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arXiv:0705.3904

Multiple Entropy Measures for Multipartite Quantum Entanglement

Dan Liu, Xin Zhao, Long, Gui Lu

Published May 26, 2007• 1 citation
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Abstract

A new entanglement measure, the multiple entropy measures (MEMS), is proposed to quantify quantum entanglement of multi-partite quantum state. The MEMS is vector-like with $m=[N/2]$, the integer part of $N/2$, components: $[S_1, S_2,..., S_m]$, and the $i$-th component $S_i$ is the geometric mean of $i$-body partial entropy of the system. The $S_i$ measures how strong an arbitrary $i$ bodies from the system are entangled with the rest of the system. The MEMS is not only transparent in physical picture, but also simple to calculate. It satisfies the conditions for a good entanglement measure. We have analyzed the entanglement properties of the GHZ-state, the W-states and cluster-states under MEMS. The cluster-state is more entangled than the GHZ-state and W-state under MEMS.

DOI
10.48550/arxiv.0705.3904

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A new entanglement measure, the multiple entropy measures (MEMS), is proposed to quantify quantum entanglement of multi-partite quantum state. The MEMS is vector-like with m=[N/2]m=[N/2]m=[N/2], the integer part of N/2N/2N/2, components: [S1,S2,...,Sm][S_1, S_2,..., S_m][S1​,S2​,...,Sm​], and the iii-th component SiS_iSi​ is the geometric mean of iii-body partial entropy of the system. The SiS_iSi​ measures how strong an arbitrary iii bodies from the system are entangled with the rest of the system. The MEMS is not only transparent in physical picture, but also simple to calculate. It satisfies the conditions for a good entanglement measure. We have analyzed the entanglement properties of the GHZ-state, the W-states and cluster-states under MEMS. The cluster-state is more entangled than the GHZ-state and W-state under MEMS.
We propose to use a set of averaged entropies, the multiple entropy measures (MEMS), to partially quantify quantum entanglement of multipartite quantum state. The MEMS is vector-like with m = [N/2] components: [S 1 , S 2 , ..., S m ], and the i-th component S i is the geometric mean of i-qubits partial entropy of the system. The S i measures how strong an arbitrary i qubits from the system are correlated with the rest of the system. It satisfies the conditions for a good entanglement measure. We have analyzed the entanglement properties of the GHZ-state, the W-states, and cluster-states under MEMS.
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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.
In this paper, we investigate the characterization of multipartite entanglement based on the unified (q,s)(q,s)(q,s)-entropy framework. For bipartite quantum states, we propose an entanglement measure Eq,sA∣B(ρ)E_{q,s}^{A|B}(ρ)Eq,sA∣B​(ρ) based on unified entropy and derive several analytical lower bounds for it using local orthonormal observables. We demonstrate with concrete examples that our lower bounds provide tighter estimates of quantum entanglement than several existing analytical bounds. For multipartite quantum states, based on unified entropy, we propose two measures Aq,sk(ρ)A^k_{q,s}(ρ)Aq,sk​(ρ) and Gq,sk(ρ)G^k_{q,s}(ρ)Gq,sk​(ρ) for quantifying kkk-nonseparability, and prove that these measures satisfy several desirable properties, such as faithfulness, local unitary invariance, and convexity. Moreover, we rigorously prove that when the parameters qqq and sss are in certain ranges, Aq,sk(ρ)A^k_{q,s}(ρ)Aq,sk​(ρ) and Gq,sk(ρ)G^k_{q,s}(ρ)Gq,sk​(ρ) also satisfy monotonicity and strong monotonicity. Furthermore, we establish the order relations of the measures Aq,sk(ρ)A^k_{q,s}(ρ)Aq,sk​(ρ) and Gq,sk(ρ)G^k_{q,s}(ρ)Gq,sk​(ρ) with respect to the parameters qqq and sss for fixed quantum states. In addition, for fixed qqq and sss, we also give their order relations for any two pure states.