  W consider the problem of testing if a given matrix in the Hilbert space
formulation of quantum mechanics or a function in the phase space formulation
of quantum theory represent a quantum state. We propose several practical
criteria to recognise states in these both versions of quantum physics. After
minor modifications they can be applied to check positivity of any operators
acting in a Hilbert space or positivity of any functions from an algebra with a
Weyl type * -- product.
