A POVM (a _positive operator valued measurement_), sometimes referred to as a _generalized measurement_ is a type of quantum measurement that we will define in terms of a set of _POVM_ elements that is expressed as the set of operators $\{E_a\}$ where $E_a = \hat{M}_a^{\dagger}\hat{M}_a$ where $\hat{M}_a$ is a [generalized measurement operator.](measurement%20operator.md) We would want to model some measurements as POVMs as opposed opposed to simpler [von Neumann measurements](von%20Neumann%20postulate.md) in cases where the number of possible measurement outcomes differs from the number of possible outcomes implied by the [Hilbert space dimensionality.](Finite%20dimensional%20Hilbert%20spaces.md) # Neumark's theorem [[Neumark's theorem]] # Von Neumann Measurement A [von Neumann measurement](von%20Neumann%20postulate.md) is a special case of a POVM measurement in which the POVM elements are the [Projection Operators](Projection%20operators%20in%20quantum%20mechanics.md) $E_a=P_a^\dagger P_a = P_a$ where here we also note [property 2](Projection%20operators%20in%20quantum%20mechanics.md#Properties%20of%20projection%20operators) of projection operators. #QuantumMechanics/QuantumMeasurement