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

TL;DR

The work addresses the subsystem-dependence of entanglement measures in all-to-all connected $N$-qubit pure states. By performing local SU(2) twirling and evaluating a four-point Haar integral, it proves the exact relation $e^{-S_W^{(2)}(\rho)} = \frac{1}{(6\pi)^N} \sum_A Tr(\rho_A^2)$, linking the Wehrl-Rényi entropy to the sum of all subsystem purities. It also provides a concrete experimental protocol to measure $S_W^{(2)}$ via two copies and derives closed-form WRE values for Haar-random, GHZ, W, and $p$-Bell states, along with asymptotic bounds that classify entanglement structures. These results establish the WRE as a subsystem-independent, experimentally accessible entanglement diagnostic applicable to strongly interacting, all-to-all coupled quantum systems, with potential connections to holographic complexity and related dynamical phenomena.

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-Rényi entropy (WRE) $S_W^{(2)}$, which is the 2nd Rényi entropy of the Husimi function of the entire system, and the purities of all possible subsystems. Specifically, we prove the relation $e^{-S_W^{(2)}} = (6π)^{-N} \sum_A \mathrm{Tr}({{\hat ρ}_A}^2)$, where $A$ denotes a subsystem with reduced density matrix ${\hat ρ}_A$, 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.

Figures (9)

• Figure 1: We present a schematic illustrating the relationship between the WRE $S_W^{(2)}$ and the sum of purities over all possible subsystems, thereby confirming that the WRE provides an efficient characterization of the overall entanglement within the entire system.
• Figure 2: As an illustration, we summarize the key steps in the proof of the exact relation between WRE and subsystem purities. Panel (a) corresponds to Eq.\ref{['eqn:main1']}; while panel (b) corresponds to Eq. \ref{['eqn:main2']} with operators $\pi_1,\pi_2\in\{\hat{I},\hat{W}\}$, where $\hat{I},\hat{W}$ represent the identity operator and the SWAP operator respectively.
• Figure 3: We provide an illustration of the experimental protocol for measuring the WRE. It consists of three steps. Step 1: preparing two identical copies of the density matrix. Step 2: performing an evolution under the Hamiltonian $\hat{V}$. Step 3: performing a projective measurement in the computational basis. More details are presented in the main text.
• Figure 4: We plot the WRE as a function of the system size $N$ using analytical expressions for representative pure states in many-body systems, including Haar-random states (averaged), the GHZ state, the W state, and the $p$-Bell state.
• Figure S1: Calculation of the two-point function. Panel (a) corresponds to Eq.\ref{['WU']}; while panel (b) corresponds to Eq. \ref{['Adelta']}.