POVM–ensemble duality for the maximally mixed state ($E_i=d{p}_i{\rho }_i$)

A POVM $\{E_i\}$ in dimension $d$ corresponds to a quantum state ensemble with $p_i=\mathrm{Tr}(E_i)/d$ and ${\rho }_i=E_i/\mathrm{Tr}(E_i)$, omitting zero effects. Its average is the maximally mixed state $I/d$. Conversely any ensemble with that average defines a POVM by $E_i=d{p}_i{\rho }_i$. These transformations are inverse. A pure POVM corresponds exactly to a pure-state ensemble decomposition of $I/d$.