Projection-valued measure

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: • \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. 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. 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\|. 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 -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. 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 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. {{math theorem|Theorem|For any bounded Borel function f on X, there exists a unique bounded operator T : H \to H such that :\langle T \xi \mid \xi \rangle = \int_X f(\lambda) \,d\mu_{\xi}(\lambda), \quad \forall \xi \in H. where \mu_{\xi} is a finite Borel measure given by :\mu_{\xi}(E) := \langle \pi(E)\xi \mid \xi \rangle, \quad \forall E \in M. Hence, (X,M,\mu) is a finite measure space.}} 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. The integral extends to an unbounded function \lambda when the spectrum of A is unbounded. 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 {Hx}x ∈ X be a μ-measurable family of separable Hilbert spaces. For every E ∈ M, let (E) be the operator of multiplication by 1E on the Hilbert space : \int_X^\oplus H_x \ d \mu(x). Then is a projection-valued measure on (X, M). Suppose , ρ are projection-valued measures on (X, M) with values in the projections of H, K. , 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 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}x ∈ X , such that is unitarily equivalent to multiplication by 1E on the Hilbert space : \int_X^\oplus H_x \ d \mu(x). The measure class 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 is homogeneous of multiplicity n if and only if the multiplicity function has constant value n. Clearly, Theorem. Any projection-valued measure 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, • 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 ==