Projection-valued measure

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

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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

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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

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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

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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

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In quantum mechanics, given a projection-valued measure of a measurable space   to the space of continuous endomorphisms upon a Hilbert space  ,

  • the projective space   of the Hilbert space   is interpreted as the set of possible (normalizable) states   of a quantum system,[11]
  • the measurable space   is the value space for some quantum property of the system (an "observable"),
  • the projection-valued measure   expresses the probability that the observable takes on various values.

A common choice for   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.),
  • the 2-point set "true" and "false" for the truth-value of an arbitrary proposition about  .

Let   be a measurable subset of   and   a normalized vector quantum state in  , so that its Hilbert norm is unitary,  . The probability that the observable takes its value in  , given the system in state  , is

 

We can parse this in two ways. First, for each fixed  , the projection   is a self-adjoint operator on   whose 1-eigenspace are the states   for which the value of the observable always lies in  , and whose 0-eigenspace are the states   for which the value of the observable never lies in  .

Second, for each fixed normalized vector state  , the association

 

is a probability measure on   making the values of the observable into a random variable.

A measurement that can be performed by a projection-valued measure   is called a projective measurement.

If   is the real number line, there exists, associated to  , a self-adjoint operator   defined on   by

 

which reduces to

 

if the support of   is a discrete subset of  .

The above operator   is called the observable associated with the spectral measure.

Generalizations

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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

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Notes

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  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.
  11. ^ Ashtekar & Schilling 1999, pp. 23–65.

References

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