Projection-valued measure

Inmathematics,particularly infunctional analysis,aprojection-valued measure(orspectral measure) is a function defined on certain subsets of a fixed set and whose values areself-adjoint projectionson a fixedHilbert space.^[1]A projection-valued measure (PVM) is formally similar to areal-valued measure,except that its values are self-adjoint projections rather than real numbers. As in the case of ordinary measures, it is possible tointegrate complex-valued functionswith respect to a PVM; the result of such an integration is alinear operatoron the given Hilbert space.

Projection-valued measures are used to express results inspectral theory,such as the importantspectral theoremforself-adjoint operators,in which case the PVM is sometimes referred to as thespectral measure.TheBorel functional calculusfor self-adjoint operators is constructed using integrals with respect to PVMs. Inquantum mechanics,PVMs are the mathematical description ofprojective measurements.^{[clarification needed]}They are generalized bypositive operator valued measures(POVMs) in the same sense that amixed stateordensity matrixgeneralizes the notion of apure state.

Definition

Let $H$ denote aseparable complex Hilbert spaceand $(X,M)$ ameasurable spaceconsisting of a set $X$ and aBorel σ-algebra $M$ on $X$ .Aprojection-valued measure $\pi$ is a map from $M$ to the set ofbounded self-adjoint operatorson $H$ satisfying the following properties:^[2]^[3]

$\pi (E)$ is anorthogonal projectionfor all $E\in M.$
$\pi (\emptyset )=0$ and $\pi (X)=I$ ,where $\emptyset$ is theempty setand $I$ theidentity operator.
If $E_{1},E_{2},E_{3},\dotsc$ in $M$ are disjoint, then for all $v\in H$ ,

\pi \left(\bigcup _{j=1}^{\infty }E_{j}\right)v=\sum _{j=1}^{\infty }\pi (E_{j})v.

$\pi (E_{1}\cap E_{2})=\pi (E_{1})\pi (E_{2})$ for all $E_{1},E_{2}\in M.$

The second and fourth property show that if $E_{1}$ and $E_{2}$ are disjoint, i.e., $E_{1}\cap E_{2}=\emptyset$ ,the images $\pi (E_{1})$ and $\pi (E_{2})$ areorthogonalto each other.

Let $V_{E}=\operatorname {im} (\pi (E))$ and itsorthogonal complement $V_{E}^{\perp }=\ker(\pi (E))$ denote theimageandkernel,respectively, of $\pi (E)$ .If $V_{E}$ is a closed subspace of $H$ then $H$ can be wrtitten as theorthogonal decomposition $H=V_{E}\oplus V_{E}^{\perp }$ and $\pi (E)=I_{E}$ is the unique identity operator on $V_{E}$ satisfying all four properties.^[4]^[5]

For every $\xi,\eta \in H$ and $E\in M$ the projection-valued measure forms acomplex-valued measureon $H$ defined as

\mu _{\xi,\eta }(E):=\langle \pi (E)\xi \mid \eta \rangle

withtotal variationat most $\|\xi \|\|\eta \|$ .^[6]It reduces to a real-valuedmeasurewhen

\mu _{\xi }(E):=\langle \pi (E)\xi \mid \xi \rangle

and aprobability measurewhen $\xi$ is aunit vector.

ExampleLet $(X,M,\mu )$ be a $σ$ -finite measure spaceand, for all $E\in M$ ,let

\pi (E):L^{2}(X)\to L^{2}(X)

be defined as

\psi \mapsto \pi (E)\psi =1_{E}\psi,

i.e., as multiplication by theindicator function $1_{E}$ onL²(X).Then $\pi (E)=1_{E}$ defines a projection-valued measure.^[6]For example, if $X=\mathbb {R}$ , $E=(0,1)$ ,and $\phi,\psi \in L^{2}(\mathbb {R} )$ there is then the associated complex measure $\mu _{\phi,\psi }$ which takes a measurable function $f:\mathbb {R} \to \mathbb {R}$ and gives the integral

\int _{E}f\,d\mu _{\phi,\psi }=\int _{0}^{1}f(x)\psi (x){\overline {\phi }}(x)\,dx

Extensions of projection-valued measures

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

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

extends to a linear map on the vector space ofstep functionsonX.In fact, it is easy to check that this map is aring homomorphism.This map extends in a canonical way to all bounded complex-valuedmeasurable functionsonX,and we have the following.

