Projection-valued measure (original) (raw)

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Mathematical operator-value measure of interest in quantum mechanics and functional analysis

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.

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

π ( ⋃ j = 1 ∞ E j ) v = ∑ j = 1 ∞ π ( E j ) v . {\displaystyle \pi \left(\bigcup _{j=1}^{\infty }E_{j}\right)v=\sum _{j=1}^{\infty }\pi (E_{j})v.} $\pi \left(\bigcup _{j=1}^{\infty }E_{j}\right)v=\sum _{j=1}^{\infty }\pi (E_{j})v.$

The second and fourth property show that if E 1 {\displaystyle E_{1}} $E_{1}$ and E 2 {\displaystyle E_{2}} $E_{2}$ are disjoint, i.e., E 1 ∩ E 2 = ∅ {\displaystyle E_{1}\cap E_{2}=\emptyset } $E_{1}\cap E_{2}=\emptyset$ , the images π ( E 1 ) {\displaystyle \pi (E_{1})} $\pi (E_{1})$ and π ( E 2 ) {\displaystyle \pi (E_{2})} $\pi (E_{2})$ are orthogonal to each other.

Let V E = im ⁡ ( π ( E ) ) {\displaystyle V_{E}=\operatorname {im} (\pi (E))} $V_{E}=\operatorname {im} (\pi (E))$ and its orthogonal complement V E ⊥ = ker ⁡ ( π ( E ) ) {\displaystyle V_{E}^{\perp }=\ker(\pi (E))} $V_{E}^{\perp }=\ker(\pi (E))$ denote the image and kernel, respectively, of π ( E ) {\displaystyle \pi (E)} $\pi (E)$ . If V E {\displaystyle V_{E}} $V_{E}$ is a closed subspace of H {\displaystyle H} $H$ then H {\displaystyle H} $H$ can be wrtitten as the orthogonal decomposition H = V E ⊕ V E ⊥ {\displaystyle H=V_{E}\oplus V_{E}^{\perp }} $H=V_{E}\oplus V_{E}^{\perp }$ and π ( E ) = I E {\displaystyle \pi (E)=I_{E}} $\pi (E)=I_{E}$ is the unique identity operator on V E {\displaystyle V_{E}} $V_{E}$ satisfying all four properties.[4][5]

For every ξ , η ∈ H {\displaystyle \xi ,\eta \in H} $\xi ,\eta \in H$ and E ∈ M {\displaystyle E\in M} $E\in M$ the projection-valued measure forms a complex-valued measure on H {\displaystyle H} $H$ defined as

μ ξ , η ( E ) := ⟨ π ( E ) ξ ∣ η ⟩ {\displaystyle \mu _{\xi ,\eta }(E):=\langle \pi (E)\xi \mid \eta \rangle } $\mu _{\xi ,\eta }(E):=\langle \pi (E)\xi \mid \eta \rangle$

with total variation at most ‖ ξ ‖ ‖ η ‖ {\displaystyle \|\xi \|\|\eta \|} $\|\xi \|\|\eta \|$ .[6] It reduces to a real-valued measure when

μ ξ ( E ) := ⟨ π ( E ) ξ ∣ ξ ⟩ {\displaystyle \mu _{\xi }(E):=\langle \pi (E)\xi \mid \xi \rangle } $\mu _{\xi }(E):=\langle \pi (E)\xi \mid \xi \rangle$

and a probability measure when ξ {\displaystyle \xi } $\xi$ is a unit vector.

Example Let ( X , M , μ ) {\displaystyle (X,M,\mu )} $(X,M,\mu )$ be a _σ_-finite measure space and, for all E ∈ M {\displaystyle E\in M} $E\in M$ , let

π ( E ) : L 2 ( X ) → L 2 ( X ) {\displaystyle \pi (E):L^{2}(X)\to L^{2}(X)} $\pi (E):L^{2}(X)\to L^{2}(X)$

be defined as

ψ ↦ π ( E ) ψ = 1 E ψ , {\displaystyle \psi \mapsto \pi (E)\psi =1_{E}\psi ,} $\psi \mapsto \pi (E)\psi =1_{E}\psi ,$

