Lower bounds of concurrence for $N$-qubit systems and the detection of $k$-nonseparability of multipartite quantum systems

Abstract: Concurrence, as one of entanglement measures, is a useful tool to characterize quantum entanglement in various quantum systems. However, the computation of the concurrence involves difficult optimizations and only for the case of two qubits an exact formula was found. We investigate the concurrence of four-qubit quantum states and derive analytical lower bound of concurrence using the multiqubit monogamy inequality. It is shown that this lower bound is able to improve the existing bounds. This approach can be generalized to arbitrary qubit systems. We present an exact formula of concurrence for some mixed quantum states. For even-qubit states, we derive an improved lower bound of concurrence using a monogamy equality for qubit systems. At the same time, we show that a multipartite state is $k$-nonseparable if the multipartite concurrence is larger than a constant related to the value of $k$, the qudit number and the dimension of the subsystems. Our results can be applied to detect the multipartite $k$-nonseparable states.