# Projection-valued measure

__: Mathematical operator-value measure of interest in quantum mechanics and functional analysis__

**Short description**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.

## Definition

Let [math]\displaystyle{ H }[/math] denote a separable complex Hilbert space and [math]\displaystyle{ (X, M) }[/math] a measurable space consisting of a set [math]\displaystyle{ X }[/math] and a Borel σ-algebra [math]\displaystyle{ M }[/math] on [math]\displaystyle{ X }[/math]. A **projection-valued measure** [math]\displaystyle{ \pi }[/math] is a map from [math]\displaystyle{ M }[/math] to the set of bounded self-adjoint operators on [math]\displaystyle{ H }[/math] satisfying the following properties:^{[2]}^{[3]}

- [math]\displaystyle{ \pi(E) }[/math] is an orthogonal projection for all [math]\displaystyle{ E \in M. }[/math]
- [math]\displaystyle{ \pi(\emptyset) = 0 }[/math] and [math]\displaystyle{ \pi(X) = I }[/math], where [math]\displaystyle{ \emptyset }[/math] is the empty set and [math]\displaystyle{ I }[/math] the identity operator.
- If [math]\displaystyle{ E_1, E_2, E_3,\dotsc }[/math] in [math]\displaystyle{ M }[/math] are disjoint, then for all [math]\displaystyle{ v \in H }[/math],

- [math]\displaystyle{ \pi\left(\bigcup_{j=1}^{\infty} E_j \right)v = \sum_{j=1}^{\infty} \pi(E_j) v. }[/math]

- [math]\displaystyle{ \pi(E_1 \cap E_2)= \pi(E_1)\pi(E_2) }[/math] for all [math]\displaystyle{ E_1, E_2 \in M. }[/math]

The second and fourth property show that if [math]\displaystyle{ E_1 }[/math] and [math]\displaystyle{ E_2 }[/math] are disjoint, i.e., [math]\displaystyle{ E_1 \cap E_2 = \emptyset }[/math], the images [math]\displaystyle{ \pi(E_1) }[/math] and [math]\displaystyle{ \pi(E_2) }[/math] are orthogonal to each other.

Let [math]\displaystyle{ V_E = \operatorname{im}(\pi(E)) }[/math] and its orthogonal complement [math]\displaystyle{ V^\perp_E=\ker(\pi(E)) }[/math] denote the image and kernel, respectively, of [math]\displaystyle{ \pi(E) }[/math]. If [math]\displaystyle{ V_E }[/math] is a closed subspace of [math]\displaystyle{ H }[/math] then [math]\displaystyle{ H }[/math] can be wrtitten as the *orthogonal decomposition* [math]\displaystyle{ H=V_E \oplus V^\perp_E }[/math] and [math]\displaystyle{ \pi(E)=I_E }[/math] is the unique identity operator on [math]\displaystyle{ V_E }[/math] satisfying all four properties.^{[4]}^{[5]}

For every [math]\displaystyle{ \xi,\eta\in H }[/math] and [math]\displaystyle{ E\in M }[/math] the projection-valued measure forms a complex-valued measure on [math]\displaystyle{ H }[/math] defined as

- [math]\displaystyle{ \mu_{\xi,\eta}(E) := \langle \pi(E)\xi \mid \eta \rangle }[/math]

with total variation at most [math]\displaystyle{ \|\xi\|\|\eta\| }[/math].^{[6]} It reduces to a real-valued measure when

- [math]\displaystyle{ \mu_{\xi}(E) := \langle \pi(E)\xi \mid \xi \rangle }[/math]

and a probability measure when [math]\displaystyle{ \xi }[/math] is a unit vector.

**Example** Let [math]\displaystyle{ (X, M, \mu) }[/math] be a *σ*-finite measure space and, for all [math]\displaystyle{ E \in M }[/math], let

- [math]\displaystyle{ \pi(E) : L^2(X) \to L^2 (X) }[/math]

be defined as

- [math]\displaystyle{ \psi \mapsto \pi(E)\psi=1_E \psi, }[/math]

i.e., as multiplication by the indicator function [math]\displaystyle{ 1_E }[/math] on *L*^{2}(*X*). Then [math]\displaystyle{ \pi(E)=1_E }[/math] defines a projection-valued measure.^{[6]} For example, if [math]\displaystyle{ X = \mathbb{R} }[/math], [math]\displaystyle{ E = (0,1) }[/math], and [math]\displaystyle{ \phi,\psi \in L^2(\mathbb{R}) }[/math] there is then the associated complex measure [math]\displaystyle{ \mu_{\phi,\psi} }[/math] which takes a measurable function [math]\displaystyle{ f: \mathbb{R} \to \mathbb{R} }[/math] and gives the integral

- [math]\displaystyle{ \int_E f\,d\mu_{\phi,\psi} = \int_0^1 f(x)\psi(x)\overline{\phi}(x)\,dx }[/math]

## Extensions of projection-valued measures

If π is a projection-valued measure on a measurable space (*X*, *M*), then the map

- [math]\displaystyle{ \chi_E \mapsto \pi(E) }[/math]

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 [math]\displaystyle{ f }[/math] on [math]\displaystyle{ X }[/math], there exists a unique bounded operator [math]\displaystyle{ T : H \to H }[/math] such that
^{[7]}^{[8]}

- [math]\displaystyle{ \langle T \xi \mid \xi \rangle = \int_X f(\lambda) \,d\mu_{\xi}(\lambda), \quad \forall \xi \in H. }[/math]

where [math]\displaystyle{ \mu_{\xi} }[/math] is a finite Borel measure given by

- [math]\displaystyle{ \mu_{\xi}(E) := \langle \pi(E)\xi \mid \xi \rangle, \quad \forall E \in M. }[/math]

Hence, [math]\displaystyle{ (X,M,\mu) }[/math] is a finite measure space.

