  We study quantum information inequalities and show that the basic inequality
between the quantum variance and the metric adjusted skew information generates
all the multi-operator matrix inequalities or Robertson type determinant
inequalities studied by a number of authors. We introduce an order relation on
the set of functions representing quantum Fisher information that renders the
set into a lattice with an involution. This order structure generates new
inequalities for the metric adjusted skew informations. In particular, the
Wigner-Yanase skew information is the maximal skew information with respect to
this order structure in the set of Wigner-Yanase-Dyson skew informations.
  Key words and phrases: Quantum covariance, metric adjusted skew information,
Robertson-type uncertainty principle, operator monotone function,
Wigner-Yanase-Dyson skew information.
