View source for Projection-valued measure

{{Short description|Mathematical operator-value measure of interest in quantum mechanics and functional analysis}}
In [[mathematics]], particularly in [[functional analysis]], a  '''projection-valued measure''' (or '''spectral measure''') is a function defined on certain subsets of a fixed set and whose values are [[self-adjoint]] [[projection (mathematics)|projection]]s on a fixed [[Hilbert space]].{{sfn | Conway | 2000 | p=41}} A projection-valued measure (PVM) is formally similar to a [[real-valued]] [[Measure (mathematics)|measure]], except that its values are self-adjoint projections rather than real numbers. As in the case of ordinary measures, it is possible to [[integration (mathematics)|integrate]] [[complex-valued function]]s 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#Spectral subspaces and projection-valued measures|spectral theorem]] for [[Self-adjoint operator#Formulation in the physics literature|self-adjoint operators]], in which case the PVM is sometimes referred to as the [[Spectral theory of ordinary differential equations#Spectral measure|spectral measure]]. 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 [[Quantum measurement|projective measurements]].{{clarify|reason=Is this a novel term? It's not defined in the linked article.|date=May 2015}}  They are generalized by [[POVM|positive operator valued measures]] (POVMs) in the same sense that a [[mixed state (physics)|mixed state]] or [[density matrix]] generalizes the notion of a [[pure state]].

== Definition ==
Let <math>H</math> denote a [[separable space|separable]] [[complex number|complex]] [[Hilbert space]] and  <math>(X, M)</math> a [[measurable space]] consisting of a set <math>X</math> and a [[Borel_set|Borel σ-algebra]]  <math>M</math> on <math>X</math>. A '''projection-valued measure''' <math>\pi</math> is a map from <math>M</math> to the set of [[Self-adjoint_operator#Bounded_self-adjoint_operators|bounded self-adjoint operators]] on <math>H</math> satisfying the following properties:{{sfn | Hall | 2013 | p=138}}{{sfn | Reed | Simon | 1980 | p=234}}
* <math>\pi(E)</math> is an [[Projection_(linear_algebra)#Orthogonal_projections|orthogonal projection]] for all <math>E \in M.</math>
* <math>\pi(\emptyset) = 0</math> and <math>\pi(X) = I</math>, where <math>\emptyset</math> is the [[empty set]] and <math>I</math> the [[identity operator]].
* If <math>E_1, E_2, E_3,\dotsc</math> in <math>M</math> are disjoint, then for all <math>v \in H</math>,
::<math>\pi\left(\bigcup_{j=1}^{\infty} E_j \right)v = \sum_{j=1}^{\infty} \pi(E_j) v.</math>
* <math>\pi(E_1 \cap E_2)= \pi(E_1)\pi(E_2)</math> for all <math>E_1, E_2 \in M.</math>

The second and fourth property show that if <math>
E_1 </math> and <math>E_2</math> are disjoint, i.e., <math>E_1 \cap E_2 = \emptyset</math>, the images <math>\pi(E_1)</math> and <math>\pi(E_2)</math> are [[orthogonal]] to each other. 

Let <math>V_E = \operatorname{im}(\pi(E))</math> and its [[orthogonal complement]] <math>V^\perp_E=\ker(\pi(E))</math> denote the [[Image_(mathematics)|image]] and  [[Kernel_(linear_algebra)|kernel]], respectively, of <math>\pi(E)</math>. If <math>V_E </math> is a closed subspace of <math>H</math> then <math>H</math> can be wrtitten as the  ''orthogonal decomposition'' <math>H=V_E \oplus V^\perp_E</math> and <math>\pi(E)=I_E</math> is the unique identity operator on <math>V_E </math> satisfying all four properties.{{sfn | Rudin | 1991 | p=308}}{{sfn | Hall | 2013 | p=541}}

For every <math>\xi,\eta\in H</math> and <math>E\in M</math> the projection-valued measure forms a [[complex measure|complex-valued measure]] on <math>H</math> defined as
:<math> \mu_{\xi,\eta}(E) := \langle \pi(E)\xi \mid \eta \rangle 
</math>
with [[total variation]] at most <math>\|\xi\|\|\eta\|</math>.{{sfn | Conway | 2000 | p=42}} It reduces to a real-valued [[Measure_(mathematics)|measure]] when
:<math> \mu_{\xi}(E) := \langle \pi(E)\xi \mid \xi \rangle 
</math>
and a [[probability measure]] when <math>\xi</math> is a [[unit vector]].

