Multipartite entanglement measures based on the thermodynamic framework

In this work, we introduce a unified method to characterize and measure multipartite entanglement using the framework of thermodynamics. A family of the new entanglement measures is proposed: \textit{ergotropic-gap concentratable entanglement}. Furthermore, we establish that ergotropic-gap concentratable entanglement constitutes a well-defined entanglement measure within a specific parameter regime, satisfying key properties including continuity, majorization monotonicity and monogamy. We demonstrate the utility of this measure by showing it effectively distinguishes between multi-qubit Greenberger-Horne-Zeilinger states and W states. It also proves effective in detecting entanglement in specific classes of four-partite star quantum network states.

💡 Research Summary

The paper introduces a unified thermodynamic framework for quantifying multipartite quantum entanglement. Building on the concepts of ergotropy (the maximal extractable work from a quantum state) and its counterpart anti‑ergotropy, the authors define the ergotropic gap Δ as the difference between global and local ergotropy, and the battery‑capacity gap Δ_cg as the difference between global and local battery capacities. They then construct two families of entanglement measures: ergotropic‑gap concentratable entanglement M(s)E and battery‑capacity‑gap concentratable entanglement M(s)B, where s denotes a chosen subset of qubits and the measures are obtained by averaging the respective gaps over all bipartitions of s, with a convex‑roof extension to mixed states.

A central result (Theorem 3) shows that for systems whose local Hamiltonians have equally spaced energy levels, the two measures are simply related by M(s)B = 2 M(s)E. This equivalence highlights that both thermodynamic quantities encode the same entanglement information under this physically relevant condition.

The authors rigorously prove that M(s)E satisfies the standard axioms of a multipartite entanglement monotone. Non‑negativity follows because fully separable states have zero ergotropic gap; monotonicity under LOCC is established by invoking known results that the ergotropic gap cannot increase under local operations and classical communication for pure states, and extending the argument to mixed states via the convex‑roof construction. Additional desirable properties are demonstrated: continuity (the measure changes at most linearly with the trace distance between states) and majorization monotonicity (if the marginal spectra of one state majorize those of another, the former’s M(s)E is larger). The paper also derives monogamy relations, showing that the entanglement quantified by M(s)E cannot be freely shared among many parties, consistent with known multipartite entanglement behavior.

To illustrate practical relevance, the authors compute M(s)E for three‑qubit GHZ and W states. GHZ states achieve the maximal ergotropic gap across all bipartitions, yielding a high M(s)E, whereas W states give a lower average value, allowing clear discrimination between these inequivalent entanglement classes. Moreover, they examine a four‑partite star‑shaped quantum network (one central node connected to three peripheral nodes) and find parameter regimes where M(s)E exceeds traditional entanglement measures, indicating enhanced sensitivity for detecting genuine multipartite entanglement in networked systems.

Overall, the work bridges quantum thermodynamics and entanglement theory, providing a physically motivated, experimentally accessible entanglement quantifier that respects all core axioms. The reliance on equally spaced Hamiltonians makes the framework directly applicable to many quantum platforms (e.g., superconducting qubits, trapped ions) where such spectra are common. Future directions suggested include extending the analysis to non‑equispaced Hamiltonians, higher‑dimensional systems, and concrete experimental implementations in near‑term quantum devices.