  The general expression with the physical significance and positive definite
condition of the eigenvalues of $4\times 4$ Hermitian and trace-one matrix are
obtained. This implies that the eigenvalue problem of the $4\times 4$ density
matrix is generally solved. The obvious expression of Peres' separability
condition for an arbitrary state of two qubits is then given out and it is very
easy to use. Furthermore, we discuss some applications to the calculation of
the entanglement, the upper bound of the entanglement, and a model of the
transfer of entanglement in a qubit chain through a noisy channel.
