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 H) and for every ξ, η ∈ H, the set-function
is a complex measure on M (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 π(E), π(F) 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 E in M,
be the operator of multiplication by the indicator function 1_{E} on L^{2}(X). Then π 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, 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 can π express 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.), 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
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 , 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.