'''Example'''  Let <math>(X, M, \mu)</math> be a [[Measure space#Important classes of measure spaces|{{math|''&sigma;''}}-finite measure space]] and, for all <math>E \in M</math>, let 
:<math>
\pi(E) : L^2(X) \to L^2 (X)
</math>
be defined as
:<math>\psi \mapsto \pi(E)\psi=1_E \psi,</math>
i.e., as multiplication by the [[indicator function]] <math>1_E</math> on [[Lp space|''L''<sup>2</sup>(''X'')]]. Then <math>\pi(E)=1_E</math> defines a projection-valued measure.{{sfn | Conway | 2000 | p=42}} For example, if <math>X = \mathbb{R}</math>, <math>E = (0,1)</math>, and <math>\phi,\psi \in L^2(\mathbb{R})</math> there is then the associated complex measure <math>\mu_{\phi,\psi}</math> which takes a measurable function <math>f: \mathbb{R} \to \mathbb{R}</math> and gives the integral
:<math>\int_E f\,d\mu_{\phi,\psi} = \int_0^1 f(x)\psi(x)\overline{\phi}(x)\,dx</math>

== Extensions of projection-valued measures ==

If {{pi}} is a projection-valued measure on a measurable space (''X'', ''M''), then the map

: <math>
 \chi_E \mapsto \pi(E)
</math>

extends to a linear map on the vector space of [[step function]]s 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 function]]s on ''X'', and we have the following.

{{math theorem|Theorem|For any bounded Borel function <math>f</math> on <math>X</math>, there exists a unique [[bounded operator]] <math> T : H \to H </math> such that
<ref>{{Citation |last=Kowalski|first=Emmanuel| year=2009|title=Spectral theory in Hilbert spaces| series = ETH Zürich lecture notes | url=https://people.math.ethz.ch/~kowalski/spectral-theory.pdf|page = 50}}</ref>{{sfn | Reed | Simon | 1980 | p=227,235}}
:<math>\langle T \xi \mid  \xi \rangle = \int_X f(\lambda) \,d\mu_{\xi}(\lambda), \quad \forall \xi \in H.</math>
where <math>\mu_{\xi}</math> is a finite [[Borel measure]] given by
:<math>\mu_{\xi}(E) := \langle \pi(E)\xi \mid \xi \rangle, \quad \forall E \in M.</math>
Hence, <math>(X,M,\mu)</math> is a [[finite measure space]].}}

The theorem is also correct for unbounded measurable functions <math>f</math> but then <math>T</math> will be an unbounded linear operator on the Hilbert space <math>H</math>.

This allows to define the [[Borel functional calculus]] for such operators and then pass to measurable functions via the [[Riesz–Markov–Kakutani representation theorem]]. That is, if <math>g:\mathbb{R}\to\mathbb{C}</math> is a measurable function, then a unique measure exists such that
:<math>g(T) :=\int_\mathbb{R} g(x) \, d\pi(x).</math>

=== Spectral theorem ===
{{see also|Self-adjoint operator#Spectral theorem}}
Let <math>H</math> be a [[separable space|separable]] [[complex number|complex]] [[Hilbert space]], <math>A:H\to H</math> be a bounded [[self-adjoint operator]] and <math>\sigma(A)</math> the [[Spectrum_(functional_analysis)|spectrum]] of <math>A</math>. Then the [[spectral theorem]] says that there exists a unique projection-valued measure <math>\pi^A</math>, defined on a [[Borel_set|Borel subset]] <math> E \subset \sigma(A)</math>, such that{{sfn | Reed | Simon | 1980 | p=235}}
:<math>A =\int_{\sigma(A)} \lambda \, d\pi^A(\lambda),</math>
where the integral extends to an unbounded function <math>\lambda</math> when the spectrum of <math>A</math> is unbounded.{{sfn | Hall | 2013 | p=205}}

