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A projection-valued measure (PVM) is formally similar to a real-valued measure, except that its values are self-adjoint projections rather than real numbers. As in the case of ordinary measures, it is possible to integrate complex-valued functions with respect to a PVM; the result of such an integration is a linear operator on the given Hilbert ...
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 ...
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 (PVMs) and, correspondingly, quantum measurements described by POVMs are a generalisation of quantum ...
The characteristic property of the von Neumann measurement scheme is that repeating the same measurement will give the same results. This is also called the projection postulate. A more general formulation replaces the projection-valued measure with a positive-operator valued measure (POVM). To illustrate, take again the finite-dimensional case.
Equivalently, a Mackey observable is a projection-valued measure on R. Theorem ( Spectral theorem ). If Q is the lattice of closed subspaces of Hilbert H , then there is a bijective correspondence between Mackey observables and densely-defined self-adjoint operators on H .
A positive operator-valued measure E then assigns each i a positive semidefinite m × m matrix . Naimark's theorem now states that there is a projection-valued measure on X whose restriction is E. Of particular interest is the special case when = where I is the identity operator.
A measure that takes values in the set of self-adjoint projections on a Hilbert space is called a projection-valued measure; these are used in functional analysis for the spectral theorem. When it is necessary to distinguish the usual measures which take non-negative values from generalizations, the term positive measure is used.
Throughout, is a fixed Hilbert space. A projection-valued measure on a measurable space (,), where is a σ-algebra of subsets of , is a mapping: such that for all , is a self-adjoint projection on (that is, () is a bounded linear operator (): that satisfies () = and () = ()) such that = (where is the identity operator of ) and for every ,, the function defined by (), is a complex measure on ...