real function on phase space corresponds to an actual quantum state. One way of identifying the legitimate functions $W_\alpha$ is simply to say they are the ones for which $\sum_\alpha W_\alpha \hat{A}_\alpha$ is a positive semidefinite matrix. Another way is to focus first on pure states. Recall the property $\hat{\rho} = \hat{\rho}^2$ of pure state density matrices. Recasting this as a discrete phase space expression, one finds that the pure states are represented by normalized functions $W_\alpha$ satisfying $$W_\alpha = \sum_{\beta\gamma} \Gamma_{\alpha\beta\gamma} W_\beta W_\gamma,$$ where $\Gamma_{\alpha\beta\gamma}$ is the three-point structure function $$\label{Gamma}
\Gamma_{\alpha\beta\gamma} = \frac{1}{N} \hbox{Tr}(\hat{A}_\alpha \hat{A}_\beta \hat{A}_\gamma).$$ Mixed states can then be identified as the convex combinations of pure states [@Wootters1987].

Later we will use the following symmetries of $\Gamma_{\alpha\beta\gamma}$: $$\begin{split}
&\hbox{(a)} \;\; \Gamma_{\alpha+\delta, \beta+\delta, \gamma+\delta} = \Gamma_{\alpha \beta\gamma}.  \\
&\hbox{(b)} \;\; \Gamma_{\alpha\beta\gamma} = \Gamma_{\gamma\alpha\beta} = \Gamma^*_{\alpha\gamma\beta},
\end{split}$$ where $\delta$ is any ordered pair $(\delta_1, \delta_2)$ and the asterisk indicates complex conjugation.

Trace-preserving quantum operations
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Consider a quantum system ${\mathcal S}$ with Hilbert space dimension $N$, possibly interacting with an environment. As long as there is no initial correlation between the system and the environment, the most general normalization-preserving transformation of ${\mathcal S}$, taking an initial density matrix 