Concentration of measure and the mean energy ensemble

          @misc{ pirsa_PIRSA:09120030,
            doi = {10.48660/09120030},
            url = {https://pirsa.org/09120030},
            author = {M{\"u}ller, Markus},
            keywords = {Quantum Information},
            language = {en},
            title = {Concentration of measure and the mean energy ensemble},
            publisher = {Perimeter Institute},
            year = {2009},
            month = {dec},
            note = {PIRSA:09120030 see, \url{https://pirsa.org}}
          }
          

Abstract

If a pure quantum state is drawn at random, this state will almost surely be almost maximally entangled. This is a well-known example for the "concentration of measure" phenomenon, which has proved to be tremendously helpful in recent years in quantum information theory. It was also used as a new method to justify some foundational aspects of statistical mechanics. In this talk, I discuss recent work with David Gross and Jens Eisert on concentration in the set of pure quantum states with fixed mean energy: We show typicality in this manifold of quantum states, and give a method to evaluate expectation values explicitly. This involves some interesting mathematics beyond Levy's Lemma, and suggests potential applications such as finding stronger counterexamples to the additivity conjecture.