Recently, a new type of symmetry for three-qubit quantum states was introduced, the so-called Greenberger-Horne-Zeilinger (GHZ) symmetry. It includes the operations which leave the three-qubit standard GHZ state unchanged. This symmetry is powerful as it yields families of mixed states that are, on the one hand, complex enough from the physics point of view and, on the other hand, simple enough mathematically so that their properties can be characterized analytically. We show that by using the properties of GHZ-symmetric states it is straightforward to derive optimal witnesses for detecting class-specific entanglement in arbitrary three-qubit states.
We propose a method of constructing the separability criteria for multipartite quantum states on the basis of entanglement witnesses. The entanglement witnesses are obtained by finding the maximal expectation values of Hermitian operators and then optimizing over all possible Hermitian operators. We derive a set of tripartite separability criteria for the four-qubit Greenberger–Horne–Zeilinger (GHZ) diagonal states. The derived criterion set contains four criteria that are necessary and sufficient for the tripartite separability of the highly symmetric four-qubit GHZ diagonal states; the criteria completely account for the numerically obtained boundaries of the tripartite separable state set. One of the criteria is just the tripartite separability criterion of the four-qubit generalized Werner states.
Recently, a new type of symmetry for three-qubit quantum states was introduced, the so-called Greenberger-Horne-Zeilinger (GHZ) symmetry. It includes the operations which leave the three-qubit standard GHZ state unchanged. This symmetry is powerful as it yields families of mixed states that are, on the one hand, complex enough from the physics point of view and, on the other hand, simple enough mathematically so that their properties can be characterized analytically. We show that by using the properties of GHZ-symmetric states it is straightforward to derive optimal witnesses for three-qubit entanglement.
Of existing entanglement witnesses that utilize only collective measurements of a spin ensemble, not all can detect genuine multipartite entanglement (GME), and none can detect Greenberger-Horne-Zeilinger (GHZ) states beyond the tripartite case. We fill this gap by introducing an entanglement witness that detects GME of spin ensembles, whose total spin is half-integer, using only collective spin measurements. Our witness is based on a nonclassicality test introduced by Tsirelson, and solely requires the measurement of total angular momentum along different directions. States detected by our witness are close to a family of GHZ-like states, which includes GHZ states of an odd number of spin-half particles. We also study the robustness of our witness under depolarizing noise, and derive exact noise bounds for detecting noisy GHZ states.
The existence and nature of tripartite entanglement of a noninteracting Fermi gas (NIFG) is investigated. Three classes of parametrized entanglement witnesses (EWs) are introduced with the aim of detecting genuine tripartite entanglement in the three-body reduced density matrix and discriminating between the presence of the two types of genuine tripartite entanglement, $W\B$ and $\mathrm{GHZ}\W$ (the convex set of
B states is comprised of mixed states of product and biseparable states; that of
W states is comprised of mixed states of
B states and
W-type pure entangled states; and the GHZ (Greenberger-Horne-Zeilinger) set contains generic mixtures of any kind for a tripartite system). By choosing appropriate EW operators, the problem of finding GHZ and
W EWs is reduced to linear programming. Specifically, we devise
W EWs based on a spin-chain model with periodic boundary conditions, and we construct a class of parametrized GHZ EWs by linearly combining projection operators corresponding to all the different state-vector types arising for a three-fermion system. A third class of EWs is provided by a GHZ stabilizer operator capable of distinguishing $W\B$ from $\mathrm{GHZ}\B$ entanglement, which is not possible with
W EWs. Implementing these classes of EWs, it is found that all states containing genuine tripartite entanglement are of
W type, and hence states containing $\mathrm{GHZ}\W$ genuine tripartite entanglement do not arise. Some genuine tripartite entangled states that have a positive partial transpose (PPT) with respect to some bipartition are detected. Finally, it is demonstrated that a NIFG does not exhibit ``pure'' $W\B$ genuine tripartite entanglement: three-party entanglement without any separable or biseparable admixture does not occur.
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
N-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
N 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+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.