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Projection-valued measure

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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.

DefinitionEdit

Let <math>H</math> denote a separable complex Hilbert space and <math>(X, M)</math> a measurable space consisting of a set <math>X</math> and a 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 bounded self-adjoint operators on <math>H</math> satisfying the following properties:Template:SfnTemplate:Sfn

  • <math>\pi(E)</math> is an 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 and 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.Template:SfnTemplate:Sfn

For every <math>\xi,\eta\in H</math> and <math>E\in M</math> the projection-valued measure forms a 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>.Template:Sfn It reduces to a real-valued 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|Template:Math-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 L2(X). Then <math>\pi(E)=1_E</math> defines a projection-valued measure.Template:Sfn For example, if <math>X = \mathbb{R}</math>, <math>E = (0,1)</math>, and <math>\varphi,\psi \in L^2(\mathbb{R})</math> there is then the associated complex measure <math>\mu_{\varphi,\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_{\varphi,\psi} = \int_0^1 f(x)\psi(x)\overline{\varphi}(x)\,dx</math>

Extensions of projection-valued measuresEdit

If Template: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 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.

Template:Math theorem

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 theoremEdit

Template:See also Let <math>H</math> be a separable complex Hilbert space, <math>A:H\to H</math> be a bounded self-adjoint operator and <math>\sigma(A)</math> the 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 subset <math> E \subset \sigma(A)</math>, such thatTemplate:Sfn

<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.Template:Sfn

Direct integralsEdit

First we provide a general example of projection-valued measure based on direct integrals. Suppose (X, M, μ) is a measure space and let {Hx}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 1E on the Hilbert space

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

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

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

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 {Hx}x ∈ X , such that Template:Pi is unitarily equivalent to multiplication by 1E on the Hilbert space

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

The measure classTemplate:Clarify of μ and the measure equivalence class of the multiplicity function x → dim Hx 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:

<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 mechanicsEdit

Template:See also 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 space <math>\mathbf{P}(H)</math> of the Hilbert space <math>H</math> is interpreted as the set of possible (normalizable) states <math>\varphi</math> of a quantum system,Template:Sfn
  • 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 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\mid\pi(E)\mid\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.

Template:AnchorA 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.

GeneralizationsEdit

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 alsoEdit

NotesEdit

Template:Reflist

ReferencesEdit

Template:Measure theory Template:Spectral theory Template:Functional analysis Template:Analysis in topological vector spaces

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