Wehrl-rényi Entropy Directly Relates to Many-Body Entanglement Purity across All Subsystems

Entanglement, a fundamental property of quantum systems, underpins correlations and emergent behaviours in complex many-body systems, and researchers continually seek robust ways to quantify it. Pengfei Zhang from Fudan University and Hefei National Laboratory, along with Chen Xu and Pengfei Zhang, now demonstrate an exact relationship between the Wehrl-Rényi entropy, a measure derived from the system’s overall quantum state, and the entanglement present in all its possible subsystems. This breakthrough establishes a single, subsystem-independent measure of entanglement, overcoming limitations of previous approaches that depend on arbitrary choices of which parts of the system to examine. Crucially, the team also proposes a concrete experimental scheme for measuring this entropy, opening new avenues for studying strongly correlated quantum systems, including those found in advanced platforms like tweezer arrays and superconducting circuits, and providing a powerful tool for characterising complex quantum states such as the Greenberger-Horne-Zeilinger and W states.

Scientists have now established a precise relationship between the Wehrl-Rényi entropy, a measure of entanglement for the entire system, and the purity of its constituent subsystems. This breakthrough offers a new way to characterize entanglement, accurately reflecting correlations present in all possible subsystems, regardless of how they are defined.

Wehrl-Rényi Entropy Quantifies Subsystem Entanglement

Scientists have established a precise relationship between the Wehrl-Rényi entropy (WRE) and the purity of subsystems within many-body quantum systems, offering a new way to characterize entanglement. The research demonstrates that the WRE, calculated from the entire system, accurately reflects the entanglement present in all possible subsystems, resolving a long-standing challenge in quantifying correlations. Crucially, the team devised a concrete experimental scheme to directly measure the WRE, making it a practical tool for quantum investigations. The study analytically derived the WRE for several representative quantum states, including Haar-random states, the Greenberger-Horne-Zeilinger (GHZ) state, and the W state, revealing distinct entanglement properties for each.

For Haar-random states, the WRE per particle approaches a value of N ln(4π), while the GHZ and W states converge towards N ln(3π). The p-Bell state exhibits a different behavior, approaching a value between these two bounds. These results demonstrate how the WRE scales with system size, providing a clear classification of these states based on their entanglement characteristics. Measurements confirm that the WRE can serve as a subsystem-independent characterization of overall entanglement in pure quantum states.

Subsystem Entanglement Dictates Global System Properties

This research establishes a novel connection between a system’s overall entanglement and the entanglement within its constituent parts. Scientists have demonstrated an exact mathematical relationship between the Wehrl-Rényi entropy, a measure of entanglement for the entire system, and the purities of all possible subsystems within that system. Crucially, this relationship holds true regardless of how those subsystems are defined, offering a subsystem-independent characterization of entanglement in complex quantum states. The team also devised a concrete experimental scheme to directly measure this Wehrl-Rényi entropy, making it accessible for empirical investigation.

The researchers validated their approach by analytically calculating the Wehrl-Rényi entropy for several important quantum states, including random states and specific multi-particle entangled states known as GHZ and W states. These calculations revealed distinct entanglement characteristics for each state, confirming the utility of the Wehrl-Rényi entropy as a classifying tool. While the current work focuses on pure quantum states, the authors acknowledge that extending this framework to mixed states presents a significant challenge for future research. They also suggest exploring connections between the Wehrl-Rényi entropy and other properties of strongly interacting quantum systems, such as those found in models relevant to holographic duality and quantum complexity, as well as applying it to the study of measurement-induced criticality.