The field of classical stochastic processes forms a major branch of mathematics.Stochastic processes are, of course, also very well studied in biology, chemistry, ecology, geology, finance, physics, and many more fields of natural and social sciences.When it comes to quantum stochastic processes, however, the topic is plagued with pathological issues that have led to fierce debates amongst researchers.Recent developments have begun to untangle these issues and paved the way for generalizing the theory of classical stochastic processes to the quantum domain without ambiguities.This tutorial details the structure of quantum stochastic processes, in terms of the modern language of quantum combs, and is aimed at students in quantum physics and quantum-information theory.We begin with the basics of classical stochastic processes and generalize the same ideas to the quantum domain.Along the way, we discuss the subtle structure of quantum physics that has led to troubles in forming an overarching theory for quantum stochastic processes.We close the tutorial by laying out many exciting problems that lie ahead in this branch of science.
The field of classical stochastic processes forms a major branch of mathematics.Stochastic processes are, of course, also very well studied in biology, chemistry, ecology, geology, finance, physics, and many more fields of natural and social sciences.When it comes to quantum stochastic processes, however, the topic is plagued with pathological issues that have led to fierce debates amongst researchers.Recent developments have begun to untangle these issues and paved the way for generalizing the theory of classical stochastic processes to the quantum domain without ambiguities.This tutorial details the structure of quantum stochastic processes, in terms of the modern language of quantum combs, and is aimed at students in quantum physics and quantum-information theory.We begin with the basics of classical stochastic processes and generalize the same ideas to the quantum domain.Along the way, we discuss the subtle structure of quantum physics that has led to troubles in forming an overarching theory for quantum stochastic processes.We close the tutorial by laying out many exciting problems that lie ahead in this branch of science.
Noncommutative mathematics is a significant new trend of mathematics. Initially motivated by the development of quantum physics, the idea of 'making theory noncommutative' has been extended to many areas of pure and applied mathematics. This book is divided into two parts. The first part provides an introduction to quantum probability, focusing on the notion of independence in quantum probability and on the theory of quantum stochastic processes with independent and stationary increments. The second part provides an introduction to quantum dynamical systems, discussing analogies with fundamental problems studied in classical dynamics. The desire to build an extension of the classical theory provides new, original ways to understand well-known 'commutative' results. On the other hand the richness of the quantum mathematical world presents completely novel phenomena, never encountered in the classical setting. This book will be useful to students and researchers in noncommutative probability, mathematical physics and operator algebras.
Currently, there is no systematic way to describe a quantum process with memory solely in terms of experimentally accessible quantities. However, recent technological advances mean we have control over systems at scales where memory effects are non-negligible. The lack of such an operational description has hindered advances in understanding physical, chemical, and biological processes, where often unjustified theoretical assumptions are made to render a dynamical description tractable. This has led to theories plagued with unphysical results and no consensus on what a quantum Markov (memoryless) process is. Here, we develop a universal framework to characterize arbitrary non-Markovian quantum processes. We show how a multitime non-Markovian process can be reconstructed experimentally, and that it has a natural representation as a many-body quantum state, where temporal correlations are mapped to spatial ones. Moreover, this state is expected to have an efficient matrix-product-operator form in many cases. Our framework constitutes a systematic tool for the effective description of memory-bearing open-system evolutions.
This book had as its main objective to discuss how (quantum) physics concepts could be used in a relevant way in the modeling of a variety of economics/finance and psychology-based processes. For those readers who have come to read this conclusion, without skipping its “backbone,” we would like to say “thank you” for keeping with us until the end. The possible claim, which says that social science driven dynamics would need analyzing with techniques from the natural sciences, may well have been tempered with this book. Although quantum physics as a theory of prediction is one of the most successful theories humankind ever devised, it is not unreasonable to suggest that the number of foundational interpretations of quantum physics is sizable. Consider now the following natural question: “How can a theory with such a wide diversity of foundational interpretations serve to inform and elucidate a non-cognate area of research (like economics or psychology)?” Indeed, we should want to stand upright and write, without hesitation, the words “non-cognate.” No theory in social science can come close to the predictive power of quantum physics. However, very few social science theories are “plagued” by foundational issues of such depth as quantum physics. As an example, consider the issue of why quantum physical wave functions do have to be complex valued. Do physicists have a satisfactory answer to this very basic question? A theory of such predictive power has a partial differential equation, the Schrödinger equation, which cannot be rigorously derived. However, in economics and finance we strive to build theoretical edifices with the utmost respect for rigor.