Theorem—For any bounded Borel function $f$ on $X$ ,there exists a uniquebounded operator $T:H\to H$ such that ^[7]^[8]

\langle T\xi \mid \xi \rangle =\int _{X}f(\lambda )\,d\mu _{\xi }(\lambda ),\quad \forall \xi \in H.

where $\mu _{\xi }$ is a finiteBorel measuregiven by

\mu _{\xi }(E):=\langle \pi (E)\xi \mid \xi \rangle,\quad \forall E\in M.

Hence, $(X,M,\mu )$ is afinite measure space.

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

This allows to define theBorel functional calculusfor such operators and then pass to measurable functions via theRiesz–Markov–Kakutani representation theorem.That is, if $g:\mathbb {R} \to \mathbb {C}$ is a measurable function, then a unique measure exists such that

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

Spectral theorem

Let $H$ be aseparable complex Hilbert space, $A:H\to H$ be a boundedself-adjoint operatorand $\sigma (A)$ thespectrumof $A$ .Then thespectral theoremsays that there exists a unique projection-valued measure $\pi ^{A}$ ,defined on aBorel subset $E\subset \sigma (A)$ ,such that^[9]

A=\int _{\sigma (A)}\lambda \,d\pi ^{A}(\lambda ),

where the integral extends to an unbounded function $\lambda$ when the spectrum of $A$ is unbounded.^[10]

Direct integrals

First we provide a general example of projection-valued measure based ondirect integrals.Suppose (X,M,μ) is a measure space and let {H_x}_x∈Xbe a μ-measurable family of separable Hilbert spaces. For everyE∈M,let $π$ (E) be the operator of multiplication by 1_Eon the Hilbert space

\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 ofH,K. $π$ ,ρ areunitarily equivalentif and only ifthere is a unitary operatorU:H→Ksuch that

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

for everyE∈M.

Theorem.If (X,M) is astandard Borel space,then for every projection-valued measure $π$ on (X,M) taking values in the projections of aseparableHilbert 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_Eon the Hilbert space

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

The measure class^{[clarification needed]}of μ and the measure equivalence class of the multiplicity functionx→ dimH_xcompletely characterize the projection-valued measure up to unitary equivalence.

A projection-valued measure $π$ ishomogeneous of multiplicitynif and only if the multiplicity function has constant valuen.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:

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

where

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

and

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

Application in quantum mechanics

In quantum mechanics, given a projection-valued measure of a measurable space $X$ to the space of continuous endomorphisms upon a Hilbert space $H$ ,

theprojective space $\mathbf {P} (H)$ of the Hilbert space $H$ is interpreted as the set of possible (normalizable) states $\varphi$ of a quantum system,^[11]
the measurable space $X$ is the value space for some quantum property of the system (an "observable" ),
the projection-valued measure $\pi$ expresses the probability that theobservabletakes on various values.

A common choice for $X$ is the real line, but it may also be

$\mathbb {R} ^{3}$ (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 $\varphi$ .

Let $E$ be a measurable subset of $X$ and $\varphi$ a normalizedvector quantum statein $H$ ,so that its Hilbert norm is unitary, $\|\varphi \|=1$ .The probability that the observable takes its value in $E$ ,given the system in state $\varphi$ ,is

P_{\pi }(\varphi )(E)=\langle \varphi \mid \pi (E)(\varphi )\rangle =\langle \varphi |\pi (E)|\varphi \rangle.

