The classical de Finetti theorem in probability theory relates symmetry under the permutation group with the independence of random variables. This result has application in quantum information. Here we study states that are invariant with respect to a natural action of the braid group, and we emphasize the pictorial formulation and interpretation of our results. We prove a new type of de Finetti theorem for the four-string, double-braid group acting on the parafermion algebra to braid qudits, a natural symmetry in the quon language for quantum information. We prove that a braid-invariant state is extremal if and only if it is a product state. Furthermore, we provide an explicit characterization of braid-invariant states on the parafermion algebra, including finding a distinction that depends on whether the order of the parafermion algebra is square free. We characterize the extremal nature of product states (an inverse de Finetti theorem).
Wenjia JingYau Mathematical Sciences Center, Tsinghua University, No 1. Tsinghua Yuan, Beijing 100084, ChinaPanagiotis E. SouganidisDepartment of Mathematics, The University of Chicago, 5734 S. University Ave., Chicago, IL 60637, USAHung V. TranDepartment of Mathematics, University of Wisconsin at Madison, 480 Lincoln Drive, Madison, WI 53706, USA
Analysis of PDEsOptimization and ControlProbabilitymathscidoc:2206.03012
Discrete and Continuous Dynamical Systems - S, 11, (5), 915-939, 2018.10
We study the averaging of fronts moving with positive oscillatory normal velocity, which is periodic in space and stationary ergodic in time. The problem can be formulated as the homogenization of coercive level set Hamilton-Jacobi equations with spatio-temporal oscillations. To overcome the difficulties due to the oscillations in time and the linear growth of the Hamiltonian, we first study the long time averaged behavior of the associated reachable sets using geometric arguments. The results are new for higher than one dimensions even in the space-time periodic setting.
Habib AmmariD ́epartement de Math ́ematiques et Applications, ́Ecole Normale Sup ́erieure, 75230 Paris Cedex05, FranceJosselin GarnierLaboratoire de Probabilit ́es et Mod`eles Al ́eatoires & Laboratoire Jacques-Louis Lions, Universit ́eParis VII, 75205 Paris Cedex 13, FranceWenjia JingD ́epartement de Math ́ematiques et Applications, ́Ecole Normale Sup ́erieure, 75230 Paris Cedex05, France
We consider reflector imaging in a weakly random waveguide. We address the situation in which the source is farther from the reflector to be imaged than the energy equipartition distance, but the receiver array is closer to the reflector to be imaged than the energy equipartition distance. As a consequence, the reflector is illuminated by a partially coherent field and the signals recorded by the receiver array are noisy. This paper shows that migration of the recorded signals cannot give a good image, but an appropriate migration of the cross correlations of the recorded signals can give a very good image. The resolution and stability analysis of this original functional shows that the reflector can be localized with an accuracy of the order of the wavelength even when the receiver array has small aperture, and that broadband sources are necessary to ensure statistical stability, whatever the aperture of the array.