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Multipartite entanglement measures based on the thermodynamic framework

Chen-Ming Bai, Yu Luo

TL;DR

This work builds a thermodynamic bridge to multipartite entanglement by introducing ergotropic-gap and battery capacity-gap concentratable entanglement measures, $M_E^{(s)}$ and $M_B^{(s)}$. It proves these quantities form valid LOCC-monotones, are continuous, satisfy majorization monotonicity, and, under equispaced energy levels, become equivalent up to a factor, enabling a unified quantification of entanglement distribution across subsystems. The authors demonstrate the framework's usefulness by deriving a sufficient GME criterion for three-qubit states, contrasting GHZ and W classes, and analyzing a four-partite star network where entanglement concentration can emerge from network structure. They provide explicit calculations for representative states (3-qubit, GHZ/W, and a star network) and compare with existing thermodynamic entanglement measures, illustrating distinct ordering and sensitivity. Overall, the paper offers a principled, operational link between quantum thermodynamics and multipartite entanglement with practical implications for state discrimination and networked quantum systems.

Abstract

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.

Multipartite entanglement measures based on the thermodynamic framework

TL;DR

This work builds a thermodynamic bridge to multipartite entanglement by introducing ergotropic-gap and battery capacity-gap concentratable entanglement measures, and . It proves these quantities form valid LOCC-monotones, are continuous, satisfy majorization monotonicity, and, under equispaced energy levels, become equivalent up to a factor, enabling a unified quantification of entanglement distribution across subsystems. The authors demonstrate the framework's usefulness by deriving a sufficient GME criterion for three-qubit states, contrasting GHZ and W classes, and analyzing a four-partite star network where entanglement concentration can emerge from network structure. They provide explicit calculations for representative states (3-qubit, GHZ/W, and a star network) and compare with existing thermodynamic entanglement measures, illustrating distinct ordering and sensitivity. Overall, the paper offers a principled, operational link between quantum thermodynamics and multipartite entanglement with practical implications for state discrimination and networked quantum systems.

Abstract

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.
Paper Structure (10 sections, 6 theorems, 72 equations, 5 figures, 2 tables)

This paper contains 10 sections, 6 theorems, 72 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminary
  3. The ergotropic gap and battery capacity gap
  4. The ergotropic-gap concentratable entanglement and battery capacity-gap concentratable entanglement
  5. Properties of ergotropic-gap concentratable entanglement
  6. Example
  7. The simplest multipartite system $(\mathbb{C}^2)^{\otimes3}$
  8. n-qubit GHZ states and W states
  9. A four-partite star quantum network
  10. Conclusion

Key Result

Theorem 3

For a system with equispaced energy level Hamiltonians $H_X=\sum_{j=0}^{d-1}j\epsilon^{X}\ket{\epsilon_j^{X}}\bra{\epsilon_j^{X}}$ and $H_{X^c}=\sum_{j=0}^{d-1}j\epsilon^{X^c}\ket{\epsilon_j^{X^c}}\bra{\epsilon_j^{X^c}}$, the battery capacity-gap and ergotropic-gap concentratable entanglement measur for any $n$-qubit pure state $\ket{\psi}$ and qubit subset $s\subseteq [n]$.

Figures (5)

  • Figure 1: $M^{(s)}_{E}(\ket{\psi}_{ABC})$ as a function of $\lambda_0$ with random sampling and $s=[3]$.
  • Figure 2: Comparison of entanglement measures between quantum states $\ket{\Psi}$ and $\ket{\Phi}$.
  • Figure 3: The difference $\Delta=M^{(s)}_E(\ket{\text{GHZ}}_n) - M^{(s)}_E(\ket{W}_n)$ for various subsystem sizes $|s|$.
  • Figure 4: The ergotropic-gap concentratable entanglements serves as an effective measure for distinguishing GHZ states from W states.
  • Figure 5: Comparison of minimum bi-separable erogotropic gap $\Delta_{\min}^{[2]}$, bi-separable average erogotropic gap $\Delta_{\rm avg}^{[2]}$, 2-erogotropic volume $\Delta_{\rm V}^{[2]}$, and the ergotropic-gap concentratable entanglement $M_{E}^{(s)}$.

Theorems & Definitions (14)

  • Definition 1: ergotropic-gap concentratable entanglement
  • Definition 2: battery capacity-gap concentratable entanglement
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Theorem 5
  • proof
  • Proposition 6
  • proof
  • ...and 4 more