  We show how group symmetries can be used to reconstruct quantum states. In
our scheme for SU(1,1) states, the input field passes through a non-degenerate
parametric amplifier and one measures the probability of finding the output
state with a certain number (usually zero) of photons in each mode. The density
matrix in the Fock basis is retrieved from the measured data by least squares
method after singular value decomposition of the design matrix. Several
illustrative examples involving the reconstruction of a pair coherent state, a
Perelomov coherent state, and a coherent superposition of pair coherent states
are considered.
