Classification of quantum multitime correlated processes

Open quantum systems are a huge field in quantum physics, specifically in quantum communication, quantum error correction, and the replication and simulation of quantum processes. All open quantum systems must deal with some noise, and can be simulated as an environment interacting with the system. When the interactions of such processes do not depend on information from past interactions, then the process is classified as a Markovian process. However, when the interactions in a process does depend on information from past interactions, then the process is classified as a non-Markovian process. Determining whether a process is Markovian or non-Markovian has been very well studied. Traditional measures of Markovianity based on quantum channel formulation can run into problems with an operational interpretation. In contrast, the process matrix formalism can determine whether a process is Markovian, and can be reconstructed operationally using process tomography between all parties communicating with one another. This formalism categorises these processes as memoryless (Markovian), or classical memory and quantum memory for non-Markovian. Memory is defined as the information encoded in the environment between each interaction, that affects future interactions. When determining the memory type consistent with a process matrix the most conservative system-environment simulation of the process matrix is chosen. That is to say, that if a process can be replicated using classical memory, then the process is classified as a classical memory process. Within this thesis a necessary and sufficient model is created for classical memory processes, as well as how to convert a continuous time process to the process matrix formalism. Both these methods are operationally defined, and each are applied to depolarising map examples. CP-divisibility, a popular traditional measure of Markovianity is discussed, and a counter example to CP-divisibility is generalised. Finally, the model developed within this thesis is connected to environment assisted error correction.