Input-Output Model Equivalence of Spin Systems: A Characterization Using Lie Algebra Homomorphisms

Abstract

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, we take as output the expectation value of a given observable or, more generally, 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 networks of spin particles with arbitrary values of the spins. In treating this problem, we give independent proofs of some instrumental results on the subalgebras of $su(N)$.