On operator valued measures

On operator valued measures

##### Author

McLaren, Darian

Plosker, Sarah

Ramsey, Christopher

##### Faculty Advisor

##### Date

2020

##### Keywords

##### Abstract (summary)

We consider positive operator valued measures whose image is
the bounded operators acting on an infinite-dimensional Hilbert space, and we
relax, when possible, the usual assumption of positivity of the operator valued
measure seen in the quantum information theory literature. We define the
Radon-Nikod´ym derivative of a positive operator valued measure with respect
to a complex measure induced by a given quantum state; this derivative does
not always exist when the Hilbert space is infinite dimensional in so much
as its range may include unbounded operators. We define integrability of a
positive quantum random variable with respect to a positive operator valued
measure. Emphasis is put on the structure of operator valued measures, and
we develop positive operator valued versions of the Lebesque decomposition
theorem and Johnson’s atomic and nonatomic decomposition theorem. Beyond
these generalizations, we make connections between absolute continuity and
the “cleanness” relation defined on positive operator valued measures as well
as to the notion of atomic and nonatomic measures.

##### Publication Information

McLaren, D., Plosker, S., and Ramsey, C. (2020). On operator valued measures. Houston Journal of Mathematics 46 (2020), 201-
226.

##### DOI

##### Notes

##### Item Type

##### Language

English

##### Rights

All Rights Reserved