$\langle \cdot, \cdot \rangle$ denotes the symplectic product $\langle \alpha, \beta \rangle = \alpha_2 \beta_1 - \alpha_1 \beta_2$. In terms of its matrix components (in the standard basis), we can write $\hat{A}_\alpha$ as $$\label{Aodd}
(\hat{A}_\alpha)_{kl} = \delta_{2\alpha_1, k+l}\, \omega^{\alpha_2 (k-l)},$$ where the matrix indices $k$ and $l$ take the values $0, 1, \ldots, N-1$, and the arithmetic in the subscript of the Kronecker delta is mod $N$.

The Hermitian operators $\hat{A}_\alpha$, which we call phase point operators (they are also called Fano operators [@Cohendet1988]), have a number of special properties:

> \(i) $ \hbox{Tr}\, \hat{A}_\alpha = 1.$

> \(ii) $ \hbox{Tr}  (\hat{A}_\alpha \hat{A}_\beta) = N \delta_{\alpha\beta}.$

> \(iii) For any striation consisting of lines $\lambda$, the operators $\hat{Q}_\lambda = (1/N)\sum_{\alpha \in \lambda} \hat{A}_\alpha$ are projection operators onto the elements of an orthonormal basis of the Hilbert space. Moreover, the bases corresponding to different striations are mutually unbiased; that is, if the lines $\lambda_1$ and $\lambda_2$ are not parallel, then $\hbox{Tr}  (\hat{Q}_{\lambda_1}\hat{Q}_{\lambda_2}) = 1/N$.

> \(iv) As follows immediately from (iii), $(1/N)\sum_\alpha \hat{A}_\alpha = \hat{I}$, where $\hat{I}$ is the identity.

The second of these statements expresses the fact that the $\hat{A}$’s constitute an orthogonal basis for the space of $N \times 