Projection-valued measure

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In mathematics, particularly in functional analysis, a projection-valued measure (PVM) is a function defined on certain subsets of a fixed set and whose values are self-adjoint projections on a fixed Hilbert space. Projection-valued measures are formally similar to real-valued measures, except that their 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. 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.[clarification needed] 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.

Formal definition[edit]

A projection-valued measure on a measurable space (X, M), where M is a σ-algebra of subsets of X, is a mapping π from M to the set of self-adjoint projections on a Hilbert space H such that

\pi(X) = \operatorname{id}_H \quad

and for every ξ, η ∈ H, the set-function

E \mapsto \langle \pi(E)\xi \mid \eta \rangle

is a complex measure on M (that is, a complex-valued countably additive function). We denote this measure by \operatorname{S}_\pi(\xi, \eta). Note that \operatorname{S}_\pi(\xi, \xi) is a real-valued measure, and a probability measure when \xi has length one.

If π is a projection-valued measure and

E \cap F = \emptyset,

then π(E), π(F) are orthogonal projections. From this follows that in general,

 \pi(E) \pi(F) = \pi(E \cap F) =  \pi(F) \pi(E),

and they commute.

Example. Suppose (X, M, μ) is a measure space. Let π(E) be the operator of multiplication by the indicator function 1E on L2(X). Then π is a projection-valued measure.

Extensions of projection-valued measures, integrals and the spectral theorem[edit]

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

 \mathbf{1}_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 is a unique bounded linear operator Tπ(f) such that

 \langle \operatorname{T}_\pi(f) \xi \mid  \eta \rangle = \int_X f(x) d \operatorname{S}_\pi (\xi,\eta)(x)

for all ξ, η ∈ H. Here, \operatorname{S}_\pi (\xi,\eta) denotes the complex measure E \mapsto \langle \pi(E)\xi \mid \eta \rangle from the definition of \pi. The map

f \mapsto \operatorname{T}_\pi(f)

is a homomorphism of rings. An integral notation is often used for \operatorname{T}_\pi(f), as in

\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 \operatorname{T}_\pi(f) will be an unbounded linear operator on the Hilbert space H.

The spectral theorem says that every self-adjoint operator A:H\to H has an associated projection-valued measure \pi_A defined on the real axis, such that

A =\int_\mathbb{R} x d \pi_A(x).

This allows to define the Borel functional calculus for such operators: if g:\mathbb{R}\to\mathbb{C} is a measurable function, we set

g(A) :=\int_\mathbb{R} g(x) d \pi_A(x).

Structure of projection-valued measures[edit]

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}xX be a μ-measurable family of separable Hilbert spaces. For every EM, let π(E) be the operator of multiplication by 1E on 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 of H, K. π, ρ are unitarily equivalent if and only if there is a unitary operator U:HK such that

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

for every EM.

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 {Hx}xX , such that π is unitarily equivalent to multiplication by 1E on 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 function x → dim Hx 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:

 \pi = \bigoplus_{1 \leq n \leq \omega} (\pi | H_n)

where

 H_n = \int_{X_n}^\oplus H_x \ d (\mu | X_n) (x)

and

 X_n = \{x \in X: \operatorname{dim} H_x = n\}.

Application in quantum mechanics[edit]

In quantum mechanics, the unit sphere 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"), and the projection-valued measure π expresses the probability that the observable takes on various values.

A common choice for X is the real numbers, but it may also be R3 (for position or momentum), a discrete set (for angular momentum, energy of a bound state, etc), or the 2-point set "true" and "false" for the truth-value of an arbitrary proposition about Φ.

Let E be a measurable subset of X and Φ a state in H, so that |Φ|=1. The probability that the observable takes its value in E given the system in state Φ is

P = \langle \phi,\pi(E)(\phi)\rangle = \langle \phi|\pi(E)|\phi\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 Φ, the association E ↦ ⟨Φ,π(⋅)Φ⟩ 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 numbers, there is associated to π a Hermitian operator A defined on H by

A(\phi) = \int_{\bold{R}} \lambda \,d\pi(\lambda)(\phi),

which takes the more readable form

A(\phi) = \sum_i \lambda_i \pi({\lambda_i})(\phi)

if the support of π is a discrete subset of R. This operator is called an observable in quantum mechanics.

Generalizations[edit]

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[clarification needed]. This generalization is motivated by applications to quantum information theory.

References[edit]

  • G. W. Mackey, The Theory of Unitary Group Representations, The University of Chicago Press, 1976
  • M. Reed and B. Simon, Methods of Mathematical Physics, vols I–IV, Academic Press 1972.
  • G. Teschl, Mathematical Methods in Quantum Mechanics with Applications to Schrödinger Operators, http://www.mat.univie.ac.at/~gerald/ftp/book-schroe/, American Mathematical Society, 2009.
  • V. S. Varadarajan, Geometry of Quantum Theory V2, Springer Verlag, 1970.