performed on a larger system. POVMs are the most general kind of measurement in quantum mechanics and can also be used in quantum field theory. They are extensively used in the field of quantum information. In the simplest case, of a POVM with a finite number of elements acting on a finite-dimensional Hilbert space, a POVM is a set of positive semi-definite matrices on a Hilbert space that sum to the identity matrix,: The POVM element is associated with the measurement outcome , such that the probability of obtaining it when making a measurement on the quantum state is given by: where is the trace operator. This is the POVM version of the Born rule. When the quantum state being measured is a pure state this formula reduces to: The Born rule, together with the unitarity of the time evolution operator (or, equivalently, the Hamiltonian being Hermitian), implies the unitarity of the theory, which is considered required for consistency. For example, unitarity ensures that the probabilities of all possible outcomes sum to 1 (though it is not the only option to get this particular requirement). History The Born rule was formulated by Born in a 1926 paper. In this paper, Born solves the Schrödinger equation for a scattering problem and, inspired by Albert Einstein and Einstein’s probabilistic rule for the photoelectric effect, concludes, in a footnote, that the Born rule gives the only possible interpretation of the solution. In 1954, together with Walther Bothe, Born was awarded the Nobel Prize in Physics for this and other work. John von Neumann discussed the application of spectral theory to Born's rule in his 1932 book. Derivation from more basic principles Gleason's theorem shows that the Born rule can be derived from the usual mathematical representation of measurements in quantum physics together with the assumption of non-contextuality. Andrew M. Gleason first proved the theorem in 1957, prompted by a question posed by George W. Mackey. This theorem was historically significant for the role it played in showing that wide classes of hidden-variable theories are inconsistent with quantum physics. Several other researchers have also tried to derive the Born rule from more basic principles. A number of derivations have been proposed in the context of the many-worlds interpretation. These include the decision-theory approach pioneered by David Deutsch and later developed by Hilary Greaves and David Wallace; and an "envariance" approach by Wojciech H. Zurek;