  We introduce the qubit representation by complex numbers on the set of
Apollonius circles with common symmetric points at $0$ and $1$, related with
$|0\rangle$ and $|1\rangle$ states. For one qubit states we find that the
Shannon entropy as a measure of randomness is a constant along Apollonius
circles. For two qubit states, the concurence as a characteristic of
entanglement is taking constant value for the states on the same Apollonius
circle. Geometrical meaning of concurence as an area and as a distance in the
Apollonius representation are found. Then we generalize our results to
arbitrary $n$-qubit Apollonius states and show that the fidelity between given
state and the symmetric one, as reflected in an axes, is a constant along
Apollonius circles. For two qubits it coinsides with the concurence. For
generic two qubit states we derived Apollonius representation by three complex
parameters and show that the determinant formula for concurence is related with
fidelity for symmetric states by two reflections in a vertical axis and
inversion in a circle. We introduce the complex concurence and an addition
formula for Apollonius states and show that for generic two qubit states its
modulus satisfies the law of cosine. Finally, we show that for two qubit
Apollonius state in bipolar coordinates, the complex concurence is decribed by
static one soliton solution of the nonlinear Schr\"odinger equation.
