A generalized definition of the monogamy relation for entanglement measures is provided. A monogamy equality rather than the usual inequality is presented based on the monogamy weight, from which monogamy relations are given satisfied by the 𝜶th (𝜶 > 0) power of the entanglement measures. Taking concurrence as an example, the significance and advantages of these relations are further demonstrated. In addition, it is shown that monogamy relations can be recovered by considering multiple copies of states for every non-additive entanglement measure that violates the inequalities. It is also demonstrated that such relations for tripartite states can be generalized to multipartite systems.

Entanglement is one of the most fascinating features of quantum mechanics and plays important roles in accelerating quantum computation. 1 Therefore, it is of significance to test the entanglement in a quantum computer. 2 The GHZ-state or Bell states are usually used to determine whether the entanglement exists in a quantum computer. In particular, the demonstration of entanglement is crucial in a noisy-intermediate-scale-quantum (NISQ) system. 3 In a recent article, 4 Ching-ray Chang's group generated and measured the GHZ-like states to justify the entanglement in a 53-qubit system by using the Mermin's polynomials. 5 They measured various combinations of multiple (2 to 7) qubits on the state-of-art IBM Rochester quantum computer and compared their results with the theoretical ones. These results show that the quantum entanglement exists for all the combinations of qubits up to 4. Despite the effects of noises, entanglement can still show up in some cases even for qubit numbers up to 7. These results may highlight further investigations on the effective NISQ quantum algorithms, with different treatments on error mitigation for different groups of qubits in the systems such as full quantum eigensolver for quantum simulation. 6

Multipartite entanglement is of important resources for quantum communication and quantum computation. The goal of this paper is to characterize general multipartite entangled states according to shallow quantum circuits. It is first proved that any genuinely multipartite entanglement on finite-dimensional spaces can be generated by using 2-layer shallow quantum circuit consisting of two biseparable quantum channels, which has the smallest nontrivial circuit depth in the shallow quantum circuit model. Further, a semi-device-independent entanglement model depending on the local connection ability in the second layer of quantum circuits is proposed. This implies a complete hierarchy of distinguishing genuinely multipartite entangled states. It shows a completely different multipartite nonlocality from the quantum network entanglement. These results show new insights for the multipartite entanglement, quantum network, and measurement-based quantum computation.