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.
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 \ensuremathϵ-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.
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.
Recently, a proper genuine multipartite entanglement measure has been found for three-qubit pure states [see Xie and Eberly, Phys. Rev. Lett. 127, 040403 (2021)PRLTAO0031-900710.1103/PhysRevLett.127.040403], but capturing useful entanglement measures for mixed states has remained an open challenge. So far, it requires not only a full tomography in experiments, but also huge calculational labor. A leading proposal was made by Gühne, Reimpell, and Werner [Phys. Rev. Lett. 98, 110502 (2007)PRLTAO0031-900710.1103/PhysRevLett.98.110502], who used expectation values of entanglement witnesses to describe a lower bound estimation of entanglement. We provide here an extension that also gives genuine upper bounds of entanglement. This advance requires only the expectation value of any Hermitian operator. Moreover, we identify a class of operators A_{1} that not only give good estimates, but also require a remarkably small number of experimental measurements. In this Letter, we define our approach and illustrate it by estimating entanglement measures for a number of pure and mixed states prepared in our recent experiments.
Recently, a technique known as quantum symmetry test has gained increasing attention for detecting bipartite entanglement in pure quantum states. In this work we show that, beyond qualitative detection, a family of well-defined measures of bipartite and multipartite entanglement can be obtained with symmetry tests. We propose and benchmark several efficient methods to estimate these measures, and derive near-optimal sampling strategies for each. Despite the nonlinearity of the methods, we demonstrate that the sampling error scales no worse than O(Ntot−1/2) with the total number of copies Ntot, which suggests experimental feasibility. By exploiting symmetries we compute our measures for large number of copies, and derive the asymptotic decay exponents for relevant states in many-body systems. Using these results we identify tradeoffs between estimation complexity and sensitivity of the presented entanglement measures, oriented to practical implementations.
Multipartite entanglement is the premier resource for quantum technologies. Yet, its exact quantification in the laboratory is notoriously challenging, typically requiring the full knowledge of high dimensional quantum states. Here, we construct observable bounds to multipartite entanglement for systems of arbitrary size, which are functions of the local and global state purities, and correlation functions. First, we derive experimentally accessible upper and lower limits to both the bipartite entanglement of formation and the squashed entanglement of bipartite systems, by leveraging cornerstone results of quantum information theory: the entropy strong subadditivity inequality and the Koashi-Winter monogamy relation. Then, we convert them into bounds to the entanglement up to degree k for arbitrary states, and to the genuine k-partite entanglement, by employing a recently proposed method. Finally, we analytically and numerically test these results, by bounding the multipartite entanglement of several relevant states and mixtures, including the important classes of GHZ, Dicke, W states, and random pure states.
Despite multipartite entanglement being a global property of a quantum state, a number of recent works have made it clear that it can be quantified using only local measurements. This is appealing because local measurements are the easiest to implement on current quantum hardware. However, it remains an open question what protocol one should use in order to minimize the resources required to estimate multipartite entanglement from local measurements alone. In this work, we construct and compare several estimators of multipartite entanglement based solely on the data from local measurements. We first construct statistical estimators for a broad family of entanglement measures using local randomized measurement (LRM) data before providing a general criterion for the construction of such estimators in terms of projective 2-designs. Importantly, this allows us to derandomize the multipartite estimation protocol based on LRMs. In particular, we show how local symmetric, informationally complete positive operator-valued measures enable multipartite entanglement quantification with only a single measurement setting. For all estimators, we provide both the classical postprocessing cost and rigorous performance guarantees in the form of analytical upper bounds on the number of measurements needed to estimate the measures to any desired precision.