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

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

• \pi(E) is an orthogonal projection for all E \in M.
• \pi(\emptyset) = 0 and \pi(X) = I, where \emptyset is the empty set and I the identity operator.
• If E_1, E_2, E_3,\dotsc in M are disjoint, then for all v \in H,

\pi\left(\bigcup_{j=1}^{\infty} E_j \right)v = \sum_{j=1}^{\infty} \pi(E_j) v.

• \pi(E_1 \cap E_2)= \pi(E_1)\pi(E_2) for all E_1, E_2 \in M.

The fourth property is a consequence of the first and third property.^[4] The second and fourth property show that if E_1 and E_2 are disjoint, i.e., E_1 \cap E_2 = \emptyset, the images \pi(E_1) and \pi(E_2) are orthogonal to each other.

Let V_E = \operatorname{im}(\pi(E)) and its orthogonal complement V^\perp_E=\ker(\pi(E)) denote the image and kernel, respectively, of \pi(E). If V_E is a closed subspace of H then H can be wrtitten as the orthogonal decomposition H=V_E \oplus V^\perp_E and \pi(E)=I_E is the unique identity operator on V_E satisfying all four properties.^[5][6]

For every \xi,\eta\in H and E\in M the projection-valued measure forms a complex-valued measure on H defined as

\mu_{\xi,\eta}(E) := \langle \pi(E)\xi \mid \eta \rangle

with total variation at most \|\xi\|\|\eta\|.^[7] It reduces to a real-valued measure when

\mu_{\xi}(E) := \langle \pi(E)\xi \mid \xi \rangle

and a probability measure when \xi is a unit vector.

Example Let (X, M, \mu) be a σ-finite measure space and, for all E \in M, let

\pi(E) : L^2(X) \to L^2 (X)

be defined as

\psi \mapsto \pi(E)\psi=1_E \psi,

i.e., as multiplication by the indicator function 1_E on L²(X). Then \pi(E)=1_E defines a projection-valued measure.^[7] For example, if X = \mathbb{R}, E = (0,1), and \varphi,\psi \in L^2(\mathbb{R}) there is then the associated complex measure \mu_{\varphi,\psi} which takes a measurable function f: \mathbb{R} \to \mathbb{R} and gives the integral

\int_E f\,d\mu_{\varphi,\psi} = \int_0^1 f(x)\psi(x)\overline{\varphi}(x)\,dx

Extensions of projection-valued measures

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

\chi_E \mapsto \pi(E)

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.

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

Spectral theorem

Let H be a separable complex Hilbert space, A:H\to H be a bounded self-adjoint operator and \sigma(A) the spectrum of A. Then the spectral theorem says that there exists a unique projection-valued measure \pi^A, defined on a Borel subset E \subset \sigma(A), such that

A =\int_{\sigma(A)} \lambda \, d\pi^A(\lambda),

and \pi^{A}(E) is called the spectral projection of A.^[3][8] The integral extends to an unbounded function \lambda when the spectrum of A is unbounded.^[9]

The spectral theorem allows us to define the Borel functional calculus for any Borel measurable function g:\mathbb{R}\to\mathbb{C} by integrating with respect to the projection-valued measure \pi^{A}:

g(A) :=\int_\mathbb{R} g(\lambda) \, d\pi^{A}(\lambda).

A similar construction holds for normal operators and measurable functions g:\mathbb{C}\to\mathbb{C}.

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 {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

\int_X^\oplus H_x \ d \mu(x).

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

\pi(E) = U^* \rho(E) U \quad

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

\int_X^\oplus H_x \ d \mu(x).

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:

\pi = \bigoplus_{1 \leq n \leq \omega} (\pi \mid H_n)

where

H_n = \int_{X_n}^\oplus H_x \ d (\mu \mid X_n) (x)

and

X_n = \{x \in X: \dim H_x = n\}.

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 projective space \mathbf{P}(H) of the Hilbert space H is interpreted as the set of possible (normalizable) states \varphi of a quantum system,^[10]
• the measurable space X is the value space for some quantum property of the system (an "observable"),
• the projection-valued measure \pi expresses the probability that the observable takes on various values.

A common choice for X is the real line, but it may also be

• \mathbb{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 \varphi.

Let E be a measurable subset of X and \varphi a normalized vector quantum state in H, so that its Hilbert norm is unitary, \|\varphi\|=1. The probability that the observable takes its value in E, given the system in state \varphi, is

P_\pi(\varphi)(E) = \langle \varphi\mid\pi(E)(\varphi)\rangle = \langle \varphi\mid\pi(E)\mid\varphi\rangle.

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

Second, for each fixed normalized vector state \varphi, the association

P_\pi(\varphi) : E \mapsto \langle\varphi\mid\pi(E)\varphi\rangle

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 \pi is called a projective measurement.

If X is the real number line, there exists, associated to \pi, a self-adjoint operator A defined on H by

A(\varphi) = \int_{\mathbb{R}} \lambda \,d\pi(\lambda)(\varphi),

which reduces to

A(\varphi) = \sum_i \lambda_i \pi({\lambda_i})(\varphi)

if the support of \pi is a discrete subset of X.

The above operator A 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

1. ^ Conway 2000, p. 41.
2. ^ Hall 2013, p. 138.
3. ^ Reed & Simon 1980, p. 234.
4. ^ Reed & Simon 1980, p. 235.
5. ^ Rudin 1991, p. 308.
6. ^ Hall 2013, p. 541.
7. ^ Conway 2000, p. 42.
8. ^ Hall 2013, pp. 125,141.
9. ^ Hall 2013, p. 205.
10. ^ Ashtekar & Schilling 1999, pp. 23–65.

References

• Ashtekar, Abhay & Schilling, Troy A. (1999). "Geometrical Formulation of Quantum Mechanics". On Einstein's Path. New York, NY: Springer New York. arXiv:gr-qc/9706069. doi:10.1007/978-1-4612-1422-9_3. ISBN 978-1-4612-7137-6.
• 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, Valter (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
• 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.
• 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.