# 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 Hilbert space. Projection-valued measures are used to express results in spectral theory, such as the spectral theorem for self-adjoint operators. In quantum mechanics, PVMs are the mathematical description of projective measurements. 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

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 .

If π is a projection-valued measure and

then π(*A*), π(*B*) are orthogonal projections. From this follows that in general,

**Example**. Suppose (*X*, *M*, μ) is a measure space. Let π(*A*) be the operator of multiplication by the indicator function 1_{A} on *L*^{2}(*X*). Then π is a projection-valued measure.

## Extensions of projection-valued measures

If π is an additive projection-valued measure on (*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. In fact this map extends in a canonical way to all bounded complex-valued Borel functions on *X*.

**Theorem**. For any bounded *M*-measurable function *f* on *X*, there is a unique bounded linear operator T_{π}(*f*) such that

for all ξ, η ∈ *H*. The map

is a homomorphism of rings.

## 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 {*H*_{x}}_{x ∈ X } be a μ-measurable family of separable Hilbert spaces. For every *A* ∈ *M*, let π(*A*) be the operator of multiplication by 1_{A} 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 *A* ∈ *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_{A} on the Hilbert space

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:

where

and

## 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. 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.