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