Abstract
Recently, there are tremendous developments on the number of controllable qubits in several quantum computing systems. For these implementations, it is crucial to determine the entanglement structure of the prepared multipartite quantum state as a basis for further information processing tasks. In reality, evaluation of a multipartite state is in general a very challenging task owing to the exponential increase of the Hilbert space with respect to the number of system components. In this work, we propose a systematic method using very few local measurements to detect multipartite entanglement structures based on the graph state—one of the most important classes of quantum states for quantum information processing. Thanks to the close connection between the Schmidt coefficient and quantum entropy in graph states, we develop a family of efficient witness operators to detect the entanglement between subsystems under any partitions and hence the entanglement intactness. We show that the number of local measurements equals to the chromatic number of the underlying graph, which is a constant number, independent of the number of qubits. In reality, the optimization problem involved in the witnesses can be challenging with large system size. For several widely used graph states, such as 1-D and 2-D cluster states and the Greenberger–Horne–Zeilinger state, by taking advantage of the area law of entanglement entropy, we derive analytical solutions for the witnesses, which only employ two local measurements. Our method offers a standard tool for entanglement-structure detection to benchmark multipartite quantum systems.
Highlights
Tremendous efforts have been devoted to the realization of multipartite entanglement in various systems,[10,11,12,13,14,15,16,17,18,19,20] which provide the foundation for small- and medium-scale quantum information processing in near future and will eventually pave the way to universal quantum computing
We propose a systematic method to witness the entanglement structure based on graph states
We propose a systematic method to construct efficient witnesses to detect entanglement structures based on graph states
Summary
Entanglement is an essential resource for many quantum information tasks,[1] such as quantum teleportation,[2] quantum cryptography,[3,4] nonlocality test,[5] quantum computing,[6] quantum simulation,[7] and quantum metrology.[8,9] Tremendous efforts have been devoted to the realization of multipartite entanglement in various systems,[10,11,12,13,14,15,16,17,18,19,20] which provide the foundation for small- and medium-scale quantum information processing in near future and will eventually pave the way to universal quantum computing. Genuine multipartite entanglement has been demonstrated and witnessed in experiment with a small amount of qubits in different realizations, such as 14-ion-trap-qubit,10 12-superconducting-qubit,[14] and 12photon-qubit systems.[17]. Even without global genuine entanglement as the target state possesses, the experimental prepared state might still have fewer-body entanglement within a subsystem and/ or among distinct subsystems.[21,22,23] The study of lower-order entanglement, which can be characterized by the detailed entanglement structures,[24,25,26] is important for quantum hardware development, because it might reveal the information on unwanted couplings to the environment and acts as a benchmark of the underlying system. The certified lower-order entanglement among several subsystems could be still useful for some quantum information tasks
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