⚛️ quantum physics

Lecture notes on classical and quantum non-Markovianity

These lecture notes introduce graduate students to classical and quantum non-Markovianity by emphasizing the correspondence between classical stochastic concepts and quantum dynamical maps, with a specific focus on characterizations based on channel divisibility and state distinguishability.

Original authors: Graeme Pleasance

Published 2026-07-09

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Graeme Pleasance

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: Memory vs. Forgetting

Imagine you are watching a movie.

Markovian (Memoryless): This is like a movie where the next scene depends only on what is happening right now. If the hero is currently running, the next scene is just "running." The movie doesn't care if the hero was sad five minutes ago or if it rained yesterday. The system has "forgotten" its past.
Non-Markovian (With Memory): This is like a movie where the next scene depends on the whole story so far. If the hero is running, it might be because they were chased by a dragon three scenes ago. The system "remembers" its history, and that memory changes what happens next.

This paper is a guide for students trying to understand how to tell the difference between these two types of behavior, specifically in the weird world of Quantum Mechanics (where particles act like waves and probabilities) compared to the Classical World (where coins flip and balls roll).

Part 1: The Classical World (The Coin Flip)

The paper starts by reviewing how we handle "normal" randomness, like flipping a coin or a drunk person walking on a grid (a "random walk").

The Rule of Thumb: In a standard "Markov" process, the future is determined solely by the present. If you know where the drunk person is right now, you don't need to know where they were an hour ago to predict where they will be in the next minute.
The "Divisibility" Test: The authors introduce a mathematical test called P-divisibility. Imagine you have a video of the drunk person walking. If you can cut the video into any two chunks (from time A to B, and B to C) and the transition from A to C is just the combination of A to B and B to C, the process is "divisible" and memoryless.
The Twist: The paper shows that sometimes a process looks divisible (you can mathematically chop it up), but it's actually non-Markovian. It's like a magic trick where the pieces fit together mathematically, but the story inside the pieces implies the character is remembering things they shouldn't.

Part 2: The Quantum World (The Spooky Box)

Now, the paper moves to Open Quantum Systems. Imagine a quantum particle (the "System") inside a box, but the box is leaking and interacting with the air outside (the "Environment").

The Problem with Quantum Memory: In the classical world, you can easily define "memory" by looking at probabilities. In the quantum world, this is tricky. If you try to measure a quantum particle to see its history, the act of measuring changes the particle. It's like trying to check if a soufflé is done by poking it; the poking ruins the soufflé. Because of this, the standard definition of "Markovian" (forgetting the past) doesn't work perfectly for quantum systems.

Part 3: Two Ways to Spot "Memory" in Quantum Systems

Since the old definition doesn't work, the paper focuses on two modern ways to detect if a quantum system is remembering its past (Non-Markovian) or forgetting it (Markovian).

1. The "Divisibility" Check (The RHP Method)

The Analogy: Imagine you have a machine that transforms a piece of paper (the quantum state).
The Test: If you can break the machine's operation into two steps (Step A then Step B) and both steps are physically valid machines that could exist on their own, the process is Markovian.
The Failure: If you try to break the process into steps, and one of those steps turns out to be "impossible" (mathematically, it would create negative probabilities or break the laws of physics), then the system has Non-Markovian memory. The system is "holding onto" information from the past that prevents the process from being cleanly chopped up.
The Paper's Claim: The authors show that if the "decay rate" (how fast the system loses energy or information) ever goes negative, the system is Non-Markovian. It's like a bank account where you suddenly get money back from the bank because you over-withdrew earlier. That "negative decay" means information is flowing back from the environment to the system.

2. The "Distinguishability" Check (The BLP Method)

The Analogy: Imagine you have two different quantum coins, Coin A and Coin B. You want to tell them apart.
The Test: In a Markovian world, as time passes, the coins get more mixed up with the environment. They become harder to tell apart. The "distance" between them shrinks steadily.
The Failure: In a Non-Markovian world, the coins might get mixed up, but then the environment "spits" some information back out. Suddenly, the coins become easier to tell apart again. The "distance" between them increases.
The Paper's Claim: If the ability to tell two quantum states apart ever increases over time, the system is Non-Markovian. This is called "Information Backflow." It's like a conversation where you forget what someone said, but then they remind you, and suddenly you remember everything clearly again.

Part 4: The Spin-Boson Example (The Real-World Test)

To prove these ideas work, the authors apply them to a famous model called the Spin-Boson model.

The Setup: Imagine a tiny magnet (a "spin" or qubit) interacting with a bath of vibrating springs (bosons).
The Result:
- If the springs are very weak and fast (like a gentle breeze), the magnet forgets its past quickly. The decay rate is always positive. Result: Markovian.
- If the springs are strong and slow (like a heavy, sticky fluid), the magnet gets "stuck" in its history. The decay rate swings back and forth, even becoming negative. Result: Non-Markovian.
The Conclusion: Both the "Divisibility" test and the "Distinguishability" test agree on this model. When the system remembers (Non-Markovian), the decay rate is negative, and the distinguishability of states grows.

Summary of Key Takeaways

Classical vs. Quantum: In classical physics, "memory" is easy to define. In quantum physics, it's hard because measuring the past changes the present.
Two Definitions: The paper focuses on two main ways to define quantum memory:
- CP-Divisibility: Can we mathematically chop the process into valid, physical steps? (If no, it has memory).
- State Distinguishability: Do two quantum states get harder to tell apart over time? (If they get easier to tell apart, information is flowing back, and it has memory).
The Connection: The paper proves that for many common systems, these two definitions are actually saying the same thing: if the system "remembers," the decay rate becomes negative, and information flows back from the environment.