=== Direct integrals===

First we provide a general example of projection-valued measure based on [[direct integral]]s. Suppose (''X'', ''M'', μ) is a measure space and let {''H''<sub>''x''</sub>}<sub>''x'' ∈ ''X'' </sub> be a μ-measurable family of separable Hilbert spaces. For every  ''E'' ∈ ''M'', let {{pi}}(''E'') be the operator of multiplication by 1<sub>''E''</sub> on the Hilbert space

:<math> \int_X^\oplus H_x \ d \mu(x). </math>

Then {{pi}} is a projection-valued measure on (''X'', ''M'').

Suppose {{pi}}, ρ  are projection-valued measures on  (''X'', ''M'') with values in the projections of ''H'', ''K''.   {{pi}},  ρ  are '''unitarily equivalent''' [[if and only if]] there is a unitary operator ''U'':''H'' → ''K'' such that

:<math> \pi(E) = U^* \rho(E) U \quad </math>

for every ''E'' ∈ ''M''.

'''Theorem'''. If (''X'', ''M'') is a [[Borel algebra#Standard Borel spaces and Kuratowski theorems|standard Borel space]], then for every projection-valued measure {{pi}} 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''<sub>''x''</sub>}<sub>''x'' ∈ ''X'' </sub>, such that {{pi}} is unitarily equivalent to multiplication by 1<sub>''E''</sub> on the Hilbert space

:<math> \int_X^\oplus H_x \ d \mu(x). </math>

The measure class{{clarify|reason=What is a measure class? A measure up to measure-preserving equivalence? Should the measure be completed?|date=May 2015}} of μ and the measure equivalence class of the multiplicity function ''x'' → dim ''H''<sub>''x''</sub> completely characterize the projection-valued measure up to unitary equivalence.

A projection-valued measure {{pi}} is ''homogeneous of multiplicity'' ''n'' if and only if the multiplicity function has constant value ''n''.  Clearly,

'''Theorem'''.  Any projection-valued measure {{pi}} taking values in the projections of a separable Hilbert space is an orthogonal direct sum of homogeneous projection-valued measures:

:<math> \pi = \bigoplus_{1 \leq n \leq \omega} (\pi \mid H_n) </math>

where

:<math> H_n = \int_{X_n}^\oplus H_x \ d (\mu \mid X_n) (x) </math>

and

:<math> X_n = \{x \in X: \dim H_x = n\}. </math>

==Application in quantum mechanics==
{{see also|Expectation value (quantum mechanics)}}
In quantum mechanics, given a projection-valued measure of a measurable space <math>X</math> to the space of continuous endomorphisms upon a Hilbert space <math>H</math>,

* the [[Projective Hilbert space|projective space]] <math>\mathbf{P}(H)</math> of the Hilbert space <math>H</math> is interpreted as the set of possible ([[Probability_amplitude#Normalization|normalizable]]) states <math>\varphi</math> of a quantum system,{{sfn | Ashtekar | Schilling | 1999 | pp=23–65}}
* the measurable space <math>X</math> is the value space for some quantum property of the system (an "observable"), 
* the projection-valued measure <math>\pi</math> expresses the probability that the [[observable]] takes on various values.

A common choice for <math>X</math> is the real line, but it may also be 

* <math>\mathbb{R}^3</math> (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 <math>\varphi</math>.

Let <math>E</math> be a measurable subset of <math>X</math> and <math>\varphi</math> a normalized [[Quantum_state|vector quantum state]] in <math>H</math>, so that its Hilbert norm is unitary, <math>\|\varphi\|=1</math>. The probability that the observable takes its value in <math>E</math>, given the system in state <math>\varphi</math>, is

:<math>
 P_\pi(\varphi)(E) = \langle \varphi\mid\pi(E)(\varphi)\rangle = \langle \varphi|\pi(E)|\varphi\rangle.</math>

We can parse this in two ways. First, for each fixed <math>E</math>, the projection <math>\pi(E)</math> is a [[self-adjoint operator]] on <math>H</math> whose 1-eigenspace are the states <math>\varphi</math> for which the value of the observable always lies in <math>E</math>, and whose 0-eigenspace are the states <math>\varphi</math> for which the value of the observable never lies in <math>E</math>.  