The theorem is also correct for unbounded measurable functions [math]\displaystyle{ f }[/math] but then [math]\displaystyle{ T }[/math] will be an unbounded linear operator on the Hilbert space [math]\displaystyle{ H }[/math].

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 [math]\displaystyle{ g:\mathbb{R}\to\mathbb{C} }[/math] is a measurable function, then a unique measure exists such that

- [math]\displaystyle{ g(T) :=\int_\mathbb{R} g(x) \, d\pi(x). }[/math]

### Spectral theorem

Let [math]\displaystyle{ H }[/math] be a separable complex Hilbert space, [math]\displaystyle{ A:H\to H }[/math] be a bounded self-adjoint operator and [math]\displaystyle{ \sigma(A) }[/math] the spectrum of [math]\displaystyle{ A }[/math]. Then the spectral theorem says that there exists a unique projection-valued measure [math]\displaystyle{ \pi^A }[/math], defined on a Borel subset [math]\displaystyle{ E \subset \sigma(A) }[/math], such that^{[9]}

- [math]\displaystyle{ A =\int_{\sigma(A)} \lambda \, d\pi^A(\lambda), }[/math]

where the integral extends to an unbounded function [math]\displaystyle{ \lambda }[/math] when the spectrum of [math]\displaystyle{ A }[/math] is unbounded.^{[10]}

### Direct integrals

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 π(*E*) be the operator of multiplication by 1_{E} on the Hilbert space

- [math]\displaystyle{ \int_X^\oplus H_x \ d \mu(x). }[/math]

Then π is a projection-valued measure on (*X*, *M*).

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

- [math]\displaystyle{ \pi(E) = U^* \rho(E) U \quad }[/math]

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 {*H*_{x}}_{x ∈ X }, such that π is unitarily equivalent to multiplication by 1_{E} on the Hilbert space

- [math]\displaystyle{ \int_X^\oplus H_x \ d \mu(x). }[/math]

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 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:

- [math]\displaystyle{ \pi = \bigoplus_{1 \leq n \leq \omega} (\pi \mid H_n) }[/math]

where

- [math]\displaystyle{ H_n = \int_{X_n}^\oplus H_x \ d (\mu \mid X_n) (x) }[/math]

and

- [math]\displaystyle{ X_n = \{x \in X: \dim H_x = n\}. }[/math]

## 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*,

- the projective space of the Hilbert space
*H*is interpreted as the set of possible states Φ of a quantum system, - the measurable space
*X*is the value space for some quantum property of the system (an "observable"), - the projection-valued measure π expresses the probability that the observable takes on various values.

A common choice for *X* is the real line, but it may also be

**R**^{3}(for position or momentum in three dimensions ),- a discrete set (for angular momentum, energy of a bound state, etc.),
- the 2-point set "true" and "false" for the truth-value of an arbitrary proposition about Φ.

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

- [math]\displaystyle{ P_\pi(\varphi)(E) = \langle \varphi\mid\pi(E)(\varphi)\rangle = \langle \varphi|\pi(E)|\varphi\rangle, }[/math]

where the latter notation is preferred in physics.

We can parse this in two ways.

First, for each fixed *E*, the projection π(*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 [math]\displaystyle{ \psi }[/math], the association

- [math]\displaystyle{ P_\pi(\psi) : E \mapsto \langle\psi\mid\pi(E)\psi\rangle }[/math]

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 π is called a **projective measurement**.

If *X* is the real number line, there exists, associated to π, a Hermitian operator *A* defined on *H* by

- [math]\displaystyle{ A(\varphi) = \int_{\mathbf{R}} \lambda \,d\pi(\lambda)(\varphi), }[/math]

which takes the more readable form

- [math]\displaystyle{ A(\varphi) = \sum_i \lambda_i \pi({\lambda_i})(\varphi) }[/math]

if the support of π is a discrete subset of **R**.

The above operator A is called the observable associated with the spectral measure.

## 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^{[clarification needed]}. This generalization is motivated by applications to quantum information theory.

## See also

## Notes

- ↑ Conway 2000, p. 41.
- ↑ Hall 2013, p. 138.
- ↑ Reed & Simon 1980, p. 234.
- ↑ Rudin 1991, p. 308.
- ↑ Hall 2013, p. 541.
- ↑
^{6.0}^{6.1}Conway 2000, p. 42. - ↑ Kowalski, Emmanuel (2009),
*Spectral theory in Hilbert spaces*, ETH Zürich lecture notes, p. 50, https://people.math.ethz.ch/~kowalski/spectral-theory.pdf - ↑ Reed & Simon 1980, p. 227,235.
- ↑ Reed & Simon 1980, p. 235.
- ↑ Hall 2013, p. 205.

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Original source: https://en.wikipedia.org/wiki/Projection-valued measure.
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