i.e., as multiplication by the indicator function 1 E {\displaystyle 1_{E}} $1_{E}$ on _L_2(X). Then π ( E ) = 1 E {\displaystyle \pi (E)=1_{E}} $\pi (E)=1_{E}$ defines a projection-valued measure.[6] For example, if X = R {\displaystyle X=\mathbb {R} } $X=\mathbb {R}$ , E = ( 0 , 1 ) {\displaystyle E=(0,1)} $E=(0,1)$ , and ϕ , ψ ∈ L 2 ( R ) {\displaystyle \phi ,\psi \in L^{2}(\mathbb {R} )} $\phi ,\psi \in L^{2}(\mathbb {R} )$ there is then the associated complex measure μ ϕ , ψ {\displaystyle \mu _{\phi ,\psi }} $\mu _{\phi ,\psi }$ which takes a measurable function f : R → R {\displaystyle f:\mathbb {R} \to \mathbb {R} } $f:\mathbb {R} \to \mathbb {R}$ and gives the integral

∫ E f d μ ϕ , ψ = ∫ 0 1 f ( x ) ψ ( x ) ϕ ¯ ( x ) d x {\displaystyle \int _{E}f\,d\mu _{\phi ,\psi }=\int _{0}^{1}f(x)\psi (x){\overline {\phi }}(x)\,dx} $\int _{E}f\,d\mu _{\phi ,\psi }=\int _{0}^{1}f(x)\psi (x){\overline {\phi }}(x)\,dx$

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

χ E ↦ π ( E ) {\displaystyle \chi _{E}\mapsto \pi (E)} $\chi _{E}\mapsto \pi (E)$

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.

Theorem — For any bounded Borel function f {\displaystyle f} $f$ on X {\displaystyle X} $X$ , there exists a unique bounded operator T : H → H {\displaystyle T:H\to H} $T:H\to H$ such that[7][8]

⟨ T ξ ∣ ξ ⟩ = ∫ X f ( λ ) d μ ξ ( λ ) , ∀ ξ ∈ H . {\displaystyle \langle T\xi \mid \xi \rangle =\int _{X}f(\lambda )\,d\mu _{\xi }(\lambda ),\quad \forall \xi \in H.} $\langle T\xi \mid \xi \rangle =\int _{X}f(\lambda )\,d\mu _{\xi }(\lambda ),\quad \forall \xi \in H.$

where μ ξ {\displaystyle \mu _{\xi }} $\mu _{\xi }$ is a finite Borel measure given by

μ ξ ( E ) := ⟨ π ( E ) ξ ∣ ξ ⟩ , ∀ E ∈ M . {\displaystyle \mu _{\xi }(E):=\langle \pi (E)\xi \mid \xi \rangle ,\quad \forall E\in M.} $\mu _{\xi }(E):=\langle \pi (E)\xi \mid \xi \rangle ,\quad \forall E\in M.$

Hence, ( X , M , μ ) {\displaystyle (X,M,\mu )} $(X,M,\mu )$ is a finite measure space.

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

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 g : R → C {\displaystyle g:\mathbb {R} \to \mathbb {C} } $g:\mathbb {R} \to \mathbb {C}$ is a measurable function, then a unique measure exists such that

g ( T ) := ∫ R g ( x ) d π ( x ) . {\displaystyle g(T):=\int _{\mathbb {R} }g(x)\,d\pi (x).} $g(T):=\int _{\mathbb {R} }g(x)\,d\pi (x).$

Let H {\displaystyle H} $H$ be a separable complex Hilbert space, A : H → H {\displaystyle A:H\to H} $A:H\to H$ be a bounded self-adjoint operator and σ ( A ) {\displaystyle \sigma (A)} $\sigma (A)$ the spectrum of A {\displaystyle A} $A$ . Then the spectral theorem says that there exists a unique projection-valued measure π A {\displaystyle \pi ^{A}} $\pi ^{A}$ , defined on a Borel subset E ⊂ σ ( A ) {\displaystyle E\subset \sigma (A)} $E\subset \sigma (A)$ , such that[9]

A = ∫ σ ( A ) λ d π A ( λ ) , {\displaystyle A=\int _{\sigma (A)}\lambda \,d\pi ^{A}(\lambda ),} $A=\int _{\sigma (A)}\lambda \,d\pi ^{A}(\lambda ),$

where the integral extends to an unbounded function λ {\displaystyle \lambda } $\lambda$ when the spectrum of A {\displaystyle A} $A$ is unbounded.[10]

