Projection-valued measure

In mathematics, particularly in functional analysis, a projection-valued measure (PVM) is a function defined on certain subsets of a fixed set and whose values are self-adjoint projections on a fixed Hilbert space. Projection-valued measures are formally similar to real-valued measures, except that their 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. 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. 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.
Formal definition.
A projection-valued measure formula_1 on a measurable space 
formula_2, where formula_3 is a σ-algebra of subsets of formula_4, is a mapping from formula_3 to the set of self-adjoint projections on a Hilbert space formula_6 (i.e. the orthogonal projections) such that
(where formula_8 is the identity operator of formula_6) and for every formula_10, the following function formula_11
is a complex measure on "formula_3" (that is, a complex-valued countably additive function).
We denote this measure by 
formula_14.
Note that formula_15 is a real-valued measure, and a probability measure when formula_16 has length one.
If formula_1 is a projection-valued measure and
then the images formula_19, formula_20 are orthogonal to each other. From this follows that in general,
and they commute.
Example. Suppose formula_22 is a measure space. Let, for every measurable subset formula_23 in formula_3, 
be the operator of multiplication by the indicator function formula_26 on "L"2("X"). Then formula_1 is a projection-valued measure.
Extensions of projection-valued measures, integrals and the spectral theorem.
If 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" "M"-"measurable function f on X, there exists" "a unique bounded linear operator" 
such that"
"for all" formula_31 "where" formula_32 "denotes the complex measure" 
"from the definition of" formula_1.
The map
is a homomorphism of rings. 
An integral notation is often used for formula_36, as in
The theorem is also correct for unbounded measurable functions "f", but then formula_36 will be an unbounded linear operator on the Hilbert space "H".
The spectral theorem says that every self-adjoint operator formula_39 has an associated projection-valued measure formula_40 defined on the real axis, such that
This allows to define the Borel functional calculus for such operators: if formula_42 is a measurable function, we set
Structure of projection-valued measures.
First we provide a general example of projection-valued measure based on direct integrals. Suppose ("X", "M", μ) is a measure space and let {"H""x"}"x" ∈ "X" be a μ-measurable family of separable Hilbert spaces. For every "E" ∈ "M", let ("E") be the operator of multiplication by 1"E" on the Hilbert space
Then is a projection-valued measure on ("X", "M").
Suppose , ρ are projection-valued measures on ("X", "M") with values in the projections of "H", "K". , 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 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 {"H""x"}"x" ∈ "X" , such that is unitarily equivalent to multiplication by 1"E" on the Hilbert space
The measure class of μ and the measure equivalence class of the multiplicity function "x" → dim "H""x" completely characterize the projection-valued measure up to unitary equivalence.
A projection-valued measure is "homogeneous of multiplicity" "n" if and only if the multiplicity function has constant value "n". Clearly,
Theorem. Any projection-valued measure taking values in the projections of a separable Hilbert space is an orthogonal direct sum of homogeneous projection-valued measures:
where
and
Application in quantum mechanics.
In quantum mechanics, given a projection valued measure of a measurable space "X" to the space of continuous endomorphisms upon a Hilbert space "H",
A common choice for "X" is the real line, but it may also be 
Let "E" be a measurable subset of the measurable space "X" and Φ a normalized vector-state in "H", so that its Hilbert norm is unitary, ||Φ|| = 1. The probability that the observable takes its value in the subset "E," given the system in state Φ, is
where the latter notation is preferred in physics.
We can parse this in two ways. 
First, for each fixed "E", the projection ("E") is a self-adjoint operator on "H" whose 1-eigenspace is the states Φ for which the value of the observable always lies in "E", and whose 0-eigenspace is the states Φ for which the value of the observable never lies in "E". 
Second, for each fixed normalized vector state formula_51, the association 
is a probability measure on "X" making the values of the observable into a random variable.
A measurement that can be performed by a projection-valued measure is called a projective measurement. 
If "X" is the real number line, there exists, associated to , a Hermitian operator "A" defined on "H" by
which takes the more readable form
if the support of is a discrete subset of R. 
The above operator A is called the observable associated with the spectral measure.
Any operator so obtained is called an observable, in quantum mechanics.
Generalizations.
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. This generalization is motivated by applications to quantum information theory.