Projection-valued measure
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.
Contents
Formal definition
A projection-valued measure on a measurable space , where is a σ-algebra of subsets of , is a mapping from to the set of self-adjoint projections on a Hilbert space (i.e. the orthogonal projections) such that
(where is the identity operator of ) and for every , the set-function
is a complex measure on (that is, a complex-valued countably additive function).
We denote this measure by .
Note that is a real-valued measure, and a probability measure when has length one.
If π is a projection-valued measure and
then the images , are orthogonal to each other. From this follows that in general,
and they commute.
Example. Suppose is a measure space. Let, for every measurable subset in ,
be the operator of multiplication by the indicator function on L^{2}(X). Then is a projection-valued measure.
Extensions of projection-valued measures, integrals and the spectral theorem
If π 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.
Theorem. For any bounded M-measurable function f on X, there exists a unique bounded linear operator
such that
for all . Where, denotes the complex measure
from the definition of .
The map
is a homomorphism of rings.
An integral notation is often used for , as in
The theorem is also correct for unbounded measurable functions f, but then will be an unbounded linear operator on the Hilbert space H.
The spectral theorem says that every self-adjoint operator has an associated projection-valued measure defined on the real axis, such that
This allows to define the Borel functional calculus for such operators: if is a measurable function, we set
Structure of projection-valued measures
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 π(E) be the operator of multiplication by 1_{E} on the Hilbert space
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:H → K such that
for every E ∈ M.
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 {H_{x}}_{x ∈ X }, such that π is unitarily equivalent to multiplication by 1_{E} on the Hilbert space
The measure class^{[clarification needed]} 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 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:
where
and
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,
- 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"),
- the projection-valued measure π 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 Φ.
Let E be a measurable subset of the measurable space X and Φ a normalized vector-state in H, so that its Hilbert norm is unitary, ||Φ|| = 1. The probability that the observable takes its value in the subset E, given the system in state Φ, is
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 normalized vector state , the association
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 number line, there exists, associated to π, a Hermitian operator A defined on H by
which takes the more readable form
if the support of π is a discrete subset of R.
The above operator A is called the observable associated with the spectral measure.
Any operator so obtained is called an observable , in quantum mechanics.
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^{[clarification needed]}. This generalization is motivated by applications to quantum information theory.
References
- Moretti, V. (2018), Spectral Theory and Quantum Mechanics Mathematical Foundations of Quantum Theories, Symmetries and Introduction to the Algebraic Formulation, 110, Springer, ISBN 978-3-319-70705-1
- Hall, B.C. (2013), Quantum Theory for Mathematicians, Graduate Texts in Mathematics, 267, Springer, ISBN 978-1461471158
- 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.