a two-level system into a photonic band gap are discussed. The power as well as the limitations of the method are demonstrated.

We give an algorithm for prediction on a quantum computer which is based on a linear regression model with least-squares optimization. In contrast to related previous contributions suffering from the problem of reading out the optimal parameters of the fit, our scheme focuses on the machine-learning task of guessing the output corresponding to a new input given examples of data points. Furthermore, we adapt the algorithm to process nonsparse data matrices that can be represented by low-rank approximations, and significantly improve the dependency on its condition number. The prediction result can be accessed through a single-qubit measurement or used for further quantum information processing routines. The algorithm's runtime is logarithmic in the dimension of the input space provided the data is given as quantum information as an input to the routine.