In mathematics, particularly in functional analysis, a projection-valued measure (or spectral measure) is a function defined on certain subsets of a fixed set and whose values are self-adjoint projections on a fixed Hilbert space.
[1] 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 space.
Projection-valued measures are used to express results in spectral theory, such as the important spectral theorem for self-adjoint operators, in which case the PVM is sometimes referred to as the spectral measure.
The Borel functional calculus for self-adjoint operators is constructed using integrals with respect to PVMs.
In quantum mechanics, PVMs are the mathematical description of projective measurements.
[clarification needed] They are generalized by positive operator valued measures (POVMs) in the same sense that a mixed state or density matrix generalizes the notion of a pure state.
denote a separable complex Hilbert space and
a measurable space consisting of a set
, let be defined as i.e., as multiplication by the indicator function
and gives the integral If π is a projection-valued measure on a measurable space (X, M), then the map extends to a linear map on the vector space of step functions on X.
In fact, it is easy to check that this map is a ring homomorphism.
This map extends in a canonical way to all bounded complex-valued measurable functions on X, and we have the following.
Theorem — For any bounded Borel function
The theorem is also correct for unbounded measurable functions
will be an unbounded linear operator on the Hilbert space
This allows to define the Borel functional calculus for such operators and then pass to measurable functions via the Riesz–Markov–Kakutani representation theorem.
Then the spectral theorem says that there exists a unique projection-valued measure
, such that[9] where the integral extends to an unbounded function
[10] First we provide a general example of projection-valued measure based on direct integrals.
Suppose (X, M, μ) is a measure space and let {Hx}x ∈ X be a μ-measurable family of separable Hilbert spaces.
Suppose π, ρ are projection-valued measures on (X, M) with values in the projections of H, K. π, ρ are unitarily equivalent if and only if there is a unitary operator U:H → K such that for every E ∈ M. Theorem.
If (X, M) is a standard Borel space, then for every projection-valued measure π on (X, M) taking values in the projections of a separable Hilbert space, there is a Borel measure μ and a μ-measurable family of Hilbert spaces {Hx}x ∈ X , such that π is unitarily equivalent to multiplication by 1E on the Hilbert space The measure class[clarification needed] of μ and the measure equivalence class of the multiplicity function x → dim Hx completely characterize the projection-valued measure up to unitary equivalence.
A projection-valued measure π is homogeneous of multiplicity n if and only if the multiplicity function has constant value n. Clearly, Theorem.
Any projection-valued measure π taking values in the projections of a separable Hilbert space is an orthogonal direct sum of homogeneous projection-valued measures: where and In quantum mechanics, given a projection-valued measure of a measurable space
making the values of the observable into a random variable.
is the real number line, there exists, associated to
is called the observable associated with the spectral measure.
The idea of a projection-valued measure is generalized by the positive operator-valued measure (POVM), where the need for the orthogonality implied by projection operators is replaced by the idea of a set of operators that are a non-orthogonal partition of unity[clarification needed].
This generalization is motivated by applications to quantum information theory.