How quantum eigenstates encode entanglement growth

Entanglement is one way to track how information spreads through a quantum system. A common measure is the Rényi entropy, calculated from powers of a subsystem’s reduced density matrix. The second Rényi entropy is especially connected to the subsystem’s purity: a measure of how close its state is to pure rather than mixed. The authors ask how the time evolution of these quantities is encoded in the eigenstates of a periodically driven system’s evolution operator, often called its Floquet operator.

Their starting point is the eigenstate thermalization hypothesis, or ETH. In broad terms, ETH describes how the eigenstates of many chaotic systems can encode thermal-looking behavior for local measurements. Senese and Calabrese adapt this idea to entanglement dynamics. Instead of studying ordinary local observables in the usual Hilbert space, they examine subsystem permutation operators in a replicated Hilbert space—a mathematical construction that helps express powers of a reduced density matrix. For the second Rényi entropy, the relevant example is the swap operator.

The authors use spectral representations of reduced-density-matrix moments. These write the time dependence in terms of eigenstate properties and phases, whose dephasing can account for the decay of purity and the growth of entanglement. For Rényi-2, they combine exact calculations from random matrix theory, established ETH results and numerical tests in one-dimensional systems. From these ingredients, they formulate ETH-like ansätze for matrix elements of the subsystem swap operator, along with their correlations with the initial state’s overlaps with eigenstates.

The abstract reports two main conclusions. First, in the thermodynamic limit, purity dynamics is determined entirely by smooth, slowly varying functions that emerge from the statistics of those initial-state overlaps. The authors describe this as parallel to generalized ETH approaches in a single replica. Second, for large subsystems, locality of the dynamics and dephasing in the bulk of the spectrum simplify the functions’ dependence: they depend on a single fundamental frequency. The framework is intended to accommodate ETH-like descriptions for both very small and large subsystems.

The result is a proposed spectral account of entanglement dynamics, not a claim that all chaotic quantum systems follow one universal formula. The work focuses on deterministic chaotic Floquet models and develops its detailed analysis around Rényi-2. Although its abstract says the models are considered in arbitrary dimensions, it specifically identifies numerical tests in one dimension. It also presents extensions to higher Rényi entropies as a subject for discussion, not as equally established results.

There are important limits to what can be assessed from the available source. The evidence here is the paper’s abstract, not its full text, so it does not provide the assumptions, equations, numerical details, tested system sizes or a detailed account of how broadly the proposed ansätze hold. The paper is listed as an arXiv preprint; the supplied information does not establish that it has undergone peer review. The results should therefore be read as a theoretical framework and supporting analysis reported by the authors, rather than as a settled general description of entanglement in every quantum system.