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Hierarchies of Gaussian multimode entanglement from thermodynamic quantifiers

Mrinmoy Samanta, Sudipta Mondal, Ayan Patra, Saptarshi Roy, Aditi Sen De

Abstract

We develop a thermodynamic characterization of multimode entanglement in pure continuous-variable systems by quantifying the gap between globally and locally extractable work (ergotropy). For arbitrary pure multimode Gaussian states, we prove that the $2$-local ergotropic gap is a faithful entanglement monotone across any bipartition and constitutes a functionally independent upper bound to the Renyi-2 entanglement entropy. We further introduce the $k$-ergotropic score, the minimum $k$-local ergotropic gap, and show that it faithfully quantifies multimode entanglement across $k$ partitions. For pure three-mode Gaussian states, we derive its closed-form relation with the geometric measure for genuine multimode entanglement $(k=2)$, and total Gaussian multimode entanglement $(k=3)$. For systems with more than three modes, the $k$-ergotropic score becomes a functionally independent measure of multimode entanglement to the standard geometric measures. Our results reveal a direct operational hierarchy linking Gaussian multimode entanglement to work extraction under locality constraints, and provide a computable and experimentally accessible thermodynamic framework for characterizing quantum correlations.

Hierarchies of Gaussian multimode entanglement from thermodynamic quantifiers

Abstract

We develop a thermodynamic characterization of multimode entanglement in pure continuous-variable systems by quantifying the gap between globally and locally extractable work (ergotropy). For arbitrary pure multimode Gaussian states, we prove that the -local ergotropic gap is a faithful entanglement monotone across any bipartition and constitutes a functionally independent upper bound to the Renyi-2 entanglement entropy. We further introduce the -ergotropic score, the minimum -local ergotropic gap, and show that it faithfully quantifies multimode entanglement across partitions. For pure three-mode Gaussian states, we derive its closed-form relation with the geometric measure for genuine multimode entanglement , and total Gaussian multimode entanglement . For systems with more than three modes, the -ergotropic score becomes a functionally independent measure of multimode entanglement to the standard geometric measures. Our results reveal a direct operational hierarchy linking Gaussian multimode entanglement to work extraction under locality constraints, and provide a computable and experimentally accessible thermodynamic framework for characterizing quantum correlations.
Paper Structure (25 sections, 12 theorems, 97 equations, 2 figures)

This paper contains 25 sections, 12 theorems, 97 equations, 2 figures.

Table of Contents

  1. Introduction
  2. A framework for $k$-ergotropic score
  3. Measuring bipartite entanglement with $2$-local ergotropic gap
  4. Connection with standard measures of entanglement: Rényi-$2$ vs $2$-local ergotropic gap
  5. Measuring multimode entanglement with $k$-local ergotropic gap
  6. Connecting $k$-mode inseparability with geometric measures of entanglement
  7. Relation between GGM and $2$-ergotropic score
  8. Linking $N$-ergotropic score to the total Gaussian multimode entanglement
  9. conclusion
  10. Preliminaries
  11. Continuous variable formalism
  12. Multimode Gaussian entanglement: Bipartite vs Multipartite
  13. Proof of Lemma \ref{['le:lemma_1']}
  14. Proof of Theorem \ref{['th:th_1']}
  15. Proof of Corollary \ref{['co:corollary2']}
  16. ...and 10 more sections

Key Result

Lemma 1

For a pure $(m_1+m_2)$-mode Gaussian state $\rho_{_N}$, bipartitioned into subsystems $\mathcal{A}_1$ (with $m_1$ modes) and $\mathcal{A}_2$ (with $m_2$ modes), having $m_1 \le m_2$, the $2$-local ergotropic gap is entirely determined by the symplectic spectrum of the reduced state on the smaller su where $\{\nu_i^{\mathcal{A}_1}\}_{i=1}^{m_1}$ denotes the set of symplectic eigenvalues of the redu

Figures (2)

  • Figure 1: Schematic of work extraction from an $N$-mode pure Gaussian state. Global Gaussian unitaries yield the maximal work (global Gaussian ergotropy), while $k$-local Gaussian unitaries yield restricted work ($k$-local ergotropy). Their difference—the $k$-local ergotropic gap minimized over all $k$ partitions ($k$-ergotropic score), quantifies hierarchies of multipartite entanglement of an $N$-mode pure Gaussian state.
  • Figure 2: Relation between thermodynamic and geometric measures of multimode entanglement. We compare multimode entanglement quantified by the $k$-ergotropic score, $\Delta^{k}(\rho_{N})$, (vertical axis) with the generalized geometric measure (GGM), $G(\rho_{N})$, and total multimode entanglement, $\mathcal{E}^{\mathcal{G}}_{N}(\rho_{N})$ (horizontal axis) at fixed total energy, $E=20$. (a) $\Delta^{2}(\rho_{N})$ with respect to $G(\rho_{N})$, for $N=3$ and $N=4$. For $N=3$, these measures exhibit a one-to-one correspondence as shown in Proposition 1 (see inset). (b) The total multimode entanglement (TME), $\mathcal{E}^{\mathcal{G}}_{N}(\rho_{N})$ against $N$-ergotropic score, $\Delta^{N}(\rho_{N})$ for $N=3,~4$. In this case also, we numerically find a one-to-one relation between TME and $N$-ergotropic score for the three-mode scenario (see inset).

Theorems & Definitions (23)

  • Definition 1: Global Gaussian ergotropy (GGE)
  • Definition 2: $k$-local Gaussian ergotropy
  • Definition 3
  • Definition 4
  • Lemma 1
  • Theorem 1
  • proof
  • Corollary 1
  • Theorem 2: $2$-local ergotropic gap bounds Rényi-$2$ entanglement
  • proof
  • ...and 13 more