Non-Markovian time evolution of quantum systems is a challenging problem, often mitigated by employing numerical methods or making simplifying assumptions. In this work, we address this problem in waveguide quantum electrodynamics (QED) by developing a diagrammatic approach, which performs fully analytical non-Markovian time evolution of single-photon states. By revisiting Fermi's two-atom problem, we tackle the impeding question of whether the rotating-wave approximation violates causality in single-photon waveguide QED. Afterward, we introduce and prove the no upper half-plane poles (no-UHP) theorem, which connects the poles of scattering parameters to the causality principle. Finally, we visualize the time-delayed coherent quantum feedback mediated by the field and discuss the Markovian limit for microscopically separated qubits where short-distance causality violations occur and the emergence of collective decay rates in this limit. Our diagrammatic approach is a method to perform exact and analytical non-Markovian time evolution of multiemitter systems in waveguide QED.
Higher-order maps account for the most general transformations in quantum mechanics—they transform input quantum maps to output quantum maps. In this thesis, I focus on the application of such higher-order maps representing two important processes: (1) a particular type of non-Markovian quantum dynamics, and (2) indefinite causal order. First, I will introduce our work on the non-Markovian evolution of quantum states. Such a situation occurs when the past system-environment interaction leads the environment to memorize the information about the system. The measure of the non-Markovianity is given by the temporal correlation among the operations on the system. One way to estimate such correlation is to employ a full tomography of the process matrix, which is computationally intractable. Instead, we utilize machine learning algorithms without using tomographically complete measurement, to estimate the measure of non-Markovianity. We apply our model to a dataset obtained from an optical experiment and show that we are able to predict the measure of non-Markovianity with 90% accuracy. I then introduce various aspects of indefinite causal order. Ordinary experience points only to a definite causal structure, but quantum mechanics admits a more general causal structure that allows for the superposition of different causal orders. That means that the statements, is in the causal past of event B, and B is in the causal past of event A can both be true. The quantum switch can physically realise such an indefinite causal structure where a control quantum system controls the order of operations acting on a target quantum system. Superposition in the control leads to the superposition of the order. We experimentally realise a quantum switch with the control qubit being the polarisation of light and the target qubit being the transverse spatial mode of light. We first verify our quantum switch by measuring a `causal witness'. We then demonstrate an advantage of the quantum switch for classical communication. Finally, I present our work on classical communication through the most general quantum causal structures, represented by the process matrix. We formulate different classical capacities of a bi-partite process matrix. We also study one-way and bidirectional communication protocols through such a process and establish the relevant communication bounds. These enable further analyses of higher-order maps that take advantage of the full suite of processes that quantum mechanics allows.
Quantum compositions of channels can grant advantages over classical compositions in quantum information processing or thermodynamical tasks. One of such quantum composition is one with indefinite causal order, where multiple quantum channels are placed in a superposition of different causal or operation orders. The advantages of indefinite causal order were demonstrated in an experimentally realizable setup known as the quantum switch and were often attributed to this new quantum resource of indefinite causality, but some doubts and disagreements still remains as compositions without indefinite causal order can achieve the same advantages. In this work, we proposed that the quantum switch's uniqueness lies in its ability to exhibit coherent non-Markovianity, where memory effects acts across different paths of superposition. We showed that the quantum switch violates completely positive or CP-divisibility and thus has intrinsic non-Markovianity in its evolution, which allows for non-Markovian backflow of information from the environment. This backflow of information can then play a role in the many advantages seen in the quantum switch, and by extending the quantum switch operation such that the non-Markovianity is controllable, we showed that these advantages depend on the presence and amount of non-Markovianity, achieving the quantum switch case when the system is fully non-Markovian. Particularly, we demonstrated this for the quantum switch's advantages in communication capacities, work extraction via daemonic ergotropy, and refrigeration via heat extraction. Other quantum compositions that are often compared to indefinite causal order, such as the superposition of independent channels and superposition of trajectories, are also captured by this extended construction of the quantum switch, and we showed that their difference with the quantum switch lies in the presence or absence of non-Markovianity. Our work suggests that non-Markovianity is an important recipe for the quantum switch or indefinite causal order, and is a nonnegligible factor for its advantages.