Complementarity, quartic polynomials and quantum states

In a previous work, based upon complementarity and inspired by a standard derivation of the x-p uncertainty inequality (via a non-negative quadratic polynomial f(2)), we explored one possible extension, through a non-negative quartic polynomial f(4), for non-commuting quantum variables. That work led to new quantum inequalities (expressed through the discriminant of the equation f(4) = 0) and to connections with quantum optics. Here, we extend the study of quartic polynomials and report classes of genuine quantum states vertical bar psi(4,n)> with interesting properties, which follow directly from f(4) = 0 for x-p. We explore vertical bar psi(4,n)> when they are generated from the coherent and the squeezed coherent states of the quantum harmonic oscillator and we display their relationships to states previously proposed by other authors through different constructions. In quantum optics, the vertical bar psi(4,n)> associated to the coherent states are just displaced Fock states of light, which have already been generated experimentally.