⚛️ quantum physics

Non-Markovian effects on informational steady states

This paper investigates non-Markovian effects on informational steady states within continuously monitored collision models, revealing that the steady-state information gain per measurement is negatively correlated with the degree of non-Markovianity and is significantly influenced by system-ancilla and ancilla-ancilla interaction parameters.

Original authors: Jacob Werner

Published 2026-10-02

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CC BY 4.0

Original authors: Jacob Werner

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

In the quantum world, where particles exist in states of probability rather than definite positions, information is not just a record of what happened; it is a physical substance that flows, accumulates, and dissipates. When a tiny quantum system interacts with its surroundings, it constantly trades information with the environment. In many simplified models, this exchange is one-way: the system leaks details into the environment and never gets them back, much like a drop of ink dispersing in a glass of water. However, the real universe is rarely so simple. Environments often have memory, meaning that information lost by a system can sometimes flow back to it, altering its future behavior. This phenomenon, known as non-Markovianity, challenges our understanding of how quantum systems evolve. A critical concept in this field is the "informational steady state," a condition where a system is being watched continuously. In this state, the information gained from each new measurement is perfectly balanced by the information lost to the environment, creating a dynamic equilibrium where the system remains constantly informative without ever settling into a static, unchanging condition.

Jacob Werner, a researcher at the University of Tokyo, set out to explore how these memory effects influence that delicate balance. Using a theoretical framework called collision models, where a quantum system interacts sequentially with a stream of simple environmental units, Werner simulated a scenario where the environment itself has internal connections. In his model, the environment consists of pairs of tiny quantum bits, or ancillas, that interact with the system and then with each other before moving on. By allowing these environmental units to talk to one another, Werner introduced the possibility of information flowing back to the system, effectively creating a memory. The goal was to see how this backflow changes the rate at which information is gained and lost when the system is under continuous observation.

The study focused on a specific setup where the system interacts with a two-qubit ancilla in a series of steps. First, the system swaps information with the first part of the ancilla, then the two parts of the ancilla interact with each other, and finally, the system interacts with the second part of the ancilla. After these interactions, a measurement is taken on the first part of the ancilla, revealing a clue about the system's state. Werner varied the strength of these interactions, essentially turning the "knobs" that control how much the system and the environment mix, and how much the two parts of the environment mix with each other. He then measured two key quantities: the degree of non-Markovianity, which tracks how much information returns to the system, and the steady-state information gain, which measures how much new information a single measurement provides once the system has settled into its long-term rhythm.

The results revealed a clear and somewhat counterintuitive relationship. As the memory effects in the environment grew stronger—meaning more information was flowing back to the system—the amount of new information gained from each measurement decreased. In other words, when the environment remembers the past and shares it with the system, the system becomes less surprising to the observer. The researchers found a statistically significant negative correlation between the strength of the memory and the information gain. This suggests that in a continuously monitored quantum system, the ability to extract fresh information is directly hindered by the environment's ability to retain and return past data. If the environment holds onto the system's secrets and whispers them back, the next measurement has less to reveal.

However, the story is not a simple linear decline. The simulations showed that the outcome depends heavily on the specific timing and strength of the interactions. When the system interacts with the environment in a very specific way, the environment's internal memory can actually help preserve correlations for a moment, leading to a temporary spike in information gain before it drops off again. This happens because the internal interactions within the environment can sometimes reorganize the information in a way that the system can still access, provided the system's own interaction with the environment is tuned correctly. But if the environmental memory is too strong or the interactions are unbalanced, the system effectively interacts with the same state over and over, causing the information gain to vanish entirely.

The study also highlighted that for these memory effects to be strong enough to matter, the internal connections within the environment must be robust. If only one part of the environmental pair interacts with the other, the flow of information is blocked, and the system behaves as if it has no memory at all. Both sides of the environmental pair must be actively connected to create a true memory effect. Furthermore, the researchers found that the system's behavior is highly sensitive to the balance between its interactions with the two parts of the environment. When the system interacts too strongly with one part and too weakly with the other, the steady state becomes predictable and uninformative.

Ultimately, this work provides a clearer picture of how memory shapes the flow of information in quantum systems. It confirms that while non-Markovian effects are often studied for their ability to preserve quantum coherence, they also fundamentally alter the thermodynamics of information. In a continuously monitored system, the presence of memory means that the observer gains less new information per step because the system is constantly being reminded of its past. This finding is crucial for the future of quantum technologies, where controlling the flow of information is essential for tasks like error correction and feedback control. By understanding how environmental memory dampens the information gain, scientists can better design systems that either minimize these effects to keep information flowing or harness them to stabilize quantum states. The research suggests that in the quantum realm, the past is never truly gone; it lingers in the environment, quietly shaping what we can learn from the present.

