Thermodynamic Probes of Multipartite Entanglement in Strongly Interacting Quantum Systems

TL;DR

This work develops a thermodynamic route to quantify genuine multipartite entanglement in strongly interacting quantum systems by using ergotropy and its global/local decompositions. By reformulating the ergotropic gap to allow interaction quenches or local measurements, the authors derive an operational measure—ergotropic volume—that captures GME across stationary and time-evolved states. They apply the framework to the Tavis-Cummings model, a three-level Dicke model, and the transverse-field Ising model, deriving analytical expressions where possible and validating with numerical results, including phase-transition signatures. Furthermore, they outline a quantum-circuit implementation using variational quantum algorithms to estimate ergotropy-based entanglement on NISQ devices, enabling practical entanglement benchmarking during quantum simulations. The approach provides a versatile, experimentally accessible tool for characterizing entanglement in near-term quantum technologies and strongly interacting many-body systems.

Abstract

Quantifying multipartite entanglement in quantum many-body systems and hybrid quantum computing architectures is a fundamental yet challenging task. In recent years, thermodynamic quantities such as the maximum extractable work from an isolated system (the ergotropy) have allowed for entanglement measures that are operationally more accessible. However, these measures can be restrictive when applied to systems governed by Hamiltonians with strong collective or interparticle interactions. Motivated by advances in quantum simulators, we propose a framework that circumvents these restrictions by evaluating global and local ergotropy either through controlled quenching of interactions or by measuring suitable local observables only. We show that this formalism allows us to correctly estimate genuine multipartite entanglement in both stationary and time-evolved states of systems with strong interactions, including parametrized quantum states simulated on a quantum circuit with varying circuit depth and noise. We demonstrate its applicability to realistic physical models, namely, the Tavis-Cummings model, the three-level Dicke model, and the transverse-field Ising model, highlighting its potential as a versatile tool for characterizing entanglement in near-term quantum simulators.

Figures (8)

• Figure 1: Visualization of ergotropy as the maximum work that can be extracted from a charged quantum battery. (a) The optimal unitary $\hat{U}^*$ discharges the battery to the passive state, which is the lowest energy state with same entropy. This is the ground state if the initial state is pure. However, for mixed initial states, the passive state is not the ground state but a mixture of low energy states. The blue box represents a device that measures the ergotropy $\mathscr{E}_{\hat{H}}(\hat{\rho})$, optimized over a set of unitaries $\hat{U}$ and Hamiltonian $\hat{H}$. (b) For a state $\rho_{AB}$, the global entropy $\mathscr{E}^g_{\hat{H}_{AB}}({\hat{\rho}_{AB}})$ is measured if the device optimizes over all global unitaries $\hat{U}_{AB}$, whereas (c) the local ergotropy is measured for the marginals $\rho_{A(B)}$, optimized over all local unitaries $\hat{U}_{A(B)}$ i.e., $\mathscr{E}^l_{\hat{H}_{AB}}({\hat{\rho}_{AB}}) = \mathscr{E}^A_{\hat{H}_{A}}({\hat{\rho}_{A}}) + \mathscr{E}^B_{\hat{H}_{B}}({\hat{\rho}_{B}})$. The local ergotropy expression comes from the simplification of Eq. \ref{['Eq:locErg']} after quenching the interaction. The figure also illustrates the strongly interacting systems we study: (d) the transverse-field Ising model, (e) the Tavis-Cummings model and (f) the three-level Dicke model.
• Figure 2: Ergotropic volume for dressed states with different number of excitations $i$. The system contains $N=100$ spins and maximum number of photons in the cavity varies from $N_{ph}=2$ to $198$, with $N_{ph}<N$ shown in red and $N_{ph}>N$ shown in blue.
• Figure 3: Phase diagram of a closed three-level system captured using ergotropic volume $\Delta^V_{\hat{H}_0^N}(\ket{\psi_N^{0}})$ of ground state $\psi_N^{0}$. The system parameters are $N=5$ and $\omega_c = \omega_a = \omega$.
• Figure 4: Ergotropic volume for ground state of transverse field Ising model. The markers represent the ergotropy computed using the analytical method for $N=10$ (blue dots), $20$ (green up-triangles), $40$ (red down-triangles) and $50$ (black cross) spins. For $N=10$ spins, the solid blue curve represents the ergotropic volume computed numerically considering all possible bipartitions.
• Figure 5: Measuring ergotropy using quantum circuits. (a) The quantum circuit for studying the entanglement dynamics in the system. (b) Unitary $\hat{U}_t$ time evolves the initial state (ground state in this case), which is implemented using Suzuki-Trotter decomposition Berry2007, where the base circuit (orange dashed box) is repeated $d$ times to reduce error. (c) Ansatz circuit for finding the optimal unitary $\hat{U}_{\mathcal{E}}^n(\theta_{opt})$ that minimizes the energy of the time-evolved state. This circuit also has a depth $d$ to reduce the error during computation. (d) Flow-chart for passive state optimization protocol, where quantum circuit computes the energy of state $E_{\theta}^n$ after implementing unitary $\hat{U}_{\mathcal{E}}^n(\theta)$. This energy is then fed to a classical optimizer which keeps on changing $\theta$ and running the quantum circuit in (a) until the optimal value $\theta_{opt}$ that minimizes the energy is found.