  In this paper, we consider the problem of model equivalence for quantum
systems. Two models are said to be (input-output) equivalent if they give the
same output for every admissible input. In the case of quantum systems, the
output is the expectation value of a given observable or, more in general, a
probability distribution for the result of a quantum measurement. We link the
input-output equivalence of two models to the existence of a homomorphism of
the underlying Lie algebra. In several cases, a Cartan decomposition of the Lie
algebra su(n) is useful to find such a homomorphism and to determine the
classes of equivalent models. We consider in detail the important cases of two
level systems with a Cartan structure and of spin networks. In the latter case,
complete results are given generalizing previous results to the case of
networks of spin particles with any value of the spin. In treating this
problem, we prove some instrumental results on the subalgebras of su(n) which
are of independent interest.
