  The products of weak values of quantum observables are shown to be of value
in deriving quantum uncertainty and complementarity relations, for both weak
and strong measurement statistics. First, a 'product representation formula'
allows the standard Heisenberg uncertainty relation to be derived from a
classical uncertainty relation for complex random variables. We show this
formula also leads to strong uncertainty relations for unitary operators, and
underlies an interpretation of weak values as optimal (complex) estimates of
quantum observables. Furthermore, we show that two incompatible observables
that are weakly and strongly measured in a weak measurement context obey a
complementarity relation under the interchange of these observables, in the
form of an upper bound on the product of the corresponding weak values.
Moreover, general tradeoff relations between weak purity, quantum purity and
quantum incompatibility, and also between weak and strong joint probability
distributions, are obtained based on products of real and imaginary components
of weak values, where these relations quantify the degree to which weak
probabilities can take anomalous values in a given context.
