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

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

• ${\displaystyle \pi (E_{1}\cap E_{2})=\pi (E_{1})\pi (E_{2})}${\displaystyle \pi (E_{1}\cap E_{2})=\pi (E_{1})\pi (E_{2})} for all ${\displaystyle E_{1},E_{2}\in M.}${\displaystyle 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 ${\displaystyle E_{1}}${\displaystyle E_{1}} and ${\displaystyle E_{2}}${\displaystyle E_{2}} are disjoint, i.e., ${\displaystyle E_{1}\cap E_{2}=\emptyset }${\displaystyle E_{1}\cap E_{2}=\emptyset }, the images ${\displaystyle \pi (E_{1})}${\displaystyle \pi (E_{1})} and ${\displaystyle \pi (E_{2})}${\displaystyle \pi (E_{2})} are orthogonal to each other.

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

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

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

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

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

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

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

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

be defined as

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

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

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

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

${\displaystyle \chi _{E}\mapsto \pi (E)}${\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 ${\displaystyle f}${\displaystyle f} but then ${\displaystyle T}${\displaystyle T} will be an unbounded linear operator on the Hilbert space ${\displaystyle H}${\displaystyle H}.

Let ${\displaystyle H}${\displaystyle H} be a separable complex Hilbert space, ${\displaystyle A:H\to H}${\displaystyle A:H\to H} be a bounded self-adjoint operator and ${\displaystyle \sigma (A)}${\displaystyle \sigma (A)} the spectrum of ${\displaystyle A}${\displaystyle A}. Then the spectral theorem says that there exists a unique projection-valued measure ${\displaystyle \pi ^{A}}${\displaystyle \pi ^{A}}, defined on a Borel subset ${\displaystyle E\subset \sigma (A)}${\displaystyle E\subset \sigma (A)}, such that $${\displaystyle A=\int _{\sigma (A)}\lambda ,円d\pi ^{A}(\lambda ),}$${\displaystyle A=\int _{\sigma (A)}\lambda ,円d\pi ^{A}(\lambda ),} and ${\displaystyle \pi ^{A}(E)}${\displaystyle \pi ^{A}(E)} is called the spectral projection of ${\displaystyle A}${\displaystyle A}.^{[3] [10]} The integral extends to an unbounded function ${\displaystyle \lambda }${\displaystyle \lambda } when the spectrum of ${\displaystyle A}${\displaystyle A} is unbounded.^[11]

The spectral theorem allows us to define the Borel functional calculus for any Borel measurable function ${\displaystyle g:\mathbb {R} \to \mathbb {C} }${\displaystyle g:\mathbb {R} \to \mathbb {C} } by integrating with respect to the projection-valued measure ${\displaystyle \pi ^{A}}${\displaystyle \pi ^{A}}: $${\displaystyle g(A):=\int _{\mathbb {R} }g(\lambda ),円d\pi ^{A}(\lambda ).}$${\displaystyle g(A):=\int _{\mathbb {R} }g(\lambda ),円d\pi ^{A}(\lambda ).} A similar construction holds for normal operators and measurable functions ${\displaystyle g:\mathbb {C} \to \mathbb {C} }${\displaystyle g:\mathbb {C} \to \mathbb {C} }.

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

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

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

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

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

where

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

and

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

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

• the projective space ${\displaystyle \mathbf {P} (H)}${\displaystyle \mathbf {P} (H)} of the Hilbert space ${\displaystyle H}${\displaystyle H} is interpreted as the set of possible (normalizable) states ${\displaystyle \varphi }${\displaystyle \varphi } of a quantum system,^[12]
• the measurable space ${\displaystyle X}${\displaystyle X} is the value space for some quantum property of the system (an "observable"),
• the projection-valued measure ${\displaystyle \pi }${\displaystyle \pi } expresses the probability that the observable takes on various values.

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

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

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

${\displaystyle P_{\pi }(\varphi )(E)=\langle \varphi \mid \pi (E)(\varphi )\rangle =\langle \varphi \mid \pi (E)\mid \varphi \rangle .}${\displaystyle 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 ${\displaystyle E}${\displaystyle E}, the projection ${\displaystyle \pi (E)}${\displaystyle \pi (E)} is a self-adjoint operator on ${\displaystyle H}${\displaystyle H} whose 1-eigenspace are the states ${\displaystyle \varphi }${\displaystyle \varphi } for which the value of the observable always lies in ${\displaystyle E}${\displaystyle E}, and whose 0-eigenspace are the states ${\displaystyle \varphi }${\displaystyle \varphi } for which the value of the observable never lies in ${\displaystyle E}${\displaystyle E}.

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

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

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

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

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

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

which reduces to

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

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

The above operator ${\displaystyle A}${\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", 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.

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

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• Linear algebra
• Measures (measure theory)
• Spectral theory

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