The main goal of these notes is to give an introduction to the mathematics of quantum noise and some of its applications in non-equilibrium statistical mechanics. We start with some reminders from the theory of classical stochastic processes. We then provide a brief overview of quantum mechanics and quantum field theory, from the viewpoint of quantum probability and adopting the language of Hudson and Parthasarathy. We introduce quantum stochastic processes on a boson Fock space and their calculus. Whenever possible, we make connections with the relevant concepts in classical probability theory. As an application of the theory, we introduce the theory of open quantum systems, with emphasis on the physics and modeling aspects of these systems.
The classical probability theory initiated by Kolmogorov and its quantum counterpart, pioneered by von Neumann, were created at about the same time in the 1930s, but development of the quantum theory has trailed far behind. Although highly appealing, the quantum theory has a steep learning curve, requiring tools from both probability and analysis and a facility for combining the two viewpoints. This book is a systematic, self-contained account of the core of quantum probability and quantum stochastic processes for graduate students and researchers. The only assumed background is knowledge of the basic theory of Hilbert spaces, bounded linear operators, and classical Markov processes. From there, the book introduces additional tools from analysis, and then builds the quantum probability framework needed to support applications to quantum control and quantum information and communication. These include quantum noise, quantum stochastic calculus, stochastic quantum differential equations, quantum Markov semigroups and processes, and large-time asymptotic behavior of quantum Markov semigroups.
[This paper is part of the Focused Collection in Investigating and Improving Quantum Education through Research.] Quantum information science and engineering (QISE) is a rapidly developing field that leverages the skills of experts from many disciplines to utilize the potential of quantum systems in a variety of applications. It requires talent from a wide variety of traditional fields, including physics, engineering, chemistry, and computer science, to name a few. To prepare students for such opportunities, it is important to give them a strong foundation in the basics of QISE, in which quantum computing plays a central role. In this study, we discuss the development, validation, and evaluation of a Quantum Interactive Learning Tutorial, on the basics and applications of quantum computing. These include an overview of key quantum mechanical concepts relevant to quantum computation (including ways a quantum computer is different from a classical computer), properties of single- and multiqubit systems, and the basics of single-qubit quantum gates. The tutorial uses guided inquiry-based teaching-learning sequences. Its development and validation involved conducting cognitive task analysis from both expert and student perspectives and using common student difficulties as a guide. For example, before engaging with the tutorial, after traditional lecture-based instruction, one reasoning primitive that was common in student responses is that a major difference between an N -bit classical and N -qubit quantum computer is that various things associated with a number N for a classical computer should be replaced with the number 2 N for a quantum computer (e.g., 2 N qubits must be initialized and 2 N bits of information are obtained as the output of the computation on the quantum computer). This type of reasoning primitive also led many students to incorrectly think that there are only N distinctly different states available when computation takes place on a classical computer. Research suggests that this type of reasoning primitive has its origins in students learning that quantum computers can provide exponential advantage for certain problems, e.g., Shor’s algorithm for factoring products of large prime numbers, and that the quantum state during the computation can be in a superposition of 2 N linearly independent states. The inquiry-based learning sequences in the tutorial provide scaffolding support to help students develop a functional understanding. The final version of the validated tutorial was implemented in two distinct courses offered by the physics department with slightly different student populations and broader course goals. Students’ understanding was evaluated after traditional lecture-based instruction on the requisite concepts and again after engaging with the tutorial. We analyze and discuss their improvement in performance on concepts covered in the tutorial. Published by the American Physical Society 2024