Quantum Stochastic Processes and Quantum non-Markovian Phenomena

Abstract

The field of classical stochastic processes forms a major branch of mathematics. They 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.

Equilibration of Non-Markovian Quantum Processes in Finite Time Intervals

“…2. Both Υ k and A k are examples of quantum combs that possess well-behaved positivity and trace properties [1,3,5,17]. The former guarantees the positivity of probabilities and the latter is crucial for ensuring the causality of a process.…”

“…We rather allow instruments to carry quantum memory, which can be operationally realized by appending a ancilla space W such that each A i acts instead on the combined space SW , as shown in Fig. 3. In such a case A k is called a tester, and is the most general way to measure a process.…”

“…Quantum processes allow the description of multitime statistics in quantum systems, providing information on the dynamics beyond the single time (quantum state) limit. Any quantum process can be represented as a single tensor Υ k ⪰ 0 [1][2][3], that can be used to compute the expectation value of a multitime measurement,…”

“…. , A 1 } (in the Schödinger picture) and Υ k is called the process tensor [1][2][3], which encapsulates the unitary dynamics (U i ) and initial state (ρ). Both of the tensors have free indices at the times k ∶= {t k , t k−1 , .…”

Inverse linear versus exponential scaling of work penalty in finite-time bit reset

“…We assume there is a Markovian thermalisation process, consistent with the stochastic thermodynamics framework. Any non-Markovian dynamics can be made Markovian by adding the necessary historical record to the microstate description [41,42]. The master equation of the bit evolution can, assuming detailed balance, always be described by the partial-swap thermalization towards the equilibrium state [37,38,43], i.e.…”