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 projectionss on a fixed Hilbert space. 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. 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

denote a separable complex Hilbert space a measurable space and a Borel σ-algebra. A projection-valued measure to the set of bounded self-adjoint operators satisfying the following properties:

• is an orthogonal projection

• is the empty set the identity operator.

• ,

The fourth property is a consequence of the first and third property. are orthogonal to each other.

and its orthogonal complement denote the image and kernel can be wrtitten as the orthogonal decomposition satisfying all four properties.

the projection-valued measure forms a complex-valued measure defined as

with total variation. It reduces to a real-valued measure when

and a probability measure is a unit vector.

Example be a σ-finite measure space, let

be defined as

i.e., as multiplication by the indicator function on L2(X) defines a projection-valued measure. and gives the integral

Extensions of projection-valued measures

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

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.

.

Spectral theorem

be a separable complex Hilbert space be a bounded self-adjoint operator the spectrum. Then the spectral theorem, defined on a Borel subset is called the spectral projection. is unbounded.

The spectral theorem allows us to define the Borel functional calculus A similar construction holds for normal operators.

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 {Hx}x ∈ X be a μ-measurable family of separable Hilbert spaces. For every E ∈ M, let (E) be the operator of multiplication by 1E on the Hilbert space

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

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

for every E ∈ M.

Theorem. If (X, M) is a standard Borel space, then for every projection-valued measure 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 is unitarily equivalent to multiplication by 1E on the Hilbert space

The measure class 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 is homogeneous of multiplicity n if and only if the multiplicity function has constant value n. Clearly,

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

where

and

Application in quantum mechanics

,

• the projective space is interpreted as the set of possible (normalizable of a quantum system,

• is the value space for some quantum property of the system (an "observable"),

• expresses the probability that the observable takes on various values.

is the real line, but it may also be

• (for position or momentum in three dimensions ),

• a discrete set (for angular momentum, energy of a bound state, etc.),

• .

a normalized vector quantum state, is

is a self-adjoint operator.

, the association

making the values of the observable into a random variable.

is called a projective measurement.

by

which reduces to

.

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

Notes

References

• Mackey, G. W., The Theory of Unitary Group Representations, The University of Chicago Press, 1976

• 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.

• Varadarajan, V. S., Geometry of Quantum Theory V2, Springer Verlag, 1970.