Technical Summary: Non-Markovian Effects on Informational Steady States

Problem Statement
Informational steady states (ISSs) occur in continuously monitored quantum systems when the rate of information gain from measurements is exactly balanced by the rate of information loss to the environment. While ISSs have been characterized in Markovian settings, the influence of memory effects (non-Markovianity) on their formation and steady-state properties remains largely unexplored. Understanding these effects is critical for realistic quantum technologies, as many environments possess memory, and because information is increasingly viewed as a physical quantity influencing entropy production and thermodynamics.

Methodology
The paper investigates ISSs within the framework of continuously monitored collision models (CMs), specifically utilizing a two-qubit ancilla model to ensure non-zero steady-state information gain and loss.

Model Structure: The environment is modeled as a sequence of ancillas, each composed of two qubits ( $Y_1$ and $Y_2$ ). The system ( $X$ ) interacts sequentially with these ancillas via partial swap unitaries parameterized by angles $\theta$ .
Non-Markovian Mechanism: Non-Markovianity is introduced through ancilla-ancilla ( $Y Y$ ) interactions ( $\theta_{Y_1Y_2}$ and $\theta_{Y_2Y_1}$ ) occurring between the system-ancilla ( $X Y$ ) interactions. This allows for information backflow from the environment to the system.
Measurement Protocol: Continuous monitoring is performed via POVMs on the $Y_1$ qubit after each time step.
Quantities of Interest:
- Non-Markovianity: Quantified using the Breuer-Laine-Piilo (BLP) measure ( $N_{BLP}$ ), calculated as the sum of increases in the trace distance between two system states over time.
- Informational Steady State: Characterized by the steady-state information gain per measurement ( $G_\infty$ ), defined as the classical quantum mutual information (CQMI) between the system and the measured ancilla in the conditional steady state. In an ISS, $G_\infty$ equals the information loss rate ( $L_\infty$ ).
Numerical Approach: The study employs numerical simulations averaging over 20 trajectories of 100 time steps, focusing on the final 30 steps to determine steady-state values.

Key Contributions and Results

Markovian Regime Characterization: In the absence of ancilla-ancilla interactions, the study confirms that steady-state information gain ( $G_\infty$ ) is maximized when the first system-ancilla interaction angle $\theta_{XY_1}$ is slightly less than $\pi/4$ and the second interaction angle $\theta_{XY_2}$ is approximately $\pi/4$ . This balance optimizes the correlation between the system and the measured ancilla while managing the steady-state population of the system.
Non-Markovianity Requirements: To achieve high degrees of non-Markovianity ( $N_{BLP}$ ), both ancilla-ancilla interaction parameters ( $\theta_{Y_1Y_2}$ and $\theta_{Y_2Y_1}$ ) must be large. If one interaction is weak, it acts as a bottleneck, preventing the accumulation of correlations necessary for information backflow.
Correlation between Non-Markovianity and Information Gain: The primary numerical finding is a statistically significant negative correlation between the degree of non-Markovianity ( $N_{BLP}$ $N_{B L P}$ ) and the steady-state information gain ( $G_\infty$ $G_{\infty}$ ).
- Mechanism: Generally, as information flows back to the system (increasing non-Markovianity), less new information is available for the measurement to extract, reducing $G_\infty$ .
- Nuance: The relationship is not strictly monotonic in all parameter regimes. For strong system-ancilla interactions ( $\theta_{XY}$ ), increasing the ancilla-ancilla interaction can initially increase $G_\infty$ by allowing the system to leverage correlations formed between the two ancilla qubits before the measurement. However, as non-Markovianity increases further, $G_\infty$ eventually decreases.
Symmetry in Trace Distance: The paper identifies a symmetry in $N_{BLP}$ with respect to the sum of system-ancilla interaction angles ( $\theta_{XY_1} + \theta_{XY_2} \approx \pi/2$ ) when ancilla-ancilla interactions are strong. In this limit, the system effectively alternates between its initial state and the ancilla's initial state, maximizing information backflow.

Significance and Scope
The paper claims to broaden the current understanding of informational steady states by explicitly incorporating memory effects. It demonstrates that while non-Markovianity is often associated with information backflow, this backflow can reduce the steady-state information gain available from continuous monitoring. The author emphasizes that the observed negative correlation does not imply a universal causal relationship, as $N_{BLP}$ is computed based on the approach to the steady state while $G_\infty$ is a steady-state property.

The work serves as a framework for understanding how non-Markovian mechanisms affect informational quantities in specific collision models. The author suggests that future work could explore alternative memory mechanisms (e.g., independent ancilla streams) and extend these findings to the thermodynamic context of ISSs, specifically regarding conditional entropy production, as originally proposed in Ref. [1]. The paper does not propose new experimental setups but rather provides a theoretical analysis of existing collision model frameworks.

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