In functional analysis and quantum information science, a positive operator-valued measure (POVM) is a measure whose values are positive semi-definite operators on a Hilbert space. POVMs are a generalization of projection-valued measures (PVM) and, correspondingly, quantum measurements described by POVMs are a generalization 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.

Let ${\displaystyle {\mathcal {H}}}$ denote a Hilbert space and ${\displaystyle (X,M)}$ a measurable space with ${\displaystyle M}$ a Borel σ-algebra on ${\displaystyle X}$. A POVM is a function ${\displaystyle F}$ defined on ${\displaystyle M}$ whose values are positive bounded self-adjoint operators on ${\displaystyle {\mathcal {H}}}$ such that for every ${\displaystyle \psi \in {\mathcal {H}}}$

is a non-negative countably additive measure on the σ-algebra ${\displaystyle M}$ and ${\displaystyle F(X)=\operatorname {I} _{\mathcal {H}}}$ is the identity operator.

In quantum mechanics, the key property of a POVM is that it determines a probability measure on the outcome space, so that ${\displaystyle \langle F(E)\psi \mid \psi \rangle }$ can be interpreted as the probability of the event ${\displaystyle E}$ when measuring a quantum state ${\displaystyle |\psi \rangle }$.

In the simplest case, in which ${\displaystyle X}$ is a finite set, ${\displaystyle M}$ is the power set of ${\displaystyle X}$ and ${\displaystyle {\mathcal {H}}}$ is finite-dimensional, a POVM is equivalently a set of positive semi-definite Hermitian matrices ${\displaystyle \{F_{i}\}}$ that sum to the identity matrix,

A POVM differs from a projection-valued measure in that, for projection-valued measures, the values of ${\displaystyle F}$ are required to be orthogonal projections.