We show that there exists, in quantum theory, a close relationship between the weak value and the correlation function, which sheds new lights on the concept of the weak value.
We experimentally demonstrate a fundamental relationship between the weak value of an observable and the complex zero of the response function of a system by employing weak measurement on spin Hall shift of a Gaussian light beam. Using this relationship, we show that extremely large weak values far beyond its upper bound in the conventional linear-response regime can be experimentally obtained from the position of the minima of the pointer intensity profile corresponding to the real part of the complex zero of the response function. The imaginary part of the complex zero, on the other hand, is related to the spatial gradient of the geometric phase of light, which in this particular scenario evolves due to the weak interaction and the pre- and postselections of the polarization states. These relationships between the weak value and the complex zeros of the response function may provide new insights on weak measurements in a wide class of physical systems. Extraction of large weak values outside the usual domain of its validity and quantification of small interaction parameters using a physically meaningful and experimentally accessible system property such as the response function may open up a new paradigm of weak measurement.
We investigate the intrinsic ambiguity in the definition of correlation functions arising from the inevitable invasiveness of quantum measurements. While algebraic correlations defined as expectation values of products of observables are widely used, their relationship to operational ones defined through actual measurement procedures remain unclear. We demonstrate that the differences among various definitions of correlation functions and those among their underlying (quasi-)joint probability distributions are bounded above by a quantitative measure of measurement invasiveness. We further obtain a lower bound on the discrepancy among operational and algebraic (quasi-)joint probability distributions, providing a new form of the uncertainty relation. In addition, we identify an equivalence condition under which operational and algebraic correlations coincide. As an application, we analyze the quantum violation of the Leggett-Garg inequality and clarify the structural origin of the equivalence among different approaches to observing the violation, including sequential projective measurements and weak-measurement. Our results provide an operational foundation for the commonly used algebraic concepts of quantum theory.