Lecture 8. Stabilizer formalism for qudit systems

In this Lecture we generalised the methods of the stabilizer formalism to study systems of particles with local dimension $d$. We discussed generalisations of such notions as the Pauli group, stabilizer groups, and Clifford groups. For systems of prime dimension $d$, the theory is essentially the same as the stabilizer theory for qubits. At the same time, in the case of composite $d$, the structure of stabilizer groups is more complicated, since in this case the $\mathbb{Z}_d$-modules are not vector spaces. In the case of odd $d$, the Pauli group is a representation of the Heisenberg group, so the stabilizer theory gains an additional interpretation in terms of the phase space $\mathbb{Z}_d^{2n}$. Each stabiliser group corresponds to some isotropic subspace $M$ with shift $v$. The discrete Wigner function can be defined as the symplectic Fourier transform of the characteristic function. The Wigner function of a pure stabilizer state is a uniform distribution over an affine Lagrangian subspace in $\mathbb{Z}_d^{2n}$, and the group of unitary Clifford gates corresponds to affine symplectic transformations of the phase space.