  A mixed quantum state is represented by a Hermitian positive semi-definite
operator $\rho$ with unit trace. The positivity requirement is responsible for
a highly nontrivial geometry of the set of quantum states. A known way to
satisfy this requirement automatically is to use the map $\rho=\tau^2 / \mathrm
{tr} \, \tau^2$, where $\tau$ can be an arbitrary Hermitian operator. We
elaborate a parametrization of the set of quantum states induced by the
parametrization of the linear space of Hermitian operators by virtue of this
map. In particular, we derive an equation for the boundary of the set. Further,
we discuss how this parametrization can be applied to a set of quantum states
constrained by some symmetry, or, more generally, some linear condition. As an
example, we consider the parametrization of sets of Werner states of qubits.
