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. 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
denote a separable complex Hilbert space and
a measurable space consisting of a set
and a Borel σ-algebra
on
. A projection-valued measure
is a map from
to the set of bounded self-adjoint operators on
satisfying the following properties:
is an orthogonal projection for all ![{\displaystyle E\in M.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
and
, where
is the empty set and
the identity operator.- If
in
are disjoint, then for all
,
![{\displaystyle \pi \left(\bigcup _{j=1}^{\infty }E_{j}\right)v=\sum _{j=1}^{\infty }\pi (E_{j})v.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
for all ![{\displaystyle E_{1},E_{2}\in M.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
The second and fourth property show that if
and
are disjoint, i.e.,
, the images
and
are orthogonal to each other.
Let
and its orthogonal complement
denote the image and kernel, respectively, of
. If
is a closed subspace of
then
can be wrtitten as the orthogonal decomposition
and
is the unique identity operator on
satisfying all four properties.
For every
and
the projection-valued measure forms a complex-valued measure on
defined as
![{\displaystyle \mu _{\xi ,\eta }(E):=\langle \pi (E)\xi \mid \eta \rangle }](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
with total variation at most
. It reduces to a real-valued measure when
![{\displaystyle \mu _{\xi }(E):=\langle \pi (E)\xi \mid \xi \rangle }](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
and a probability measure when
is a unit vector.
Example Let
be a σ-finite measure space and, for all
, let
![{\displaystyle \pi (E):L^{2}(X)\to L^{2}(X)}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
be defined as
![{\displaystyle \psi \mapsto \pi (E)\psi =1_{E}\psi ,}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
i.e., as multiplication by the indicator function
on L2(X). Then
defines a projection-valued measure. For example, if
,
, and
there is then the associated complex measure
which takes a measurable function
and gives the integral
![{\displaystyle \int _{E}f\,d\mu _{\phi ,\psi }=\int _{0}^{1}f(x)\psi (x){\overline {\phi }}(x)\,dx}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Extensions of projection-valued measures
If π is a projection-valued measure on a measurable space (X, M), then the map
![{\displaystyle \chi _{E}\mapsto \pi (E)}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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
on
, there exists a unique bounded operator
such that[7]
![{\displaystyle \langle T\xi \mid \xi \rangle =\int _{X}f(\lambda )\,d\mu _{\xi }(\lambda ),\quad \forall \xi \in H.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
where
is a finite Borel measure given by
![{\displaystyle \mu _{\xi }(E):=\langle \pi (E)\xi \mid \xi \rangle ,\quad \forall E\in M.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Hence,
is a finite measure space.
The theorem is also correct for unbounded measurable functions
but then
will be an unbounded linear operator on the Hilbert space
.
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
is a measurable function, then a unique measure exists such that
![{\displaystyle g(T):=\int _{\mathbb {R} }g(x)\,d\pi (x).}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Spectral theorem
Let
be a separable complex Hilbert space,
be a bounded self-adjoint operator and
the spectrum of
. Then the spectral theorem says that there exists a unique projection-valued measure
, defined on a Borel subset
, such that
![{\displaystyle A=\int _{\sigma (A)}\lambda \,d\pi ^{A}(\lambda ),}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
where the integral extends to an unbounded function
when the spectrum of
is unbounded.
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 π(E) be the operator of multiplication by 1E on the Hilbert space
![{\displaystyle \int _{X}^{\oplus }H_{x}\ d\mu (x).}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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 }](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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 {Hx}x ∈ X , such that π is unitarily equivalent to multiplication by 1E on the Hilbert space
![{\displaystyle \int _{X}^{\oplus }H_{x}\ d\mu (x).}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
The measure class[clarification needed] 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 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})}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
where
![{\displaystyle H_{n}=\int _{X_{n}}^{\oplus }H_{x}\ d(\mu \mid X_{n})(x)}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
and
![{\displaystyle X_{n}=\{x\in X:\dim H_{x}=n\}.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Application in quantum mechanics
In quantum mechanics, given a projection-valued measure of a measurable space
to the space of continuous endomorphisms upon a Hilbert space
,
- the projective space
of the Hilbert space
is interpreted as the set of possible (normalizable) states
of a quantum system, - the measurable space
is the value space for some quantum property of the system (an "observable"), - the projection-valued measure
expresses the probability that the observable takes on various values.
A common choice for
is the real line, but it may also be
(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
.
Let
be a measurable subset of
and
a normalized vector quantum state in
, so that its Hilbert norm is unitary,
. The probability that the observable takes its value in
, given the system in state
, is
![{\displaystyle P_{\pi }(\varphi )(E)=\langle \varphi \mid \pi (E)(\varphi )\rangle =\langle \varphi |\pi (E)|\varphi \rangle .}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
We can parse this in two ways. First, for each fixed
, the projection
is a self-adjoint operator on
whose 1-eigenspace are the states
for which the value of the observable always lies in
, and whose 0-eigenspace are the states
for which the value of the observable never lies in
.
Second, for each fixed normalized vector state
, the association
![{\displaystyle P_{\pi }(\varphi ):E\mapsto \langle \varphi \mid \pi (E)\varphi \rangle }](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
is a probability measure on
making the values of the observable into a random variable.
A measurement that can be performed by a projection-valued measure
is called a projective measurement.
If
is the real number line, there exists, associated to
, a self-adjoint operator
defined on
by
![{\displaystyle A(\varphi )=\int _{\mathbb {R} }\lambda \,d\pi (\lambda )(\varphi ),}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
which reduces to
![{\displaystyle A(\varphi )=\sum _{i}\lambda _{i}\pi ({\lambda _{i}})(\varphi )}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
if the support of
is a discrete subset of
.
The above operator
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[clarification needed]. This generalization is motivated by applications to quantum information theory.
See also
Notes
- ^ Kowalski, Emmanuel (2009), Spectral theory in Hilbert spaces (PDF), ETH Zürich lecture notes, p. 50
References
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