Projection-valued measure

Template:Short description 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 projections on a fixed Hilbert space.Template:Sfn A projection-valued measure (PVM) is formally similar to a real-valued measure, except that its 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, in which case the PVM is sometimes referred to as the 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 projective measurements.Template:Clarify 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.

Definition

Let $H$ denote a separable complex Hilbert space and $(X,M)$ a measurable space consisting of a set $X$ and a Borel σ-algebra $M$ on $X$. A projection-valued measure $\pi$ is a map from $M$ to the set of bounded self-adjoint operators on $H$ satisfying the following properties:Template:SfnTemplate:Sfn

• $\pi (E)$ is an orthogonal projection for all $E\in M.$
• $\pi (\mathrm{\varnothing })=0$ and $\pi (X)=I$, where $\mathrm{\varnothing }$ is the empty set and $I$ the identity operator.
• If $E_1,E_2,E_3,\dots$ in $M$ are disjoint, then for all $v\in H$,

$\pi \left(\bigcup _{j=1}^{\mathrm{\infty }}{E}_j\right)v=\sum _{j=1}^{\mathrm{\infty }}\pi (E_j)v.$

• $\pi (E_1\cap E_2)=\pi (E_1)\pi (E_2)$ for all $E_1,E_2\in M.$

The fourth property is a consequence of the first and third property.Template:Sfn The second and fourth property show that if $E_1$ and $E_2$ are disjoint, i.e., $E_1\cap E_2=\mathrm{\varnothing }$, the images $\pi (E_1)$ and $\pi (E_2)$ are orthogonal to each other.

Let $V_E=\mathrm{im}(\pi (E))$ and its orthogonal complement $V_E^{\perp }=\ker (\pi (E))$ denote the image and kernel, respectively, of $\pi (E)$. If $V_E$ is a closed subspace of $H$ then $H$ can be wrtitten as the orthogonal 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.Template:SfnTemplate:Sfn

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

${\mu }_{\xi ,\eta }(E):=⟨\pi (E)\xi \mid \eta ⟩$

with total variation at most $\Vert \xi \Vert \Vert \eta \Vert$.Template:Sfn It reduces to a real-valued measure when

${\mu }_{\xi }(E):=⟨\pi (E)\xi \mid \xi ⟩$

and a probability measure when $\xi$ is a unit vector.

Example Let $(X,M,\mu )$ be a [[Measure space#Important classes of measure spaces|Template:Math-finite measure space]] and, 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 the indicator function $1_E$ on L²(X). Then $\pi (E)=1_E$ defines a projection-valued measure.Template:Sfn For example, if $X=ℝ$, $E=(0,1)$, and $\phi ,\psi \in L^2(ℝ)$ there is then the associated complex measure ${\mu }_{\phi ,\psi }$ which takes a measurable function $f:ℝ\to ℝ$ and gives the integral

${\int }_{E}fd{\mu }_{\phi ,\psi }={\int }_0^{1}f(x)\psi (x)\bar{\phi }(x)dx$

Extensions of projection-valued measures

If Template:Pi 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 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.

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

Spectral theorem

Template:See also Let $H$ be a separable complex Hilbert space, $A:H\to H$ be a bounded self-adjoint operator and $\sigma (A)$ the spectrum of $A$. Then the spectral theorem says that there exists a unique projection-valued measure ${\pi }^A$, defined on a Borel subset $E\subset \sigma (A)$, such that $$A={\int }_{\sigma (A)}\lambda d{\pi }^A(\lambda ),$$ and ${\pi }^A(E)$ is called the spectral projection of $A$.Template:SfnTemplate:Sfn The integral extends to an unbounded function $\lambda$ when the spectrum of $A$ is unbounded.Template:Sfn

The spectral theorem allows us to define the Borel functional calculus for any Borel measurable function $g:ℝ\to ℂ$ by integrating with respect to the projection-valued measure ${\pi }^A$: $$g(A):={\int }_{ℝ}g(\lambda )d{\pi }^A(\lambda ).$$ A similar construction holds for normal operators and measurable functions $g:ℂ\to ℂ$.

Direct integrals

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 Template:Pi(E) be the operator of multiplication by 1_E on the Hilbert space

${\int }_X^{\oplus }{H}_{x}d\mu (x).$

Then Template:Pi is a projection-valued measure on (X, M).

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

$\pi (E)=U^{\ast }\rho (E)U$

for every E ∈ M.

Theorem. If (X, M) is a standard Borel space, then for every projection-valued measure Template: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_x}_{x ∈ X} , such that Template:Pi is unitarily equivalent to multiplication by 1_E on the Hilbert space

${\int }_X^{\oplus }{H}_{x}d\mu (x).$

The measure classTemplate:Clarify 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 Template:Pi is homogeneous of multiplicity n if and only if the multiplicity function has constant value n. Clearly,

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

$\pi =\underset{1\le n\le \omega }{⨁}(\pi \mid H_n)$

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

Template:See also 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 $𝐏(H)$ of the Hilbert space $H$ is interpreted as the set of possible (normalizable) states $\phi$ of a quantum system,Template:Sfn
• 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 the observable takes on various values.

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

• ${ℝ}^3$ (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 $\phi$.

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

$P_{\pi }(\phi )(E)=⟨\phi \mid \pi (E)(\phi )⟩=⟨\phi \mid \pi (E)\mid \phi ⟩.$

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

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

$P_{\pi }(\phi ):E\mapsto ⟨\phi \mid \pi (E)\phi ⟩$

is a probability measure on $X$ making the values of the observable into a random variable.

Template:AnchorA measurement that can be performed by a projection-valued measure $\pi$ is called a projective measurement.

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

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

which reduces to

$A(\phi )=\sum _i{\lambda }_i\pi ({\lambda }_i)(\phi )$

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 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", i.e. a set of positive semi-definite Hermitian operators that sum to the identity. 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

References

• Template:Cite book
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• Mackey, G. W., The Theory of Unitary Group Representations, The University of Chicago Press, 1976
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• Template:Narici Beckenstein Topological Vector Spaces
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• Template:Schaefer Wolff Topological Vector Spaces
• 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.
• Template:Trèves François Topological vector spaces, distributions and kernels
• Varadarajan, V. S., Geometry of Quantum Theory V2, Springer Verlag, 1970.