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arXiv:2104.06923 · Physical Review Letters

Computable and Operationally Meaningful Multipartite Entanglement Measures

Jacob L. Beckey, N. Gigena, Patrick J. Coles, Marco Cerezo

Published Sep 27, 2021• 68 citations
arXivHTMLPDF

Abstract

Multipartite entanglement is an essential resource for quantum communication, quantum computing, quantum sensing, and quantum networks. The utility of a quantum state |ψ⟩ for these applications is often directly related to the degree or type of entanglement present in |ψ⟩. Therefore, efficiently quantifying and characterizing multipartite entanglement is of paramount importance. In this work, we introduce a family of multipartite entanglement measures, called concentratable entanglements. Several well-known entanglement measures are recovered as special cases of our family of measures, and hence we provide a general framework for quantifying multipartite entanglement. We prove that the entire family does not increase, on average, under local operations and classical communications. We also provide an operational meaning for these measures in terms of probabilistic concentration of entanglement into Bell pairs. Finally, we show that these quantities can be efficiently estimated on a quantum computer by implementing a parallelized SWAP test, opening up a research direction for measuring multipartite entanglement on quantum devices.

DOI
10.1103/physrevlett.127.140501

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Multipartite entanglement is an essential resource for quantum communication, quantum computing, quantum sensing, and quantum networks. The utility of a quantum state |ψ⟩ for these applications is often directly related to the degree or type of entanglement present in |ψ⟩. Therefore, efficiently quantifying and characterizing multipartite entanglement is of paramount importance. In this work, we introduce a family of multipartite entanglement measures, called concentratable entanglements. Several well-known entanglement measures are recovered as special cases of our family of measures, and hence we provide a general framework for quantifying multipartite entanglement. We prove that the entire family does not increase, on average, under local operations and classical communications. We also provide an operational meaning for these measures in terms of probabilistic concentration of entanglement into Bell pairs. Finally, we show that these quantities can be efficiently estimated on a quantum computer by implementing a parallelized SWAP test, opening up a research direction for measuring multipartite entanglement on quantum devices.
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.
We show how to estimate a broad class of multipartite entanglement measures from Bell-basis measurement data. In addition to lowering the experimental requirements relative to previously known methods of estimating these measures, our proposed scheme also enables a simpler analysis of the number of measurement repetitions required to achieve an $\ensuremath{\epsilon}$-close approximation of the measures, which we provide for each. We focus our analysis on the recently introduced concentratable entanglements [Beckey et al., Phys. Rev. Lett. 127, 140501 (2021)] because many other well-known multipartite entanglement measures are recovered as special cases of this family of measures. We extend the definition of the concentratable entanglements to mixed states and show how to construct lower bounds on the mixed state concentratable entanglements that can also be estimated using only Bell-basis measurement data. Finally, we demonstrate the feasibility of our methods by classically simulating their implementation on a noisy Rydberg atom quantum computer.
ABSTRACT Multipartite entanglement is widely recognized as a fundamental resource for various quantum protocols, including those in quantum communication and quantum computing. However, a unified characterization of multipartite entanglement remains elusive. In this paper, we introduce ‐Concentratable Entanglements, a one‐parameter family of measures for multipartite entanglement. This framework generalizes the Concentratable Entanglements framework, recovering the original measure at the limit of . We prove that ‐Concentratable Entanglements is a valid entanglement measure satisfying local operations and classical communication monotonicity and continuity. While linear cluster states are known to exhibit higher concentratable entanglements than Greenberger‐Horne‐Zeilinger (GHZ) states at the specific point , To demonstrate the versatility of the ‐Concentratable Entanglements framework, we performed a comprehensive analysis of several paradigmatic multipartite states. Our comparison of ‐qubit GHZ and W states shows that the measure can distinguish them well, while the investigation of the 4‐partite star network illustrates the role of as a tunable parameter for probing structural subtleties. Most significantly, by analyzing the 4‐qubit linear cluster state () and the 4‐qubit GHZ states, we find that the hierarchical ordering is a robust feature in the parameter range .
Multipartite entanglement is an essential resource for quantum information theory and technologies, but its quantification has been a persistent challenge. Concentratable entanglement (CE), introduced recently, can be estimated from just two copies of a quantum state. Here, we propose generalized concentratable entanglement (GCE), a broader class of multipartite entanglement measures naturally tied to quantum Tsallis entropies, and present a parallelized protocol for estimating GCE across multiple state copies. Increasing the number of copies yields an improved error bound in the presence of imperfections. We prove that GCE is a well-defined entanglement monotone and conjecture some new entropic inequalities. Moreover, we demonstrate the concentration of entanglement into W states using three-state copies. Our results contribute to more robust and versatile characterizations of multipartite entanglement.
Multipartite entanglement is an essential resource for quantum information theory and technologies, but its quantification has been a persistent challenge. Concentratable entanglement (CE), introduced recently, can be estimated from just two copies of a quantum state. Here, we propose generalized concentratable entanglement (GCE), a broader class of multipartite entanglement measures naturally tied to quantum Tsallis entropies, and present a parallelized protocol for estimating GCE across multiple state copies. Increasing the number of copies yields an improved error bound in the presence of imperfections. We prove that GCE is a well-defined entanglement monotone and conjecture some new entropic inequalities. Moreover, we demonstrate the concentration of entanglement into W states using three-state copies. Our results contribute to more robust and versatile characterizations of multipartite entanglement.
Abstract Entanglement is a striking feature of quantum mechanics, and it has a key property called unextendibility. In this paper, we present a framework for quantifying and investigating the unextendibility of general bipartite quantum states. First, we define the unextendible entanglement, a family of entanglement measures based on the concept of a state-dependent set of free states. The intuition behind these measures is that the more entangled a bipartite state is, the less entangled each of its individual systems is with a third party. Second, we demonstrate that the unextendible entanglement is an entanglement monotone under two-extendible quantum operations, including local operations and one-way classical communication as a special case. Normalization and faithfulness are two other desirable properties of unextendible entanglement, which we establish here. We further show that the unextendible entanglement provides efficiently computable benchmarks for the rate of exact entanglement or secret key distillation, as well as the overhead of probabilistic entanglement or secret key distillation.