The controlled SWAP test for detecting and quantifying entanglement applied to pure qubit states is robust to small errors in the states and efficient for large multi-qubit states (Foulds et al . 2021 Quantum Sci. Technol . 6 , 035002 (doi: 10.1088/2058-9565/abe458 )). We extend this, and the related measure concentratable entanglement (CE), to enable important practical applications in quantum information processing. We investigate the lower bound of concentratable entanglement given in (Beckey et al . 2023 Phys. Rev. A 107 , 062425 (doi: 10.1103/physreva.107.062425 )) and conjecture an upper bound of the mixed-state concentratable entanglement that is robust to c-SWAP test errors. Since experimental states are always slightly mixed, our work makes the c-SWAP test and CE measure suitable for application in experiments to characterize entanglement. We further present the CE of some key higher-dimensional states such as qudit states and entangled optical states to validate the CE as a higher-dimensional measure of entanglement. This article is part of the theme issue ‘The quantum theory of light’.
The controlled SWAP test for detecting and quantifying entanglement applied to pure qubit states is robust to small errors in the states and efficient for large multi-qubit states (Foulds et al . 2021 Quantum Sci. Technol . 6 , 035002 (doi: 10.1088/2058-9565/abe458 )). We extend this, and the related measure concentratable entanglement (CE), to enable important practical applications in quantum information processing. We investigate the lower bound of concentratable entanglement given in (Beckey et al . 2023 Phys. Rev. A 107 , 062425 (doi: 10.1103/physreva.107.062425 )) and conjecture an upper bound of the mixed-state concentratable entanglement that is robust to c-SWAP test errors. Since experimental states are always slightly mixed, our work makes the c-SWAP test and CE measure suitable for application in experiments to characterize entanglement. We further present the CE of some key higher-dimensional states such as qudit states and entangled optical states to validate the CE as a higher-dimensional measure of entanglement. This article is part of the theme issue ‘The quantum theory of light’.
Abstract Quantum entanglement plays a key role in the field of quantum computing and quantum communication. The controlled SWAP test has been used to detect pure state entanglement ( Foulds S. et al. Quantum Sci. Technol. , 6 (2021) 035002; Beckey J. L. et al. Phys. Rev. Lett. , 127 (2021) 140501). In this paper, we investigate that quantum circuits for controlled SWAP test can be used to detect the entanglement of multipartite mixed state. Based on the measured probability, we propose entanglement criteria for detecting multipartite entanglement and genuine entanglement of pure states. In addition, we give representation of the entanglement measures based on the measured probability of the controlled SWAP tests. Numerical examples are used to illustrate the efficiency of the entanglement criteria.
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 one of the hallmarks of quantum mechanics and is central to quantum information processing. In this work we show that concentratable entanglement (CE), an operationally motivated entanglement measure, induces a hierarchy upon pure states from which different entanglement structures can be experimentally certified. In particular, we find that nearly all genuine multipartite entangled states can be verified through the CE. Interestingly, GHZ states prove to be far from maximally entangled according to this measure. Instead we find the exact maximal value and corresponding states for up to 18 qubits and show that these correspond to extremal quantum error correcting codes. The latter allows us to unravel a deep connection between CE and coding theory. Finally, our results also offer an alternative proof, on up to 31 qubits, that absolutely maximally entangled states do not exist. Published by the American Physical Society 2024
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 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.