  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.
