In mathematics, particularly in functional analysis, a projectionvalued measure (or spectral measure) is a function defined on certain subsets of a fixed set and whose values are selfadjoint projections on a fixed Hilbert space.^{[1]} A projectionvalued measure (PVM) is formally similar to a realvalued measure, except that its values are selfadjoint projections rather than real numbers. As in the case of ordinary measures, it is possible to integrate complexvalued functions with respect to a PVM; the result of such an integration is a linear operator on the given Hilbert space.
Projectionvalued measures are used to express results in spectral theory, such as the important spectral theorem for selfadjoint operators, in which case the PVM is sometimes referred to as the spectral measure. The Borel functional calculus for selfadjoint 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.
Let denote a separable complex Hilbert space and a measurable space consisting of a set and a Borel σalgebra on . A projectionvalued measure is a map from to the set of bounded selfadjoint operators on satisfying the following properties:^{[2]}^{[3]}
The second and fourth property show that if and are disjoint, i.e., , the images and are orthogonal to each other.
Let and its orthogonal complement denote the image and kernel, respectively, of . If is a closed subspace of then can be wrtitten as the orthogonal decomposition and is the unique identity operator on satisfying all four properties.^{[4]}^{[5]}
For every and the projectionvalued measure forms a complexvalued measure on defined as
with total variation at most .^{[6]} It reduces to a realvalued measure when
and a probability measure when is a unit vector.
Example Let be a σfinite measure space and, for all , let
be defined as
i.e., as multiplication by the indicator function on L^{2}(X). Then defines a projectionvalued measure.^{[6]} For example, if , , and there is then the associated complex measure which takes a measurable function and gives the integral
If π is a projectionvalued 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 complexvalued measurable functions on X, and we have the following.
Theorem — For any bounded Borel function on , there exists a unique bounded operator such that ^{[7]}^{[8]}
where is a finite Borel measure given by
Hence, is a finite measure space.
The theorem is also correct for unbounded measurable functions but then 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. That is, if is a measurable function, then a unique measure exists such that
Let be a separable complex Hilbert space, be a bounded selfadjoint operator and the spectrum of . Then the spectral theorem says that there exists a unique projectionvalued measure , defined on a Borel subset , such that^{[9]}
where the integral extends to an unbounded function when the spectrum of is unbounded.^{[10]}
First we provide a general example of projectionvalued measure based on direct integrals. Suppose (X, M, μ) is a measure space and let {H_{x}}_{x ∈ X } be a μmeasurable family of separable Hilbert spaces. For every E ∈ M, let π(E) be the operator of multiplication by 1_{E} on the Hilbert space
Then π is a projectionvalued measure on (X, M).
Suppose π, ρ are projectionvalued 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 projectionvalued measure π on (X, M) taking values in the projections of a separable Hilbert space, there is a Borel measure μ and a μmeasurable family of Hilbert spaces {H_{x}}_{x ∈ X }, such that π is unitarily equivalent to multiplication by 1_{E} on the Hilbert space
The measure class^{[clarification needed]} of μ and the measure equivalence class of the multiplicity function x → dim H_{x} completely characterize the projectionvalued measure up to unitary equivalence.
A projectionvalued measure π is homogeneous of multiplicity n if and only if the multiplicity function has constant value n. Clearly,
Theorem. Any projectionvalued measure π taking values in the projections of a separable Hilbert space is an orthogonal direct sum of homogeneous projectionvalued measures:
where
and
In quantum mechanics, given a projectionvalued measure of a measurable space to the space of continuous endomorphisms upon a Hilbert space ,
A common choice for is the real line, but it may also be
Let be a measurable subset of and a normalized vector quantum state in , so that its Hilbert norm is unitary, . The probability that the observable takes its value in , given the system in state , is
We can parse this in two ways. First, for each fixed , the projection is a selfadjoint operator on whose 1eigenspace are the states for which the value of the observable always lies in , and whose 0eigenspace are the states for which the value of the observable never lies in .
Second, for each fixed normalized vector state , the association
is a probability measure on making the values of the observable into a random variable.
A measurement that can be performed by a projectionvalued measure is called a projective measurement.
If is the real number line, there exists, associated to , a selfadjoint operator defined on by
which reduces to
if the support of is a discrete subset of .
The above operator is called the observable associated with the spectral measure.
The idea of a projectionvalued measure is generalized by the positive operatorvalued 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 nonorthogonal partition of unity^{[clarification needed]}. This generalization is motivated by applications to quantum information theory.
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