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.

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

• ${\displaystyle \pi (E)}$ is an orthogonal projection for all ${\displaystyle E\in M.}$
• ${\displaystyle \pi (\emptyset )=0}$ and ${\displaystyle \pi (X)=I}$, where ${\displaystyle \emptyset }$ is the empty set and ${\displaystyle I}$ the identity operator.
• If ${\displaystyle E_{1},E_{2},E_{3},\dotsc }$ in ${\displaystyle M}$ are disjoint, then for all ${\displaystyle v\in H}$,

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

• ${\displaystyle \pi (E_{1}\cap E_{2})=\pi (E_{1})\pi (E_{2})}$ for all ${\displaystyle E_{1},E_{2}\in M.}$

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

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

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

${\displaystyle \mu _{\xi ,\eta }(E):=\langle \pi (E)\xi \mid \eta \rangle }$

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

${\displaystyle \mu _{\xi }(E):=\langle \pi (E)\xi \mid \xi \rangle }$

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

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

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

be defined as

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

i.e., as multiplication by the indicator function ${\displaystyle 1_{E}}$ on L²(X). Then ${\displaystyle \pi (E)=1_{E}}$ defines a projection-valued measure.^[6] For example, if ${\displaystyle X=\mathbb {R} }$, ${\displaystyle E=(0,1)}$, and ${\displaystyle \phi ,\psi \in L^{2}(\mathbb {R} )}$ there is then the associated complex measure ${\displaystyle \mu _{\phi ,\psi }}$ which takes a measurable function ${\displaystyle f:\mathbb {R} \to \mathbb {R} }$ and gives the integral

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

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

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

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 ${\displaystyle f}$ but then ${\displaystyle T}$ will be an unbounded linear operator on the Hilbert space ${\displaystyle H}$.

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

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

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

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

where the integral extends to an unbounded function ${\displaystyle \lambda }$ when the spectrum of ${\displaystyle A}$ 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

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

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

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

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

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

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:

${\displaystyle \pi =\bigoplus _{1\leq n\leq \omega }(\pi \mid H_{n})}$

where

${\displaystyle H_{n}=\int _{X_{n}}^{\oplus }H_{x}\ d(\mu \mid X_{n})(x)}$

and

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

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

• the projective space ${\displaystyle \mathbf {P} (H)}$ of the Hilbert space ${\displaystyle H}$ is interpreted as the set of possible (normalizable) states ${\displaystyle \varphi }$ of a quantum system,^[11]
• the measurable space ${\displaystyle X}$ is the value space for some quantum property of the system (an "observable"),
• the projection-valued measure ${\displaystyle \pi }$ expresses the probability that the observable takes on various values.

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

• ${\displaystyle \mathbb {R} ^{3}}$ (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 ${\displaystyle \varphi }$.

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

${\displaystyle P_{\pi }(\varphi )(E)=\langle \varphi \mid \pi (E)(\varphi )\rangle =\langle \varphi |\pi (E)|\varphi \rangle .}$

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

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

${\displaystyle P_{\pi }(\varphi ):E\mapsto \langle \varphi \mid \pi (E)\varphi \rangle }$

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

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

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

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

which reduces to

${\displaystyle A(\varphi )=\sum _{i}\lambda _{i}\pi ({\lambda _{i}})(\varphi )}$

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

The above operator ${\displaystyle A}$ 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.

• Spectral theorem
• Spectral theory of compact operators
• Spectral theory of normal C*-algebras

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