Projection-valued measure

In mathematics, particularly in functional analysis, a projection-valued measure (PVM) is a function defined on certain subsets of a fixed set and whose values are self-adjoint projections on a Hilbert space. Projection-valued measures are used to express results in spectral theory, such as the spectral theorem for self-adjoint operators. In quantum mechanics, PVMs are the mathematical description of projective measurements. 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.

Formal definition

A projection-valued measure on a measurable space (X, M), where M is a σ-algebra of subsets of X, is a mapping π from M to the set of self-adjoint projections on a Hilbert space H such that

${\displaystyle \pi (X)=\operatorname {id} _{H}\quad }$

and for every ξ, η ∈ H, the set-function

${\displaystyle A\mapsto \langle \pi (A)\xi \mid \eta \rangle }$

is a complex measure on M (that is, a complex-valued countably additive function). We denote this measure by ${\displaystyle \operatorname {S} _{\pi }(\xi ,\eta )}$.

If π is a projection-valued measure and

${\displaystyle A\cap B=\emptyset ,}$

then π(A), π(B) are orthogonal projections. From this follows that in general,

${\displaystyle \pi (A)\pi (B)=\pi (A\cap B).}$

Example. Suppose (X, M, μ) is a measure space. Let π(A) be the operator of multiplication by the indicator function 1A on L2(X). Then π is a projection-valued measure.

Extensions of projection-valued measures

If π is an additive projection-valued measure on (X, M), then the map

${\displaystyle \mathbf {1} _{A}\mapsto \pi (A)}$

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. In fact this map extends in a canonical way to all bounded complex-valued Borel functions on X.

Theorem. For any bounded M-measurable function f on X, there is a unique bounded linear operator Tπ(f) such that

${\displaystyle \langle \operatorname {T} _{\pi }(f)\xi \mid \eta \rangle =\int _{X}f(x)d\operatorname {S} _{\pi }(\xi ,\eta )(x)}$

for all ξ, η ∈ H. The map

${\displaystyle f\mapsto \operatorname {T} _{\pi }(f)}$

Structure of projection-valued measures

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}xX be a μ-measurable family of separable Hilbert spaces. For every AM, let π(A) be the operator of multiplication by 1A 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:HK such that

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

for every AM.

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}xX , such that π is unitarily equivalent to multiplication by 1A on the Hilbert space

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

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 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:

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

where

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

and

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

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. This generalization is motivated by applications to quantum information theory.

References

• G. W. Mackey, The Theory of Unitary Group Representations, The University of Chicago Press, 1976
• M. Reed and B. Simon, Methods of Mathematical Physics, vols I–IV, Academic Press 1972.
• G. Teschl, Mathematical Methods in Quantum Mechanics with Applications to Schrödinger Operators, http://www.mat.univie.ac.at/~gerald/ftp/book-schroe/, American Mathematical Society, 2009.
• V. S. Varadarajan, Geometry of Quantum Theory V2, Springer Verlag, 1970.