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.[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.

## Formal definition

A projection-valued measure $\pi$ on a measurable space $(X,M)$ , where $M$ is a σ-algebra of subsets of $X$ , is a mapping from $M$ to the set of self-adjoint projections on a Hilbert space $H$ (i.e. the orthogonal projections) such that

$\pi (X)=\operatorname {id} _{H}\quad$ (where $\operatorname {id} _{H)$ is the identity operator of $H$ ) and for every $\xi ,\eta \in H$ , the following function $M\to \mathbb {C}$ $E\mapsto \langle \pi (E)\xi \mid \eta \rangle$ is a complex measure on $M$ (that is, a complex-valued countably additive function).

We denote this measure by $\operatorname {S} _{\pi }(\xi ,\eta )$ .

Note that $\operatorname {S} _{\pi }(\xi ,\xi )$ is a real-valued measure, and a probability measure when $\xi$ has length one.

If $\pi$ is a projection-valued measure and

$E\cap F=\emptyset ,$ then the images $\pi (E)$ , $\pi (F)$ are orthogonal to each other. From this follows that in general,

$\pi (E)\pi (F)=\pi (E\cap F)=\pi (F)\pi (E),$ and they commute.

Example. Suppose $(X,M,\mu )$ is a measure space. Let, for every measurable subset $E$ in $M$ ,

$\pi (E):L^{2}(\mu )\to L^{2}(\mu ):\psi \mapsto \chi _{E}\psi$ be the operator of multiplication by the indicator function $1_{E)$ on L2(X). Then $\pi$ is a projection-valued measure. For example, if $X=\mathbb {R}$ , $E=(0,1)$ , and $\phi ,\psi \in L^{2}(\mathbb {R} )$ there is then the associated complex measure $S_{(0,1)}(\phi ,\psi )$ which takes a measurable function $f:\mathbb {R} \to \mathbb {R}$ and gives the integral

$S_{(0,1)}(\phi ,\psi )(f)=\int _{(0,1)}f(x)\psi (x){\overline {\phi ))(x)dx$ ## Extensions of projection-valued measures, integrals and the spectral theorem

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

$\chi _{E}\mapsto \pi (E)$ 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

$\mathrm {T} _{\pi }(f):H\to H$ such that

$\langle \operatorname {T} _{\pi }(f)\xi \mid \eta \rangle =\int _{X}f\ d\operatorname {S} _{\pi }(\xi ,\eta )$ for all $\xi ,\eta \in H,$ where $\operatorname {S} _{\pi }(\xi ,\eta )$ denotes the complex measure

$E\mapsto \langle \pi (E)\xi \mid \eta \rangle$ from the definition of $\pi$ .

The map

${\mathcal {BM))(X,M)\to {\mathcal {L))(H):f\mapsto \operatorname {T} _{\pi }(f)$ An integral notation is often used for $\operatorname {T} _{\pi }(f)$ , as in

$\operatorname {T} _{\pi }(f)=\int _{X}f(x)\,d\pi (x)=\int _{X}f\,d\pi .$ The theorem is also correct for unbounded measurable functions f, but then $\operatorname {T} _{\pi }(f)$ will be an unbounded linear operator on the Hilbert space H.

The spectral theorem says that every self-adjoint operator $A:H\to H$ has an associated projection-valued measure $\pi _{A)$ defined on the real axis, such that

$A=\int _{\mathbb {R} }x\,d\pi _{A}(x).$ This allows to define the Borel functional calculus for such operators: if $g:\mathbb {R} \to \mathbb {C}$ is a measurable function, we set

$g(A):=\int _{\mathbb {R} }g(x)\,d\pi _{A}(x).$ ## 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 {Hx}xX be a μ-measurable family of separable Hilbert spaces. For every EM, let π(E) be the operator of multiplication by 1E on the Hilbert space

$\int _{X}^{\oplus }H_{x}\ d\mu (x).$ 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:HK such that

$\pi (E)=U^{*}\rho (E)U\quad$ for every EM.

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}xX , such that π is unitarily equivalent to multiplication by 1E on the Hilbert space

$\int _{X}^{\oplus }H_{x}\ d\mu (x).$ 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:

$\pi =\bigoplus _{1\leq n\leq \omega }(\pi \mid H_{n})$ where

$H_{n}=\int _{X_{n))^{\oplus }H_{x}\ d(\mu \mid X_{n})(x)$ and

$X_{n}=\{x\in X:\dim H_{x}=n\}.$ ## 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

• R3 (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

$P_{\pi }(\varphi )(E)=\langle \varphi \mid \pi (E)(\varphi )\rangle =\langle \varphi |\pi (E)|\varphi \rangle ,$ 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 $\psi$ , the association

$P_{\pi }(\psi ):E\mapsto \langle \psi \mid \pi (E)\psi \rangle$ 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

$A(\varphi )=\int _{\mathbf {R} }\lambda \,d\pi (\lambda )(\varphi ),$ which takes the more readable form

$A(\varphi )=\sum _{i}\lambda _{i}\pi ({\lambda _{i)))(\varphi )$ if the support of π 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[clarification needed]. This generalization is motivated by applications to quantum information theory.