We can parse this in two ways. First, for each fixed $E$ ,the projection $\pi (E)$ is aself-adjoint operatoron $H$ whose 1-eigenspace are the states $\varphi$ for which the value of the observable always lies in $E$ ,and whose 0-eigenspace are the states $\varphi$ for which the value of the observable never lies in $E$ .

Second, for each fixed normalized vector state $\varphi$ ,the association

P_{\pi }(\varphi ):E\mapsto \langle \varphi \mid \pi (E)\varphi \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 $\pi$ is called aprojective measurement.

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

A(\varphi )=\int _{\mathbb {R} }\lambda \,d\pi (\lambda )(\varphi ),

which reduces to

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

if the support of $\pi$ is a discrete subset of $X$ .

The above operator $A$ is called the observable associated with the spectral measure.

Generalizations

The idea of a projection-valued measure is generalized by thepositive 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 toquantum information theory.

Notes

^Conway 2000,p. 41.
^Hall 2013,p. 138.
^Reed & Simon 1980,p. 234.
^Rudin 1991,p. 308.
^Hall 2013,p. 541.
^^a ^bConway 2000,p. 42.
^Kowalski, Emmanuel (2009),Spectral theory in Hilbert spaces(PDF),ETH Zürich lecture notes, p. 50
^Reed & Simon 1980,p. 227,235.
^Reed & Simon 1980,p. 235.
^Hall 2013,p. 205.
^Ashtekar & Schilling 1999,pp. 23–65.

References

Ashtekar, Abhay; Schilling, Troy A. (1999). "Geometrical Formulation of Quantum Mechanics".On Einstein's Path.New York, NY: Springer New York.arXiv:gr-qc/9706069.doi:10.1007/978-1-4612-1422-9_3.ISBN 978-1-4612-7137-6.*Conway, John B. (2000).A course in operator theory.Providence (R.I.): American mathematical society.ISBN 978-0-8218-2065-0.
Hall, Brian C. (2013).Quantum Theory for Mathematicians.New York: Springer Science & Business Media.ISBN 978-1-4614-7116-5.
Mackey, G. W.,The Theory of Unitary Group Representations,The University of Chicago Press, 1976
Moretti, Valter (2017),Spectral Theory and Quantum Mechanics Mathematical Foundations of Quantum Theories, Symmetries and Introduction to the Algebraic Formulation,vol. 110, Springer,Bibcode:2017stqm.book.....M,ISBN 978-3-319-70705-1
Narici, Lawrence; Beckenstein, Edward (2011).Topological Vector Spaces.Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press.ISBN 978-1584888666.OCLC 144216834.
Reed, M.;Simon, B.(1980).Methods of Modern Mathematical Physics: Vol 1: Functional analysis.Academic Press.ISBN 978-0-12-585050-6.
Rudin, Walter (1991).Functional Analysis.Boston, Mass.: McGraw-Hill Science, Engineering & Mathematics.ISBN 978-0-07-054236-5.
Schaefer, Helmut H.;Wolff, Manfred P. (1999).Topological Vector Spaces.GTM.Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer.ISBN 978-1-4612-7155-0.OCLC 840278135.
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(2006) [1967].Topological Vector Spaces, Distributions and Kernels.Mineola, N.Y.: Dover Publications.ISBN 978-0-486-45352-1.OCLC 853623322.
Varadarajan, V. S.,Geometry of Quantum TheoryV2, Springer Verlag, 1970.

[FOOTNOTEConway200041-1] Conway 2000,p. 41.

[FOOTNOTEHall2013138-2] Hall 2013,p. 138.

[FOOTNOTEReedSimon1980234-3] Reed & Simon 1980,p. 234.

[FOOTNOTERudin1991308-4] Rudin 1991,p. 308.

[FOOTNOTEHall2013541-5] Hall 2013,p. 541.

[FOOTNOTEConway200042-6] Conway 2000,p. 42.

