# On discrete Wigner transforms

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Abstract

In this work, we derive a discrete analog of the Wigner transform over the
space $(\mathbb{C}^p)^{\otimes N}$ for any prime $p$ and any positive integer
$N$. We show that the Wigner transform over this space can be constructed as
the inverse Fourier transform of the standard Pauli matrices for $p=2$ or more
generally of the Heisenberg-Weyl group elements for $p > 2$. We connect our
work to a previous construction by Wootters of a discrete Wigner transform by
showing that for all $p$, Wootters' construction corresponds to taking the
inverse symplectic Fourier transform instead of the inverse Fourier transform.
Finally, we discuss some implications of these results for the numerical
simulation of many-body quantum spin systems.

Type

Journal articlePermalink

https://hdl.handle.net/10161/17797Collections

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Show full item record### Scholars@Duke

#### Jianfeng Lu

Associate Professor of Mathematics

Jianfeng Lu is an applied mathematician interested in mathematical analysis and algorithm
development for problems from computational physics, theoretical chemistry, materials
science and other related fields.More specifically, his current research focuses include:Electronic
structure and many body problems; quantum molecular dynamics; multiscale modeling
and analysis; rare events and sampling techniques.

#### Kevin Stubbs

Student

Doctor of Philosophy (PhD) CandidateMy interests lie at the intersection between mathematics,
signal processing, and physics with a particular focus on efficient algorithms for
high dimensional problems. My current work focuses on tensor network representations
and quantum information theory. EducationDoctor of PhilosophyDuke University (Durham,
NC, USA) 2015-Bachelor of Science<br

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