Robertson Intelligent States

Diagonalization of uncertainty matrix and minimization of Robertson inequality for n observables are considered. It is proved that for even n this relation is minimized in states which are eigenstates of n/2 independent complex linear combinations of the observables. In case of canonical observables this eigenvalue condition is also necessary. Such minimizing states are called Robertson intelligent states (RIS). It is shown that group related coherent states (CS) with maximal symmetry (for semisimple Lie groups) are particular case of RIS for the quadratures of Weyl generators. Explicit constructions of RIS are considered for operators of su(1, 1), su(2), h N and sp(N, R) algebras. Unlike the group related CS, RIS can exhibit strong squeezing of group generators. Multimode squared amplitude squeezed states are naturally introduced as sp(N, R) RIS. It is shown that the uncertainty matrices for quadratures of q-deformed boson operators a q,j (q > 0) and of any k power of a j = a 1,j are positive definite and can be diagonalized by symplectic linear transformations.