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arXiv:1902.07869 · Physical Review A

Efficient and robust detection of multipartite Greenberger-Horne-Zeilinger-like states

Qi Zhao, Gerui Wang, Xiao Yuan, Xiongfeng Ma

Published May 28, 2019• 11 citations
arXivHTMLPDF

Abstract

Entanglement is a key resource for quantum information processing. A widely used tool for detecting entanglement is the entanglement witness, where the measurement of the witness operator is guaranteed to be positive for all separable states and can be negative for certain entangled states. In reality, due to exponentially increasing the Hilbert-space dimension with respect to the system size, it is very challenging to construct an efficient entanglement witness for general multipartite entangled states. For $N$-partite Greenberger-Horne-Zeilinger (GHZ)-like states, the most robust witness scheme requires $N+1$ local measurement settings and can tolerate up to $1/2$ white noise. As a comparison, the most efficient witness for GHZ-like states only needs two local measurement settings and can tolerate up to $1/3$ white noise. There is a tradeoff between the realization efficiency, the number of measurement settings, and the detection robustness, the maximally tolerable white noise. In this paper, we study this tradeoff by proposing a family of entanglement witnesses with $k$ ($2\ensuremath{\le}k\ensuremath{\le}N+1$) local measurement settings. Considering symmetric local measurements, we calculate the maximal tolerable noise for any given number of measurement settings. Consequently, we design the optimal witness with a minimal number of settings for any given level of white noise. Our theoretical analysis can be applied to other multipartite entangled states with a strong symmetry. Our witnesses can be easily implemented in experiment and applied in practical multipartite entanglement detection under different noise conditions.

DOI
10.1103/physreva.99.052349

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Entanglement is a key resource for quantum information processing. A widely used tool for detecting entanglement is the entanglement witness, where the measurement of the witness operator is guaranteed to be positive for all separable states and can be negative for certain entangled states. In reality, due to exponentially increasing the Hilbert-space dimension with respect to the system size, it is very challenging to construct an efficient entanglement witness for general multipartite entangled states. For NNN-partite Greenberger-Horne-Zeilinger (GHZ)-like states, the most robust witness scheme requires N+1N+1N+1 local measurement settings and can tolerate up to 1/21/21/2 white noise. As a comparison, the most efficient witness for GHZ-like states only needs two local measurement settings and can tolerate up to 1/31/31/3 white noise. There is a tradeoff between the realization efficiency, the number of measurement settings, and the detection robustness, the maximally tolerable white noise. In this paper, we study this tradeoff by proposing a family of entanglement witnesses with kkk (2\ensuremath≤k\ensuremath≤N+12\ensuremath{\le}k\ensuremath{\le}N+12\ensuremath≤k\ensuremath≤N+1) local measurement settings. Considering symmetric local measurements, we calculate the maximal tolerable noise for any given number of measurement settings. Consequently, we design the optimal witness with a minimal number of settings for any given level of white noise. Our theoretical analysis can be applied to other multipartite entangled states with a strong symmetry. Our witnesses can be easily implemented in experiment and applied in practical multipartite entanglement detection under different noise conditions.
Entanglement is an essential resource in many quantum information tasks and the entanglement witness is a widely used tool for its detection. In experiments the prepared state generally deviates from the target state due to some noise. Normally the white-noise model is applied to quantifying such a derivation and at the same time reveals the robustness of the witness. However, there may exist other kinds of noise, in which the coherent noise can dramatically ``rotate'' the prepared state. In this way, coherent noise is likely to lead to a failure of detection, even though the underlying state is actually entangled. In this work, we propose an efficient entanglement detection protocol for NNN-partite Greenberger-Horne-Zeilinger (GHZ)--like states. The protocol can eliminate the effect of coherent noise and at the same time feedback the corresponding noise parameters, which are beneficial to further improvements of the experiment system. In particular, we consider two experiment-relevant coherent noise models; one is from the unconscious phase accumulation on NNN qubits, and the other is from the rotation on the control qubit. The protocol effectively realizes a family of entanglement witnesses by postprocessing the measurement results from N+2N+2N+2 local measurement settings, which adds only one more setting than the original witness specialized for the GHZ state. Moreover, by considering the trade-off between the detection efficiency and the white-noise robustness, we further reduce the number of local measurements to three without altering the performance under the coherent noise. Our protocol can enhance entanglement detection under coherent noises and act as a benchmark for state-of-the-art quantum devices.
Certifying entanglement of a multipartite state is generally considered a demanding task. Since an N qubit state is parametrized by 4^{N}-1 real numbers, one might naively expect that the measurement effort of generic entanglement detection also scales exponentially with N. Here, we introduce a general scheme to construct efficient witnesses requiring a constant number of measurements independent of the number of qubits for states like, e.g., Greenberger-Horne-Zeilinger states, cluster states, and Dicke states. For four qubits, we apply this novel method to experimental realizations of the aforementioned states and prove genuine four-partite entanglement with two measurement settings only.
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.
We construct an entanglement witness for many-qubit systems, based on symmetric two-body correlations with two measurement settings. This witness is able to detect the entanglement of some Dicke states for any number of particles and such detection exhibits some robustness against white noise and thermal noise under the Lipkin-Meshkov-Glick Hamiltonian. In addition, it detects the entanglement of spin-squeezed states, with a detection strength that approaches the maximal value for sufficiently large numbers of particles. As spin-squeezed states can be experimentally generated, the properties of the witness with respect to these states may be amenable to experimental investigation. We show that while the witness is unable to detect Greenberger-Horne-Zeilinger (GHZ) states, it is instead able to detect superpositions of Dicke states with GHZ states.
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.
A Multipartite entangled state has many different kinds of entanglements specified by the number of partitions. The most essential example of multipartite entanglement is the entanglement of multi-qubit Greenberger–Horne–Zeilinger (GHZ) state in white noise. We explicitly construct the entanglement witnesses for these states with stabilizer generators of the GHZ states. For an [Formula: see text] qubit GHZ state in white noise, we demonstrate the necessary and sufficient criterion of separability when it is divided into [Formula: see text] parties with [Formula: see text] for arbitrary [Formula: see text] and [Formula: see text]. The criterion covers more than a half of all kinds of partial entanglements for [Formula: see text]-qubit GHZ states in white noise. For the rest of multipartite entanglement problems, we present a method to obtain the sufficient conditions of separability. As an application, we consider [Formula: see text] qubit GHZ state as a codeword of the degenerate quantum code passing through depolarizing channel. We find that the output state is neither genuinely entangled nor fully separable when the quantum channel capacity reduces from positive to zero.