Family of two-parameter multipartite entanglement measures

TL;DR

This work introduces unified-entropy concentratable entanglements, a two-parameter family $E^{(s)}_{\\alpha,\\beta}$ that unifies bipartite and multipartite entanglement measures for mixed states. It proves these quantities are entanglement monotones under LOCC (and multipartite separable operations) and shares desirable properties like subadditivity and continuity, while interpolating among Rényi, Tsallis, von Neumann, and original concentratable entanglements. The authors establish ordering relations among variants, provide practical bounds via the original concentratable entanglement, and demonstrate the approach on GHZ vs. W states, a four-partite star network, and four-qubit Dicke states, showing improved sensitivity in certain regimes. The framework enables efficient estimation on near-term devices and offers a versatile tool for characterizing multipartite entanglement through bipartite quantities, with implications for quantum networks and sensing.

Abstract

Multipartite entanglement is regarded as a crucial physical resource in quantum network communication. However, due to the intrinsic complexity of quantum many-body systems, identifying a multipartite entanglement measure that is both efficiently computable and capable of accurately characterizing entanglement remains a challenging problem. To address these issues, we propose a family of two-parameter multipartite entanglement measures for mixed states, termed unified-entropy concentratable entanglements. Many well-known multipartite entanglement measures are recovered as special cases of this family of measures, such as the entanglement of formation and the concentratable entanglements introduced in [Phys. Rev. Lett. 127, 140501 (2021)]. We demonstrate that the unified-entropy concentratable entanglements constitutes a well-defined entanglement monotones, and establish several desirable properties it satisfies, such as subadditivity and continuity. We further investigate the ordering relations of unified-entropy concentratable entanglements and discuss how these quantities can be efficiently estimated on near-term quantum devices. As an application, we demonstrate that the unified-entropy concentratable entanglements can effectively distinguish between multi-qubit Greenberger-Horne-Zeilinger (GHZ) states and W states. The ordering relations of these entanglement measures are further validated using four-partite star quantum network states and four-qubit Dicke states. Moreover, we find that the unified-entropy concentratable entanglements exhibit greater sensitivity than the original concentratable entanglements in detecting certain four-partite star quantum network states.

Figures (7)

• Figure 1: The difference $\Delta=E^{(s)}(\ket{\text{GHZ}}) - E^{(s)}(\ket{W})$ for various subsystem sizes $|s|$.
• Figure 2: The von Neumann concentratable entanglements serves as an effective measure for distinguishing GHZ states from W states.
• Figure 3: The Rényi concentratable entanglements serves as an effective measure for distinguishing GHZ states from W states.
• Figure 4: The Tsallis concentratable entanglements serves as an effective measure for distinguishing GHZ states from W states.
• Figure 5: The concentratable entanglements serves as an effective measure for distinguishing GHZ states from W states.

Theorems & Definitions (11)

• Definition 1
• Definition 2: unified-entropy concentratable entanglements
• Theorem 3
• Proposition 4
• Proposition 5
• Corollary 6
• Corollary 7
• Proposition 8
• Theorem 9
• Theorem 10