# 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.[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 on a measurable space (X, M), where M is a σ-algebra of subsets of X, is a mapping π from M to the set of self-adjoint projections on a Hilbert space H such that

$\pi(X) = \operatorname{id}_H \quad$

and for every ξ, η ∈ H, the set-function

$E \mapsto \langle \pi(E)\xi \mid \eta \rangle$

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 π is a projection-valued measure and

$E \cap F = \emptyset,$

then π(E), π(F) are orthogonal projections. 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, μ) is a measure space. Let π(E) be the operator of multiplication by the indicator function 1E on L2(X). Then π is a projection-valued measure.

## Extensions of projection-valued measures, integrals and the spectral theorem

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

$\mathbf{1}_E \mapsto \pi(E)$

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 is a unique bounded linear operator Tπ(f) such that

$\langle \operatorname{T}_\pi(f) \xi \mid \eta \rangle = \int_X f(x) d \operatorname{S}_\pi (\xi,\eta)(x)$

for all ξ, η ∈ H. Here, $\operatorname{S}_\pi (\xi,\eta)$ denotes the complex measure $E \mapsto \langle \pi(E)\xi \mid \eta \rangle$ from the definition of $\pi$. The map

$f \mapsto \operatorname{T}_\pi(f)$

is a homomorphism of rings. 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:

$\pi = \bigoplus_{1 \leq n \leq \omega} (\pi | H_n)$

where

$H_n = \int_{X_n}^\oplus H_x \ d (\mu | X_n) (x)$

and

$X_n = \{x \in X: \operatorname{dim} H_x = n\}.$

## Application in quantum mechanics

In quantum mechanics, the unit sphere 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"), and the projection-valued measure π expresses the probability that the observable takes on various values.

A common choice for X is the real numbers, but it may also be R3 (for position or momentum), a discrete set (for angular momentum, energy of a bound state, etc), or the 2-point set "true" and "false" for the truth-value of an arbitrary proposition about Φ.

Let E be a measurable subset of X and Φ a state in H, so that |Φ|=1. The probability that the observable takes its value in E given the system in state Φ is

$P = \langle \phi,\pi(E)(\phi)\rangle = \langle \phi|\pi(E)|\phi\rangle,$

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 Φ, the association E ↦ ⟨Φ,π(⋅)Φ⟩ 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 numbers, there is associated to π a Hermitian operator A defined on H by

$A(\phi) = \int_{\bold{R}} \lambda \,d\pi(\lambda)(\phi),$

which takes the more readable form

$A(\phi) = \sum_i \lambda_i \pi({\lambda_i})(\phi)$

if the support of π is a discrete subset of R. This operator 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.

## References

• G. W. Mackey, The Theory of Unitary Group Representations, The University of Chicago Press, 1976
• M. Reed and B. Simon, Methods of Mathematical Physics, vols I–IV, Academic Press 1972.
• G. Teschl, Mathematical Methods in Quantum Mechanics with Applications to Schrödinger Operators, http://www.mat.univie.ac.at/~gerald/ftp/book-schroe/, American Mathematical Society, 2009.
• V. S. Varadarajan, Geometry of Quantum Theory V2, Springer Verlag, 1970.