Applications of algebraic methods in quantum physics

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This proposal runs along two parallel tracks.  One deals with the construction and properties of quantum mechanical states that have the property of saturating the Heisenberg uncertainty relation.  Such states are called intelligent, and have numerous applications, particularly in optical interferometry.  They also have an important theoretical role in the detection of entanglement, a feature essential to quantum information theory.A second track deals with complementarity in finite dimensional quantum systems.  Here, the objective is to study the construction of sets of states that describe maximally incompatible preparations of a quantum system.  Such states are useful for the optimal implementation of various protocols of quantum information.  The intention is to consider cases, as in a quantum system of dimenion 6, where the dimension is not a prime number or a power of a prime number.  In the latter cases, sets of complementary states are already known and constructed.