The description of the evolution of open quantum systems is a topic of great interest both for its wide range of technological applications and for the understanding of fundamental aspects of quantum theory. In this context, the quantum dynamical mappings formalism is a widely used tool for generally characterizing open systems dynamics. Nonetheless, for this formalism there is no agreement on how to define Markovianity, which is an extremely important concept for most informational analysis of quantum systems. In this undergraduate thesis we compare the quantum dynamical mappings framework with the stochastic processes theory, which has a precise definition of Markovianity, and show that there is a fundamental difference between them that prevents us from properly defining Markovianity in the quantum case. In the end, we briefly present an alternative approach, the process tensor framework, in which is possible to consistently define quantum Markovianity. We then argue that, for this reason, the process tensor framework might be more adequate than quantum dynamical mappings for generally describing the evolution of open quantum systems
The description of the evolution of open quantum systems is a topic of great interest both for its wide range of technological applications and for the understanding of fundamental aspects of quantum theory. In this context, the quantum dynamical mappings formalism is a widely used tool for generally characterizing open systems dynamics. Nonetheless, for this formalism there is no agreement on how to define Markovianity, which is an extremely important concept for most informational analysis of quantum systems. In this undergraduate thesis we compare the quantum dynamical mappings framework with the stochastic processes theory, which has a precise definition of Markovianity, and show that there is a fundamental difference between them that prevents us from properly defining Markovianity in the quantum case. In the end, we briefly present an alternative approach, the process tensor framework, in which is possible to consistently define quantum Markovianity. We then argue that, for this reason, the process tensor framework might be more adequate than quantum dynamical mappings for generally describing the evolution of open quantum systems
Markovian approximation is a widely-employed idea in the descriptions of the dynamics of open quantum systems. Although it is usually claimed to be a concept inspired by classical Markovianity, the physical meaning of this concept, however, is vague. In this thesis, I compare the descriptions of classical stochastic processes and open quantum systems, and show that those inherent differences lead to the non-trivial problem of characterizing quantum non-Markovianity. With various examples, I argue that the current most often used definitions of quantumMarkovianity in the literature do not fully capture the memoryless property of OQSs. To address this issue, I take a different approach by studying a host of Markov-related concepts in the quantum regime. Some of these concepts have long been used in quantum theory, while others are first proposed in this thesis. I define all these concepts under a unified framework, which allows one to rigorously build hierarchy relations among them. In this sense, quantum non-Markovianity is highly context-dependent. The results in this thesis, summarized as a hierarchy figure, are expected to bring clarity to the nature of quantumnon-Markovianity.
The paradigm of considering open quantum systems—i.e., focusing only on the system of interest, and treating the rest of the world as an effective environment—has proven to be a highly effective way to understand a range of quantum systems, across areas of study such as quantum optics, cold atoms, superconducting qubits, and impurities in solid-state systems. A common approach in many of these contexts has been to consider simplified approaches based on the Born and Markov approximations. While these approximations are indeed often appropriate in contexts such as quantum optics, the widespread application of these approximations has been driven more by simplicity than by accuracy. In particular, these Markovian treatments will fail in many cases, such as when coupling to the environment is not weak, when the environment is structured and has resonances, when the system couples to low-frequency modes of the environment, or when the questions of interest involve the propagation of information through the environment. Despite the fact that many real problems are non-Markovian, the Markov approximation is still widely used, as it is often assumed that a fully non-Markovian treatment is too complex to be practical. In this Perspective we discuss a recently developed set of techniques that address this challenge. Centering our discussion around the notion of the process tensor, we demonstrate that the generality of the process tensor concept, coupled with efficient tensor-network methods, opens the door to the description of a wide range of observable non-Markovian processes in a wide range of open quantum systems.
Understanding whether the features of open quantum dynamics are genuinely quantum remains a central challenge in quantum dynamics. Even though the non-Markovian behavior of quantum dynamics has been widely investigated across different settings, there is still no consensus on which properties of a dynamics reflect genuine quantum features and which arise from classical or non-genuine quantum sources. In this review, we provide detailed information on recent developments in characterizing quantum non-Markovianity based on information backflow and the nature of its origin. We also present a survey on how various approaches separate classical and quantum contributions, as well as how they define operational tasks that reveal genuine quantum non-Markovianity. We analyze several frameworks, including state-distinguishability -based, channel-based (``CP-divisibility''), and process-tensor methods. For each framework, we outline the underlying physical motivation, the criteria proposed to distinguish genuine quantum non-Markovianity from practical or apparent memory effects. We further compare different approaches and their strengths and limitations. The review aims to clarify the conceptual and operational aspects of quantum non-Markovian processes based on their nature and to provide a foundation for future research on quantum non-Markovianity and its role in advancing quantum information science and technology.
Generic open quantum dynamics can be described by two seemingly very distinct approaches: a top down approach by considering an (unknown) environment coupled to the system and affects the observed dynamics of the system; or a bottom up approach which tries to build an open quantum evolution model from the observed data. The process tensor framework describes all the possible observations one could possibly make on a quantum system, however it is computationally inefficient and not predictive. Here we define the purified process tensor which 1) allows efficient tomography as well as prediction for future process and 2) naturally defines a stationary quantum process as well as a quantitative and easy-to-evaluate definition of the memory complexity, or the degree of non-Markovianity, for it. As such it allows to uncover the minimal open quantum evolution model hidden in the observed data, completing the second approach for understanding open quantum dynamics. The intimate connection between quantum processes and classical stochastic processes is drawn in the end.
During the last ten years, the studies on non-Markovian open system dynamics has become increasingly popular and having contributions from diverse sets of research communities. This interest has arisen due to fundamental problematics as to how to define and quantify memory effects in the quantum domain, how to exploit and develop applications based on them, and also due to the following question: what are the ultimate limits for controlling open system dynamics? We give here a simple theoretical introduction to the basic approaches to define and quantify quantum non-Markovianity —also highlighting their connections and differences. In addition to the importance of the development for open quantum systems studies, we also discuss the implications of the progress for other fields including, e.g. , formal studies of stochastic processes and quantum information science, and conclude with possible future directions the recent developments open.
The probabilistic description of the time evolution of a physical system can take two conceptually distinct forms: a trajectory of probabilities, which specifies how probabilities evolve over time, and a probability on trajectories, which assigns probabilities to possible histories. A lack of a clear distinction between these two probabilistic descriptions has given rise to a number of conceptual difficulties, particularly in recent analyses of stochastic-quantum correspondence. This paper provides a systematic account of their relationship. We define probability dynamics and stochastic process families together with a precise notion of implementation that connects the two descriptions. We show that implementations are generically non-unique, that every probability dynamics admits a Markovian implementation, and characterize when non-Markovian implementations are possible. We expose fallacies in common arguments for the linearity of probability dynamics based on the law of total probability and clarify the proper interpretation of ``transition matrices'' by distinguishing dynamics-level maps from the conditional probability matrices of implementing processes. We further introduce decomposability as the appropriate general notion of stepwise evolution for (possibly nonlinear) probability dynamics, relate it to divisibility in the linear case -- showing that the two can come apart -- and disentangle both notions from Markovianity and time-homogeneity. Finally, we connect these results to what we call statistical dynamics, in which linearity is indeed physically motivated, and contrast the framework with quantum mechanics.