Genuine multipartite entanglement represents the strongest type of entanglement, which is an essential resource for quantum information processing. Standard methods to detect genuine multipartite entanglement, e.g., entanglement witnesses, state tomography, or quantum state verification, require full knowledge of the Hilbert space dimension and precise calibration of measurement devices, which are usually difficult to acquire in an experiment. The most radical way to overcome these problems is to detect entanglement solely based on the Bell-like correlations of measurement outcomes collected in the experiment, namely, device independently. However, it is difficult to certify genuine entanglement of practical multipartite states in this way, and even more difficult to quantify it, due to the difficulty in identifying optimal multipartite Bell inequalities and protocols tolerant to state impurity. In this Letter, we explore a general and robust device-independent method that can be applied to various realistic multipartite quantum states in arbitrary finite dimension, while merely relying on bipartite Bell inequalities. Our method allows us both to certify the presence of genuine multipartite entanglement and to quantify it. Several important classes of entangled states are tested with this method, leading to the detection of genuinely entangled states. We also certify genuine multipartite entanglement in weakly entangled Greenberger-Horne-Zeilinger states, showing that the method applies equally well to less standard states.
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.
We develop a necessary condition of quantum correlation. It is utilized to construct a
d-level bipartite Bell-type inequality which is strongly resistant to noise and requires only analyses of
O(d) measurement outcomes compared to the previous result
O(d2). Remarkably, a connection between the arbitrary high-dimensional bipartite Bell-type inequality and entanglement witnesses is found. Through the necessary condition of quantum correlation, we propose that the witness operators to detect truly multipartite entanglement for a generalized Greenberger-Horne-Zeilinger (GHZ) state with two local measurement settings and a four-qubit singlet state with three settings. Moreover, we also propose a robust entanglement witness to detect a four-level tripartite GHZ state with only two local measurement settings.
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.