Projection-valued measure

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

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

(where ${\displaystyle \operatorname {id} _{H}}$ is the identity operator of ${\displaystyle H}$) and for every ${\displaystyle \xi ,\eta \in H}$, the following function ${\displaystyle M\to \mathbb {C} }$

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

is a complex measure on ${\displaystyle M}$ (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 ${\displaystyle \pi }$ is a projection-valued measure and

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

then the images ${\displaystyle \pi (E)}$, ${\displaystyle \pi (F)}$ 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 ${\displaystyle (X,M,\mu )}$ is a measure space. Let, for every measurable subset ${\displaystyle E}$ in ${\displaystyle M}$,

${\displaystyle \pi (E):L^{2}(\mu )\to L^{2}(\mu ):\psi \mapsto \chi _{E}\psi }$

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

${\displaystyle S_{(0,1)}(\phi ,\psi )(f)=\int _{(0,1)}f(x)\psi (x){\overline {\phi }}(x)dx}$

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

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

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

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

such that

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

for all ${\displaystyle \xi ,\eta \in H,}$ where ${\displaystyle \operatorname {S} _{\pi }(\xi ,\eta )}$ denotes the complex measure

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

from the definition of ${\displaystyle \pi }$.

The map

${\displaystyle {\mathcal {BM}}(X,M)\to {\mathcal {L}}(H):f\mapsto \operatorname {T} _{\pi }(f)}$

is a homomorphism of rings.

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

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

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

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

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

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

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

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

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:H → K such that

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

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

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

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:

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

where

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

and

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

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

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

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 ${\displaystyle \psi }$, the association

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

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

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

which takes the more readable form

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

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.

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.

• Spectral theorem
• Spectral theory of compact operators
• Spectral theory of normal C*-algebras

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