# 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 (i.e. the orthogonal projections) such that

${\displaystyle \pi (X)=\operatorname {id} _{H}\quad }$

(where ${\displaystyle \operatorname {id} _{H}}$ is the identity operator of H) and for every ξ, η ∈ H, the set-function

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

is a complex measure on M (that is, a complex-valued countably additive function). We denote this measure by ${\displaystyle \operatorname {S} _{\pi }(\xi ,\eta )}$. Note that ${\displaystyle \operatorname {S} _{\pi }(\xi ,\xi )}$ is a real-valued measure, and a probability measure when ${\displaystyle \xi }$ has length one.

If π is a projection-valued measure and

${\displaystyle E\cap F=\emptyset ,}$

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

${\displaystyle \pi (E)\pi (F)=\pi (E\cap F)=\pi (F)\pi (E),}$

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

${\displaystyle \mathbf {1} _{E}\mapsto \pi (E)}$

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

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

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

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

is a homomorphism of rings. An integral notation is often used for ${\displaystyle \operatorname {T} _{\pi }(f)}$, as in

${\displaystyle \operatorname {T} _{\pi }(f)=\int _{X}f(x)d\pi (x)=\int _{X}fd\pi .}$

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

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

${\displaystyle A=\int _{\mathbb {R} }xd\pi _{A}(x).}$

This allows to define the Borel functional calculus for such operators: if ${\displaystyle g:\mathbb {R} \to \mathbb {C} }$ is a measurable function, we set

${\displaystyle g(A):=\int _{\mathbb {R} }g(x)d\pi _{A}(x).}$

## 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

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

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

${\displaystyle \pi (E)=U^{*}\rho (E)U\quad }$

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

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

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:

${\displaystyle \pi =\bigoplus _{1\leq n\leq \omega }(\pi |H_{n})}$

where

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

and

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

## 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

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

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

${\displaystyle A(\phi )=\int _{\mathbf {R} }\lambda \,d\pi (\lambda )(\phi ),}$

which takes the more readable form

${\displaystyle A(\phi )=\sum _{i}\lambda _{i}\pi ({\lambda _{i}})(\phi )}$

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

• 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.