Uncertainty and Complementarity Relations with Weak values

          @misc{ pirsa_PIRSA:16060054,
            doi = {10.48660/16060054},
            url = {https://pirsa.org/16060054},
            author = {Pati, Arun},
            keywords = {Quantum Foundations},
            language = {en},
            title = {Uncertainty and Complementarity Relations with Weak values},
            publisher = {Perimeter Institute},
            year = {2016},
            month = {jun},
            note = {PIRSA:16060054 see, \url{https://pirsa.org}}
          }
          

Abstract

The products of weak values of quantum observables have interesting implications in deriving quantum uncertainty and complementarity relations for both weak and strong measurement statistics. We show that a product representation formula allows the standard Heisenberg uncertainty relation to be derived from a classical uncertainty relation for complex random variables. This formula also leads to a strong uncertainty relation for unitary operators which displays a new preparation uncertainty relation for quantum systems. Furthermore, the two system observables that are weakly and strongly measured in a weak measurement context are shown to 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, we derive general tradeoff relations, between weak purity, quantum purity and quantum incompatibility using the weak value formalism. Our results may open up new ways of thinking about uncertainty and complementarity relations using products of weak values.