Projection-valued measure

Summary

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.[1] 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.[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.

Definition edit

Let   denote a separable complex Hilbert space and   a measurable space consisting of a set   and a Borel σ-algebra   on  . A projection-valued measure   is a map from   to the set of bounded self-adjoint operators on   satisfying the following properties:[2][3]

  •   is an orthogonal projection for all  
  •   and  , where   is the empty set and   the identity operator.
  • If   in   are disjoint, then for all  ,
 
  •   for all  

The second and fourth property show that if   and   are disjoint, i.e.,  , the images   and   are orthogonal to each other.

Let   and its orthogonal complement   denote the image and kernel, respectively, of  . If   is a closed subspace of   then   can be wrtitten as the orthogonal decomposition   and   is the unique identity operator on   satisfying all four properties.[4][5]

For every   and   the projection-valued measure forms a complex-valued measure on   defined as

 

with total variation at most  .[6] It reduces to a real-valued measure when

 

and a probability measure when   is a unit vector.

Example Let   be a σ-finite measure space and, for all  , let

 

be defined as

 

i.e., as multiplication by the indicator function   on L2(X). Then   defines a projection-valued measure.[6] For example, if  ,  , and   there is then the associated complex measure   which takes a measurable function   and gives the integral

 

Extensions of projection-valued measures edit

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 Borel function   on  , there exists a unique bounded operator   such that [7][8]

 

where   is a finite Borel measure given by

 

Hence,   is a finite measure space.

The theorem is also correct for unbounded measurable functions   but then   will be an unbounded linear operator on the Hilbert space  .

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   is a measurable function, then a unique measure exists such that

 

Spectral theorem edit

Let   be a separable complex Hilbert space,   be a bounded self-adjoint operator and   the spectrum of  . Then the spectral theorem says that there exists a unique projection-valued measure  , defined on a Borel subset  , such that[9]

 

where the integral extends to an unbounded function   when the spectrum of   is unbounded.[10]

Direct integrals 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

 

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

 

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

 

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:

 

where

 

and

 

Application in quantum mechanics edit

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 projective space 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

  • R3 (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.

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.

See also edit

Notes edit

  1. ^ Conway 2000, p. 41.
  2. ^ Hall 2013, p. 138.
  3. ^ Reed & Simon 1980, p. 234.
  4. ^ Rudin 1991, p. 308.
  5. ^ Hall 2013, p. 541.
  6. ^ a b Conway 2000, p. 42.
  7. ^ Kowalski, Emmanuel (2009), Spectral theory in Hilbert spaces (PDF), ETH Zürich lecture notes, p. 50
  8. ^ Reed & Simon 1980, p. 227,235.
  9. ^ Reed & Simon 1980, p. 235.
  10. ^ Hall 2013, p. 205.

References edit

  • Conway, John B. (2000). A course in operator theory. Providence (R.I.): American mathematical society. ISBN 978-0-8218-2065-0.
  • Hall, Brian C. (2013). Quantum Theory for Mathematicians. New York: Springer Science & Business Media. ISBN 978-1-4614-7116-5.
  • Mackey, G. W., The Theory of Unitary Group Representations, The University of Chicago Press, 1976
  • Moretti, V. (2017), Spectral Theory and Quantum Mechanics Mathematical Foundations of Quantum Theories, Symmetries and Introduction to the Algebraic Formulation, vol. 110, Springer, Bibcode:2017stqm.book.....M, ISBN 978-3-319-70705-1
  • Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
  • Reed, M.; Simon, B. (1980). Methods of Modern Mathematical Physics: Vol 1: Functional analysis. Academic Press. ISBN 978-0-12-585050-6.
  • Rudin, Walter (1991). Functional Analysis. Boston, Mass.: McGraw-Hill Science, Engineering & Mathematics. ISBN 978-0-07-054236-5.
  • Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
  • 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.
  • Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
  • Varadarajan, V. S., Geometry of Quantum Theory V2, Springer Verlag, 1970.