Exact Relation Between Wehrl-Rényi Entropy and Many-Body Entanglement

Abstract: Quantum entanglement is key to understanding correlations and emergent phenomena in quantum many-body systems. For $N$ qubits (distinguishable spin-$1/2$ particles) in a pure quantum state, many-body entanglement can be characterized by the purity of the reduced density matrix of a subsystem, defined as the trace of the square of this reduced density matrix. Nevertheless, this approach depends on the choice of subsystem. In this letter, we establish an exact relation between the Wehrl-R\'enyi entropy (WRE) $S_W^{(2)}$, which is the 2nd R\'enyi entropy of the Husimi function of the entire system, and the purities of all possible subsystems. Specifically, we prove the relation $e^{-S_W^{(2)}} = (6\pi)^{-N} \sum_A \mathrm{Tr}({{\hat \rho}_A}^2)$, where $A$ denotes a subsystem with reduced density matrix ${\hat \rho}_A$, and the summation runs over all $2^N$ possible subsystems. Furthermore, we show that the WRE can be experimentally measured via a concrete scheme. Therefore, the WRE is a subsystem-independent and experimentally measurable characterization of the overall entanglement in pure states of $N$ qubits. It can be applied to the study of strongly correlated spin systems, particularly those with all-to-all couplings that do not have a natural subsystem division, such as systems realized with natural atoms in optical tweezer arrays or superconducting quantum circuits. We also analytically derive the WRE for several representative many-body states, including Haar-random states, the Greenberger-Horne-Zeilinger (GHZ) state, and the W state.