Robertson intelligent states

The diagonalization of the uncertainty matrix and the 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 the case of canonical observables, this eigenvalue condition is also necessary. Such minimizing states are called Robertson intelligent states (RIS). The group-related coherent states (CS) with maximal symmetry (for semisimple Lie groups) are a 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 aq,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.