In functional analysis and quantum measurement theory, a positive operator-valued measure (POVM) is a measure whose values are positive semi-definite operators on a Hilbert space. POVMs are a generalisation of projection-valued measures (PVM) and, correspondingly, quantum measurements described by POVMs are a generalisation of quantum measurement described by PVMs (called projective measurements). In rough analogy, a POVM is to a PVM what a mixed state is to a pure state. Mixed states are needed to specify the state of a subsystem of a larger system (see purification of quantum state); analogously, POVMs are necessary to describe the effect on a subsystem of a projective measurement 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. Definition 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 Hermitian matrices on a Hilbert space that sum to the identity matrix, In quantum mechanics, 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. When the quantum state being measured is a pure state this formula reduces to . The simplest case of a POVM generalises the simplest case of a PVM, which is a set of orthogonal projectors that sum to the identity matrix: The probability formulas for a PVM are the same as for the POVM. An important difference is that the elements of a POVM are not necessarily orthogonal. As a consequence, the number of elements of the POVM can be larger than the dimension of the Hilbert space they act in. On the other hand, the number of elements of the PVM is at most the dimension of the Hilbert space. In general, POVMs can also be defined in situations where the number of elements and the dimension of the Hilbert space is not finite: Definition. Let be measurable space; that is is a σ-algebra of subsets of . A POVM is a function defined on whose values are bounded non-negative self-adjoint operators on a Hilbert space such that and for every , is a non-negative countably additive measure on the σ-algebra .