[7] Kowalski, Emmanuel (2009),Spectral theory in Hilbert spaces(PDF),ETH Zürich lecture notes, p. 50

[FOOTNOTEReedSimon1980227,235-8] Reed & Simon 1980,p. 227,235.

[FOOTNOTEReedSimon1980235-9] Reed & Simon 1980,p. 235.

[FOOTNOTEHall2013205-10] Hall 2013,p. 205.

[FOOTNOTEAshtekarSchilling199923–65-11] Ashtekar & Schilling 1999,pp. 23–65.

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v t e Spectral theoryand^*-algebras
Basic concepts	Involution/-algebra Banach algebra B-algebra C-algebra Noncommutative topology Projection-valued measure Spectrum Spectrum of a C-algebra Spectral radius Operator space
Main results	Gelfand–Mazur theorem Gelfand–Naimark theorem Gelfand representation Polar decomposition Singular value decomposition Spectral theorem Spectral theory of normal C*-algebras
Special Elements/Operators	Isospectral Normal operator Hermitian/Self-adjoint operator Unitary operator Unit
Spectrum	Krein–Rutman theorem Normal eigenvalue Spectrum of a C*-algebra Spectral radius Spectral asymmetry Spectral gap
Decomposition	Decomposition of a spectrum Continuous Point Residual Approximate point Compression Direct integral Discrete Spectral abscissa
Spectral Theorem	Borel functional calculus Min-max theorem Positive operator-valued measure Projection-valued measure Riesz projector Rigged Hilbert space Spectral theorem Spectral theory of compact operators Spectral theory of normal C*-algebras
Special algebras	Amenable Banach algebra With anApproximate identity Banach function algebra Disk algebra Nuclear C*-algebra Uniform algebra Von Neumann algebra Tomita–Takesaki theory
Finite-Dimensional	Alon–Boppana bound Bauer–Fike theorem Numerical range Schur–Horn theorem
Generalizations	Dirac spectrum Essential spectrum Pseudospectrum Structure space(Shilov boundary)
Miscellaneous	Abstract index group Banach algebra cohomology Cohen–Hewitt factorization theorem Extensions of symmetric operators Fredholm theory Limiting absorption principle Schröder–Bernstein theorems for operator algebras Sherman–Takeda theorem Unbounded operator
Examples	Wiener algebra
Applications	Almost Mathieu operator Corona theorem Hearing the shape of a drum(Dirichlet eigenvalue) Heat kernel Kuznetsov trace formula Lax pair Proto-value function Ramanujan graph Rayleigh–Faber–Krahn inequality Spectral geometry Spectral method Spectral theory of ordinary differential equations Sturm–Liouville theory Superstrong approximation Transfer operator Transform theory Weyl law Wiener–Khinchin theorem

v t e Analysisintopological vector spaces
Basic concepts	Abstract Wiener space Classical Wiener space Bochner space Convex series Cylinder set measure Infinite-dimensional vector function Matrix calculus Vector calculus
Derivatives	Differentiable vector–valued functions from Euclidean space Differentiation in Fréchet spaces Fréchet derivative Total Functional derivative Gateaux derivative Directional Generalizations of the derivative Hadamard derivative Holomorphic Quasi-derivative
Measurability	Besov measure Cylinder set measure Canonical Gaussian Classical Wiener measure Measurelikeset functions infinite-dimensional Gaussian measure Projection-valued Vector Bochner/Weakly/Strongly measurable function Radonifying function
Integrals	Bochner Direct integral Dunford Gelfand–Pettis/Weak Regulated Paley–Wiener
Results	Cameron–Martin theorem Inverse function theorem Nash–Moser theorem Feldman–Hájek theorem No infinite-dimensional Lebesgue measure Sazonov's theorem Structure theorem for Gaussian measures
Related	Crinkled arc Covariance operator
Functional calculus	Borel functional calculus Continuous functional calculus Holomorphic functional calculus
Applications	Banach manifold(bundle) Convenient vector space Choquet theory Fréchet manifold Hilbert manifold