Southwest Quantum Information and Technology

Group representations and generalized phase spaces

Presenting Author: Christopher Jackson, University of New Mexico CQuIC
Contributing Author(s): Carl Caves, Ivan Deutsch, Akimasa Miyake, Ninnat Dangniam, Ezad Shojaee, Gopi Muraleedharan, Adrian Chapman, Mitchell Brickson

Given a group and representation, a generalized phase space is the orbit of some fiducial state. For example, the most studied phase space of boson quadratures can be described as the Fock space representation of the Weyl-Heisenberg group acting on the vacuum state. In general, if the representation is irreducible, then Wigner functions over the phase space provide an alternative (but equivalent) way to represent quantum information. If the orbit is generated from a state of highest weight, then the phase space can be represented by a single constraint, quadratic in the density operator. With a better understanding of generalized phase spaces, we have done many things such as: 1) Measure the rank of an unknown state with Haar random measurements and generalize the Porter-Thomas distribution. 2) Show that an independent sequence of Haar random weak measurements limit to the POVM consisting of coherent state projectors. 3) Easily prove that an independent sequence of random group elements limit to a Haar random element and calculate the rate of convergence. 4) Calculate Wigner functions relative to Fermion Gaussian states/measurements and show they are not positive, contrary to some expectations.

(Session 5 : Thursday from 5:00pm - 7:00 pm)