# 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.

## Contents

## Formal definition[edit]

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

(where is the identity operator of *H*) and for every ξ, η ∈ *H*, the set-function

is a complex measure on *M* (that is, a complex-valued countably additive function).

We denote this measure by .

Note that is a real-valued measure, and a probability measure when has length one.

If π is a projection-valued measure and

then the images π(*E*), π(*F*) are orthogonal to each other. From this follows that in general,

and they commute.

**Example**. Suppose is a measure space. Let, for every measurable subset E in M,

be the operator of multiplication by the indicator function 1_{E} on *L*^{2}(*X*). Then π is a projection-valued measure.

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

If π 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* . *Where,* *denotes the complex measure*

*from the definition of* .

The map

is a homomorphism of rings.

An integral notation is often used for , as in

The theorem is also correct for unbounded measurable functions *f*, but then will be an unbounded linear operator on the Hilbert space *H*.

The spectral theorem says that every self-adjoint operator has an associated projection-valued measure defined on the real axis, such that

This allows to define the Borel functional calculus for such operators: if is a measurable function, we set

## Structure of projection-valued measures[edit]

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

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

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

The measure class^{[clarification needed]} 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:

where

and

## Application in quantum mechanics[edit]

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 **R**^{3} (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

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

which takes the more readable form

if the support of π is a discrete subset of **R**. This operator is called an observable in quantum mechanics.

## Generalizations[edit]

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[edit]

- Moretti, V. (2018),
*Spectral Theory and Quantum Mechanics Mathematical Foundations of Quantum Theories, Symmetries and Introduction to the Algebraic Formulation*,**110**, Springer, ISBN 978-3-319-70705-1 - Hall, B.C. (2013),
*Quantum Theory for Mathematicians*, Graduate Texts in Mathematics,**267**, Springer, ISBN 978-1461471158 - 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.