Its key property is that it determines a probability measure on the outcome space, so that can be interpreted as the probability (density) of outcome when making a measurement on the quantum state . This definition should be contrasted with that of the projection-valued measure, which is similar, except that for projection-valued measures, the values of are required to be projection operators. Naimark's dilation theorem Note: An alternate spelling of this is "Neumark's Theorem" Naimark's dilation theorem shows how POVMs can be obtained from PVMs acting on a larger space. This result is of critical importance in quantum mechanics, as it gives a way to physically realize POVM measurements. In the simplest case, of a POVM with a finite number of elements acting on a finite-dimensional Hilbert space, Naimark's theorem says that if is a POVM acting on a Hilbert space of dimension , then there exists a PVM acting on a Hilbert space of dimension and an isometry such that for all , One way to construct such a PVM and isometry is to let , , and The probability of obtaining outcome with this PVM, and the state suitably transformed by the isometry, is the same as the probability of obtaining it with the original POVM: This construction can be turned into a recipe for a physical realisation of the POVM by extending the isometry into a unitary , that is, finding such that This can always be done. The recipe for realizing the POVM measurement described by on a quantum state is then to prepare an ancilla in the state , evolve it together with through the unitary , and make the projective measurement on the ancilla described by the PVM . Note that in this construction the dimension of the larger Hilbert space is given by . This is not the minimum possible, as a more complicated construction gives for rank-1 POVMs. Post-measurement state The post-measurement state is not determined by the POVM itself, but rather by the PVM that physically realizes it. Since there are infinitely many different PVMs that realize the same POVM, the operators alone do not determine what the post-measurement state will be. To see that, note that for any unitary the operators will also have the property that , so that using the isometry in the above construction will also implement the same POVM. In the case where the state being