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The generalized measurement postulate specifies operators $M_i$ with ${\sum }_i{M}_i^{†}{M}_i=I$. On a density matrix $\rho$, the outcome probability and conditional state areThe probabilities are nonnegative and sum to one. More generally an outcome may contain several Kraus operators $M_{i\alpha }$, in which case one sums over $\alpha$ in both formulas. A projective measurement with the usual Lüders update is the special case $M_i=P_i$, where $P_i=P_i^{†}$, $P_i{P}_j={\delta }_{ij}{P}_i$ and ${\sum }_i{P}_i=I$. Projection-valued effects alone fix the projective outcome probabilities, but an additional outcome-dependent unitary can change the conditional state; for the standard projective update the measurement operators themselves are the projectors.

A POVM is a family of positive effects $E_i$ summing to $I$. Its probabilities are $p_i=\mathrm{Tr}(\rho E_i)$. A generalized measurement gives effects $E_i=M_i^{†}{M}_i$, or $E_i={\sum }_{\alpha }{M}_{i\alpha }^{†}{M}_{i\alpha }$. Conversely $M_i=\sqrt{E_i}$ realizes any POVM. The POVM does not determine the post-measurement state: different operators with these same effects have the same probabilities and can have different state updates.

A pure POVM has rank-one nonzero effects, $E_i=w_i|{\psi }_i⟩⟨{\psi }_i|$, with normalized pure states $|{\psi }_i⟩$ and positive $w_i$. Here pure means rank-one effects, not the different property of extremality in the convex set of POVMs.

The POVM–ensemble duality for the maximally mixed state is a bijection between POVMs in dimension $d$ and ensemble decompositions of $I/d$. Given effects $E_i$, setomitting zero effects. Each ${\sigma }_i$ is a density matrix, ${\sum }_i{p}_i=\mathrm{Tr}I/d=1$, andConversely, if an ensemble satisfies ${\sum }_i{p}_i{\sigma }_i=I/d$, then $E_i=d{p}_i{\sigma }_i$ are positive and sum to $I$, hence form a POVM. Taking traces recovers the original probabilities, so the transformations are inverse. Pure POVMs correspond exactly to pure-state ensemble decompositions of the maximally mixed state. For such a POVM, $p_i=w_i/d$ and ${\sum }_i{w}_i=d$; it need not be an orthogonal measurement.