Quantum Fourier transform, Heisenberg groups and quasi-probability distributions

TY - JOUR

T1 - Quantum Fourier transform, Heisenberg groups and quasi-probability distributions

AU - Patra, Manas Kumar

AU - Braunstein, Sam

PY - 2011/6/9

Y1 - 2011/6/9

N2 - This paper aims to explore the inherent connection between Heisenberg groups, quantum Fourier transform (QFT) and (quasi-probability) distribution functions. Distribution functions for continuous and finite quantum systems are examined from three perspectives and all of them lead to Weyl-Gabor-Heisenberg groups. The QFT appears as the intertwining operator of two equivalent representations arising out of an automorphism of the group. Distribution functions correspond to certain distinguished sets in the group algebra. The marginal properties of a particular class of distribution functions (Wigner distributions) arise from a class of automorphisms of the group algebra of the Heisenberg group. We then study the reconstruction of the Wigner function from the marginal distributions via inverse Radon transform giving explicit formulae. We consider some applications of our approach to quantum information processing and quantum process tomography.

AB - This paper aims to explore the inherent connection between Heisenberg groups, quantum Fourier transform (QFT) and (quasi-probability) distribution functions. Distribution functions for continuous and finite quantum systems are examined from three perspectives and all of them lead to Weyl-Gabor-Heisenberg groups. The QFT appears as the intertwining operator of two equivalent representations arising out of an automorphism of the group. Distribution functions correspond to certain distinguished sets in the group algebra. The marginal properties of a particular class of distribution functions (Wigner distributions) arise from a class of automorphisms of the group algebra of the Heisenberg group. We then study the reconstruction of the Wigner function from the marginal distributions via inverse Radon transform giving explicit formulae. We consider some applications of our approach to quantum information processing and quantum process tomography.

UR - http://www.scopus.com/inward/record.url?scp=79959646344&partnerID=8YFLogxK

U2 - 10.1088/1367-2630/13/6/063013

DO - 10.1088/1367-2630/13/6/063013

M3 - Article

SN - 1367-2630

VL - 13

JO - New Journal of Physics

JF - New Journal of Physics

M1 - 063013

ER -