= POVM–ensemble duality for the maximally mixed state
{c}
{title2=$E_i=d p_i\rho_i$}
A <POVM> $\{E_i\}$ in dimension $d$ corresponds to a <quantum state ensemble> with $p_i=\operatorname{Tr}(E_i)/d$ and $\rho_i=E_i/\operatorname{Tr}(E_i)$, omitting zero effects. Its average is the <maximally mixed state> $I/d$. Conversely any ensemble with that average defines a POVM by $E_i=dp_i\rho_i$. These transformations are inverse. A <pure POVM> corresponds exactly to a pure-state ensemble decomposition of $I/d$.
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