Second, for each fixed normalized vector state <math>\varphi</math>, the association 

:<math>
P_\pi(\varphi) :
E \mapsto  \langle\varphi\mid\pi(E)\varphi\rangle
</math>
is a probability measure on <math>X</math> making the values of the observable into a random variable.

{{Anchor|Projective measurement}}A measurement that can be performed by a projection-valued measure <math>\pi</math> is called a '''projective measurement'''. 

If <math>X</math> is the real number line, there exists, associated to <math>\pi</math>, a self-adjoint operator <math>A</math> defined on <math>H</math> by

:<math>A(\varphi) = \int_{\mathbb{R}} \lambda \,d\pi(\lambda)(\varphi),</math>

which reduces to

:<math>A(\varphi) = \sum_i \lambda_i \pi({\lambda_i})(\varphi)</math>

if the support of <math>\pi</math> is a discrete subset of <math>X</math>. 

The above operator <math>A</math> is called the observable associated with the spectral measure.

==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{{clarify|reason=Partition of unity in the operator sense is not defined in the article "partition of unity".|date=May 2015}}. This generalization is motivated by applications to [[quantum information theory]].

== See also ==
* [[Spectral theorem]]
* [[Spectral theory of compact operators]]
* [[Spectral theory of normal C*-algebras]]

==Notes==
{{reflist}}

==References==
* {{cite book | last1=Ashtekar | first1=Abhay | last2=Schilling | first2=Troy A. | title=On Einstein's Path | chapter=Geometrical Formulation of Quantum Mechanics | publisher=Springer New York | publication-place=New York, NY | year=1999 | isbn=978-1-4612-7137-6 | doi=10.1007/978-1-4612-1422-9_3 | arxiv=gr-qc/9706069 }}* {{cite book | last=Conway | first=John B. | title=A course in operator theory | publisher=American mathematical society | publication-place=Providence (R.I.) | date=2000 | isbn=978-0-8218-2065-0}}
* {{cite book | last=Hall | first=Brian C. | title=Quantum Theory for Mathematicians | publisher=Springer Science & Business Media | publication-place=New York | date=2013 | isbn=978-1-4614-7116-5}}
* Mackey, G. W., ''The Theory of Unitary Group Representations'', The University of Chicago Press, 1976
* {{citation |last=Moretti |first=Valter |title=Spectral Theory and Quantum Mechanics Mathematical Foundations of Quantum Theories, Symmetries and Introduction to the Algebraic Formulation |volume=110 |year=2017 |publisher=Springer |bibcode=2017stqm.book.....M |isbn=978-3-319-70705-1}}
* {{Narici Beckenstein Topological Vector Spaces|edition=2}} <!-- {{sfn | Narici | 2011 | p=}} -->
* {{cite book | last1=Reed | first1=M. | author-link=Michael C. Reed | last2=Simon | first2=B. | author-link2=Barry Simon| title=Methods of Modern Mathematical Physics: Vol 1: Functional analysis | publisher=Academic Press | year=1980 | isbn=978-0-12-585050-6}}
* {{cite book | last=Rudin | first=Walter | title=Functional Analysis | publisher=McGraw-Hill Science, Engineering & Mathematics | publication-place=Boston, Mass. | date=1991 | isbn=978-0-07-054236-5}}
* {{Schaefer Wolff Topological Vector Spaces|edition=2}} <!-- {{sfn | Schaefer | 1999 | p=}} -->
* [[Gerald Teschl|G. Teschl]], ''Mathematical Methods in Quantum Mechanics with Applications to Schrödinger Operators'', https://www.mat.univie.ac.at/~gerald/ftp/book-schroe/, American Mathematical Society, 2009.
* {{Trèves François Topological vector spaces, distributions and kernels}} <!-- {{sfn | Treves | 2006 | p=}} -->
* Varadarajan, V. S., ''Geometry of Quantum Theory'' V2, Springer Verlag, 1970.

{{Measure theory}}
{{Spectral theory}}
{{Functional analysis}}
{{Analysis in topological vector spaces}}

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