  The states of the qubit, the basic unit of quantum information, are $2 \times
2$ positive semi-definite Hermitian matrices with trace 1. We contribute to the
program to axiomatize quantum mechanics by characterizing these states in terms
of an entropic uncertainty principle formulated on an eight-point phase space.
We do this by employing R\'enyi entropy (a generalization of Shannon entropy)
suitably defined for the signed phase-space probability distributions that
arise in representing quantum states.