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

∫ X ⊕ H x d μ ( x ) . {\displaystyle \int _{X}^{\oplus }H_{x}\ d\mu (x).} $\int _{X}^{\oplus }H_{x}\ d\mu (x).$

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

π ( E ) = U ∗ ρ ( E ) U {\displaystyle \pi (E)=U^{*}\rho (E)U\quad } $\pi (E)=U^{*}\rho (E)U\quad$

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

∫ X ⊕ H x d μ ( x ) . {\displaystyle \int _{X}^{\oplus }H_{x}\ d\mu (x).} $\int _{X}^{\oplus }H_{x}\ d\mu (x).$

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:

π = ⨁ 1 ≤ n ≤ ω ( π ∣ H n ) {\displaystyle \pi =\bigoplus _{1\leq n\leq \omega }(\pi \mid H_{n})} $\pi =\bigoplus _{1\leq n\leq \omega }(\pi \mid H_{n})$

where

H n = ∫ X n ⊕ H x d ( μ ∣ X n ) ( x ) {\displaystyle H_{n}=\int _{X_{n}}^{\oplus }H_{x}\ d(\mu \mid X_{n})(x)} $H_{n}=\int _{X_{n}}^{\oplus }H_{x}\ d(\mu \mid X_{n})(x)$

and

X n = { x ∈ X : dim ⁡ H x = n } . {\displaystyle X_{n}=\{x\in X:\dim H_{x}=n\}.} $X_{n}=\{x\in X:\dim H_{x}=n\}.$

Application in quantum mechanics

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In quantum mechanics, given a projection-valued measure of a measurable space X {\displaystyle X} $X$ to the space of continuous endomorphisms upon a Hilbert space H {\displaystyle H} $H$ ,

A common choice for X {\displaystyle X} $X$ is the real line, but it may also be

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

P π ( φ ) ( E ) = ⟨ φ ∣ π ( E ) ( φ ) ⟩ = ⟨ φ | π ( E ) | φ ⟩ . {\displaystyle P_{\pi }(\varphi )(E)=\langle \varphi \mid \pi (E)(\varphi )\rangle =\langle \varphi |\pi (E)|\varphi \rangle .} $P_{\pi }(\varphi )(E)=\langle \varphi \mid \pi (E)(\varphi )\rangle =\langle \varphi |\pi (E)|\varphi \rangle .$

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

Second, for each fixed normalized vector state φ {\displaystyle \varphi } $\varphi$ , the association

P π ( φ ) : E ↦ ⟨ φ ∣ π ( E ) φ ⟩ {\displaystyle P_{\pi }(\varphi ):E\mapsto \langle \varphi \mid \pi (E)\varphi \rangle } $P_{\pi }(\varphi ):E\mapsto \langle \varphi \mid \pi (E)\varphi \rangle$

is a probability measure on X {\displaystyle X} $X$ making the values of the observable into a random variable.

A measurement that can be performed by a projection-valued measure π {\displaystyle \pi } $\pi$ is called a projective measurement.

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

A ( φ ) = ∫ R λ d π ( λ ) ( φ ) , {\displaystyle A(\varphi )=\int _{\mathbb {R} }\lambda \,d\pi (\lambda )(\varphi ),} $A(\varphi )=\int _{\mathbb {R} }\lambda \,d\pi (\lambda )(\varphi ),$

which reduces to

A ( φ ) = ∑ i λ i π ( λ i ) ( φ ) {\displaystyle A(\varphi )=\sum _{i}\lambda _{i}\pi ({\lambda _{i}})(\varphi )} $A(\varphi )=\sum _{i}\lambda _{i}\pi ({\lambda _{i}})(\varphi )$

if the support of π {\displaystyle \pi } $\pi$ is a discrete subset of X {\displaystyle X} $X$ .

The above operator A {\displaystyle A} $A$ is called the observable associated with the spectral measure.

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.

^ Conway 2000, p. 41.
^ Hall 2013, p. 138.
^ Reed & Simon 1980, p. 234.
^ Rudin 1991, p. 308.
^ Hall 2013, p. 541.
^ a b Conway 2000, p. 42.
^ Kowalski, Emmanuel (2009), Spectral theory in Hilbert spaces (PDF), ETH Zürich lecture notes, p. 50
^ Reed & Simon 1980, p. 227,235.
^ Reed & Simon 1980, p. 235.
^ Hall 2013, p. 205.
^ Ashtekar & Schilling 1999, pp. 23–65.

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