[Paper Review] Bridging the gap between classical and quantum many-body information dynamics

Andrea Pizzi, Daniel Malz+2 more|arXiv (Cornell University)|Apr 6, 2022

Cellular Automata and Applications4 citations

TL;DR

This paper demonstrates that key features of quantum many-body information dynamics—such as unbounded linear growth of entanglement entropy, asymptotic extensivity, and measurement-induced phase transitions—emerge naturally in classical systems when the full exponentially large classical probability distribution is explicitly accounted for. By introducing a classical analog of entanglement entropy (cEE), the authors show that classical information spreading exhibits the same rich, complex behavior as in quantum systems, revealing deep structural parallels between classical and quantum information dynamics.

ABSTRACT

The fundamental question of how information spreads in closed quantum many-body systems is often addressed through the lens of the bipartite entanglement entropy, a quantity that describes correlations in a comprehensive (nonlocal) way. Among the most striking features of the entanglement entropy are its unbounded linear growth in the thermodynamic limit, its asymptotic extensivity in finite-size systems, and the possibility of measurement-induced phase transitions, all of which have no obvious classical counterpart. Here, we show how these key qualitative features emerge naturally also in classical information spreading, as long as one treats the classical many-body problem on par with the quantum one, that is, by explicitly accounting for the exponentially large classical probability distribution. Our analysis is supported by extensive numerics on prototypical cellular automata and Hamiltonian systems, for which we focus on the classical mutual information and also introduce a `classical entanglement entropy'. Our study sheds light on the nature of information spreading in classical and quantum systems, and opens new avenues for quantum-inspired classical approaches across physics, information theory, and statistics.

Motivation & Objective

• To understand the origin of striking quantum information dynamics features—such as unbounded entanglement growth and measurement-induced phase transitions—by treating classical and quantum systems on equal footing.
• To close the conceptual gap between classical and quantum information spreading by showing that analogous features emerge in classical systems when the full phase space is considered.
• To introduce and validate a classical analog of bipartite entanglement entropy (cEE) that captures nonlocal correlations in classical many-body systems.
• To demonstrate that classical systems, when treated with full probability distribution complexity, can exhibit the same qualitative richness as quantum systems in information dynamics.
• To open new research avenues in classical physics, information theory, and statistics by providing a quantum-inspired tool for characterizing classical correlations.

Proposed method

• The authors define a classical entanglement entropy (cEE) based on the classical mutual information, using the full probability distribution over the exponentially large phase space.
• They analyze prototypical classical systems—cellular automata and Hamiltonian systems—under time-reversible, local dynamics to study information spreading.
• The cEE is computed numerically across various system sizes and time evolutions to observe its scaling behavior and growth dynamics.
• The study draws parallels between classical and quantum systems by comparing the cEE to the quantum entanglement entropy (EE), highlighting shared features such as unbounded linear growth and extensivity.
• They introduce a classical measurement protocol analogous to quantum measurement-induced phase transitions, showing that classical systems can also exhibit volume- and area-law scaling of cEE.
• The framework treats classical and quantum systems symmetrically, avoiding semiclassical approximations, and emphasizes intrinsic similarities rather than quantitative emulation.

Experimental results

Research questions

• RQ1 Can classical many-body systems exhibit unbounded linear growth of information measures similar to quantum systems?
• RQ2 Do classical systems display measurement-induced phase transitions analogous to those in quantum systems?
• RQ3 What is the classical analog of quantum entanglement entropy, and how does it scale in time and system size?
• RQ4 How do classical correlations in many-body systems compare to quantum correlations in terms of nonlocality and complexity?
• RQ5 What are the classical counterparts of quantum information-theoretic quantities such as out-of-time-order correlators and prethermalization features?

Key findings

• The classical mutual information and the introduced classical entanglement entropy (cEE) both exhibit unbounded linear growth in the thermodynamic limit, mirroring the behavior of quantum entanglement entropy.
• In finite-size systems, the cEE saturates to an extensive value at long times, just as in the quantum case, indicating asymptotic extensivity.
• Classical systems under local, time-reversible dynamics can display both volume-law and area-law scaling of cEE depending on measurement protocols, demonstrating classical analogs of measurement-induced phase transitions.
• The cEE captures features missed by standard mutual information, such as extensivity at infinite temperature, making it a more comprehensive measure of classical correlations.
• The emergence of complex information dynamics in classical systems is rooted in the combination of chaos and incompressibility of the probability distribution, leading to effective randomization similar to quantum dephasing.
• The study establishes that the complexity of information spreading is not inherently quantum but arises from the exponential structure of the underlying state space, shared by both classical and quantum systems.