Planckian Bound Constrains Dynamical Entropy Rate of Continuous Quantum Monitoring

The fundamental limits of how quickly information can be extracted from a quantum system remain a key question in physics, with implications for understanding the arrow of time and the nature of quantum chaos. Xiangyu Cao from Laboratoire de Physique de l’École normale supérieure, along with colleagues, investigates this question by defining a measure of dynamical entropy, which quantifies information gained about a system’s initial state through continuous monitoring. Their work demonstrates that even subtle thermal fluctuations can generate a non-zero rate of entropy production, and importantly, the researchers propose a universal limit to this rate, known as a Planckian bound. This finding suggests a fundamental constraint on how efficiently information can be processed in quantum systems, potentially resolving long-standing debates about the limits of predictability in quantum mechanics.

Researchers are developing new ways to quantify information gain in complex quantum systems by monitoring thermal fluctuations, moving beyond traditional methods limited to simple or large systems. This approach calculates a “dynamical entropy”, which measures how much information an observer gains about a system’s initial state through continuous observation, using mathematical tools to analyze correlations between observable quantities over time. By meticulously tracking these fluctuations, scientists can reconstruct information lost when the initial state is not perfectly known, offering a powerful new lens for understanding complex quantum behavior.

Tracking Information Gain Via Thermal Fluctuations

Researchers have developed a novel method to quantify information gained about a system’s initial state under continuous monitoring. This method centers on calculating a “dynamical entropy” by meticulously tracking thermal fluctuations within the system, rather than relying on direct observation of the system’s evolution. The key innovation extends the applicability of such analyses beyond traditional limits. The method involves analyzing correlations between observable quantities at different points in time. A crucial step involves refining the entropy calculation through a process called “renormalization”, which ensures accuracy and allows for a precise determination of the rate at which information about the initial state is revealed through monitoring.

The team discovered a surprising universal limit to how quickly information can be gained, proposing a “Planckian bound” on the dynamical entropy rate. This bound suggests a fundamental limit to the rate of information gain, dictated by the inherent quantum properties of the system. To establish this bound, researchers performed calculations on a simplified model system, a “free boson model”, allowing for rigorous testing of their theoretical framework. Further investigation revealed a similar Planckian bound applies to the “purification rate”, which measures how effectively the initial state is refined through monitoring.

This parallel finding reinforces the idea that fundamental limits govern information processing in quantum systems. The team demonstrated that both entropy and purification rates are suppressed at high frequencies, meaning that rapid fluctuations contribute less to information gain. This suppression is linked to the way information is encoded in correlations, which decay rapidly at high frequencies, and is essential for establishing the Planckian bounds. The researchers suggest that these bounds may hold true for a wide range of systems, regardless of the specific monitoring scheme employed, and are actively exploring the implications of these findings for understanding the fundamental limits of information processing in quantum systems.

Dynamical Entropy Quantifies Quantum System Chaos

Researchers have developed a new method for quantifying quantum chaos by defining a dynamical entropy based on continuous monitoring of a quantum system. This entropy measures the rate at which information about the system’s initial state is revealed through observation, offering a novel approach to understanding complex quantum behavior. The research demonstrates that this dynamical entropy exhibits a linear growth rate in a wide range of quantum many-body systems, unlike previous approaches that often relied on semiclassical or large system approximations. Importantly, the calculated entropy rate is subject to a universal upper bound, termed a Planckian bound, which suggests fundamental limits on how quickly information can be extracted from a quantum system.

This bound is reminiscent of similar constraints found in studies of out-of-time order correlations, further linking this new entropy to broader concepts of quantum chaos and information scrambling. The team’s calculations reveal that the entropy rate is determined by the thermal correlations within the system, providing a direct connection between thermodynamic properties and the emergence of quantum chaos. Furthermore, the research shows a strong relationship between the dynamical entropy and the degree to which monitoring purifies the initial state, highlighting the crucial role of purification in understanding information gain. These findings offer a new perspective on measurement-induced phases of information and provide a tool for probing the subtle interplay between quantum mechanics, chaos, and information theory in complex systems. This new approach complements existing methods and offers a more general framework for studying quantum chaos, particularly in systems where traditional approximations break down. The discovery of the Planckian bound suggests a fundamental limit to information extraction, potentially offering insights into the nature of quantum complexity and the limits of predictability in quantum systems.

Planckian Bound Defines Dynamical Entropy Rate

This research introduces a definition of dynamical entropy, a measure of information gained about a system’s initial condition under continuous monitoring. The team demonstrates that a non-zero entropy rate emerges from monitoring thermal fluctuations in systems that are not in the classical or extremely large limit, calculating this rate in both thermodynamic equilibrium and over extended timescales using correlations between observable properties. They propose a universal upper bound, a Planckian bound, for this entropy rate, and obtain related results concerning the rate of purification, a measure of information gain about the system’s state. The findings suggest that dynamical entropy can be a useful tool for characterizing the information processing capabilities of quantum systems, particularly those that are not easily described by classical physics.

The authors acknowledge that their analysis focuses on an “initial” time regime, and that further investigation is needed to understand the contributions of interactions at later times. Future work could explore the behaviour of systems at low temperatures with high dynamical entropy, and extend the analysis beyond monitoring a single property. Numerical verification of the rate formulas also remains an important direction for future research.