One of our [postulates](Postulates%20of%20Quantum%20Mechanics.md) pertaining [quantum measurement](Quantum%20measurement%20(index).md) states that  The set of measurement operators is defined as, $\{\hat{M}_a\}$, where the subscript $a$ refers to a possible measurement outcome that may occur in an experiment and thus there's a particular measurement operator associated with each possible [observed quantity.](Observable.md#Physical%20meaning%20of%20_Observables_) A [projection operator](Projection%20operators%20in%20quantum%20mechanics.md) is a type of measurement operator in accordance with the [von Neumann postulate](von%20Neumann%20postulate.md) and more complicated measurement operators describe [generalized measurements.](POVM.md) # States following quantum measurement Given a measurement operator, $\hat{M}_a$, an initial [[State vector]], $|\psi\rangle$, after a measurement is given by $|\phi\rangle = \frac{\hat{M}_a|\psi\rangle}{\sqrt{\langle\psi|\hat{M}_a^{\dagger}\hat{M}_a|\psi\rangle}}$ where the probability of a particular outcome is $P_a=\langle\psi|\hat{M}_a^{\dagger}\hat{M}_a|\psi\rangle.$ This expression is a [generalization of the von Neumann postulate](von%20Neumann%20postulate.md#Generalization%20of%20the%20von%20Neumann%20postulate) that also gives resulting quantum state from a [generalized measurement.](POVM.md) # Projections onto one dimensional subspaces The most elementary measurement operator is the [projections onto a one-dimensional subspace](Projection%20operators%20in%20quantum%20mechanics.md#Projections%20onto%20one-dimensional%20subspaces), which transforms state vector being measured in accordance with the [von Neumann postulate.](von%20Neumann%20postulate.md). [(... see more on one dimensional projections)](Projection%20operators%20in%20quantum%20mechanics.md#Projections%20onto%20one-dimensional%20subspaces) [(... see more on von Neumann measurements)](von%20Neumann%20postulate.md) # Properties of measurement operators 1. $\sum_a \hat{M}_a^{\dagger}\hat{M}_a=\hat{\mathbb{1}}$ (completeness relation) --- # Recommended reading For an introduction to the notion of measurement operators as a type of operator that also includes [projection operators](Projection%20operators%20in%20quantum%20mechanics.md) see * [Nielson, M. A., Chuang, I. L. _Quantum Computation and Quantum Information_, Cambridge University Press, 2010](Nielsen,%20M.%20A.,%20Chuang,%20I.%20L.%20Quantum%20Computation%20and%20Quantum%20Information,%20Cambridge%20University%20Press,%202010.md) pgs 84-85. Here the notion of measurement operators is introduced as part of the measurement postulate in quantum mechanics. #QuantumMechanics/QuantumMeasurement