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.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.

arXiv:1802.05834Feb 16, 20182 citations

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.