What the paper does NOT do:
The paper is a theoretical lecture note. It does not discuss clinical applications, medical uses, or specific future technologies. It strictly defines the mathematical rules for identifying memory in quantum systems and proves how these rules apply to a specific model of a magnet interacting with vibrations.

Technical Summary: Lecture Notes on Classical and Quantum Non-Markovianity

Problem Statement
The study of non-Markovian quantum processes has generated significant interest due to the existence of several competing notions of quantum non-Markovianity. A fundamental challenge arises because the classical definition of Markovianity, rooted in the independence of future states from past history given the present, is incompatible with the statistical foundations of quantum mechanics. Specifically, quantum processes generally violate the Kolmogorov consistency conditions required to define a classical stochastic hierarchy, primarily due to the generation of coherences and the disturbance caused by measurements. Consequently, a precise definition of what constitutes "quantum non-Markovianity" remains an open and complex topic. These notes address this by reviewing introductory concepts to characterize non-Markovianity in both classical and quantum stochastic processes, focusing on the theory of open quantum systems where noise emerges from system-environment interactions.

Methodology
The paper adopts a pedagogical approach, bridging classical probability theory and open quantum system theory to establish a correspondence between classical and quantum concepts.

Classical Foundation: The notes begin by reviewing classical stochastic processes, defining Markov chains via the Chapman-Kolmogorov equation (CKE) and introducing the concept of $P$ -divisible processes (where intermediate transition maps are positive and trace-preserving). The relationship between the Kolmogorov distance and the contractivity of stochastic maps is established.
Quantum Framework: The theory of open quantum systems is reviewed, focusing on Completely Positive Trace-Preserving (CPTP) maps (dynamical maps) and the Kraus representation. The concept of divisibility is extended to the quantum domain, distinguishing between $P$ -divisibility (positive intermediate maps) and CP-divisibility (completely positive intermediate maps).
Characterization of Markovianity: Two primary intrinsic approaches to defining quantum Markovianity are analyzed:
- CP-Divisibility: A process is Markovian if the dynamical map $\Phi_t$ can be decomposed into a sequence of CPTP intermediate maps $\Lambda_{t,s}$ for all $t \ge s$ . This is linked to the non-negativity of decay rates in the time-local master equation (GKSL form).
- Monotonically Decreasing State Distinguishability (MDSD): A process is Markovian if the trace distance between any pair of system states decreases monotonically over time. This is interpreted as a continuous loss of information from the system to the environment, whereas an increase signifies information backflow (non-Markovianity).
Application: These measures are applied to the spin-boson model (a qubit interacting with a bath of harmonic oscillators). The exact solution for the reduced density matrix is utilized to derive time-dependent decay rates and evaluate the non-Markovianity measures.

Key Contributions and Results

Hierarchy of Classical Processes: The notes clarify the hierarchy $Markov \subset CKE \subset P\text{-divisible}$ . While all Markov processes are $P$ -divisible, the converse is not strictly true for higher-order correlations, though they are operationally equivalent when restricted to one-point probabilities.
Quantum Definitions:
- CP-Divisibility (RHP measure): Defined by Wolf, Cirac, Rivas, Huelga, and Plenio, this criterion relies on the complete positivity of the intermediate map. The paper derives the measure $N_{RHP}(\Phi)$ based on the Choi matrix norm, showing it is non-zero if and only if the decay rates in the master equation become negative.
- MDSD (BLP measure): Defined by Breuer, Laine, and Piilo, this criterion relies on the trace distance. The measure $N_{BLP}(\Phi)$ quantifies the total amount of information backflow.
Relationship between Criteria: The paper establishes that CP-divisibility implies MDSD (i.e., if a process is CP-divisible, the trace distance is monotonically decreasing), but the converse is not necessarily true. However, for quantum dynamical semigroups (time-homogeneous processes with GKSL generators), the two measures coincide and both predict Markovian behavior.
Spin-Boson Model Analysis:
- The model is solved exactly, yielding a time-local master equation with time-dependent Lamb shift $S(t)$ and decay rate $\gamma(t)$ .
- The condition for CP-divisibility is shown to be $\int_0^t \gamma(s) ds \ge 0$ .
- Non-Markovianity is identified when $\gamma(t) < 0$ .
- For a Lorentzian spectral density, the paper demonstrates that strong coupling ( $g \gg \Gamma$ ) leads to oscillatory decay rates that become negative, indicating non-Markovian behavior (coherent oscillations and information backflow). Conversely, weak coupling ( $g \ll \Gamma$ ) results in a positive, constant decay rate, recovering the Markovian limit consistent with the Born-Markov approximations.
- The analysis confirms that for this specific single-channel model, the RHP and BLP measures yield identical predictions regarding the onset of non-Markovianity.

Significance
The paper serves as an introductory resource for graduate students, aiming to demystify the various definitions of quantum non-Markovianity by grounding them in classical probability theory. It emphasizes the correspondence between classical stochastic matrices and quantum dynamical maps, specifically highlighting how the violation of classical consistency conditions (Kolmogorov conditions) necessitates alternative quantum characterizations. By focusing on intrinsic criteria (CP-divisibility and state distinguishability), the notes provide a clear framework for distinguishing between memoryless (Markovian) and memory-influenced (non-Markovian) dynamics without requiring explicit modeling of the environment's global state. The application to the spin-boson model illustrates how these abstract measures translate into physical phenomena, such as the transition from exponential decay to coherent oscillations depending on the coupling strength and spectral density of the environment.