 
Summary: Almost Sharp Quantum Effects
Alvaro Arias and Stan Gudder
Department of Mathematics
The University of Denver
Denver, Colorado 80208
April 15, 2004
Abstract
Quantum effects are represented by operators on a Hilbert space satis
fying 0 A I, and sharp quantum effects are represented by projection
operators. We say that an effect A is almost sharp if A = PQP for projec
tions P and Q. We give simple characterizations of almost sharp effects.
We also characterize effects that can be written as longer products of pro
jections. For generality we first work in the formalism of von Neumann
algebras. We then specialize to the full operator algebra B (H) and to
finite dimensional Hilbert spaces.
1 Introduction
Let H be a complex Hilbert space that represents the state space of a quantum
system S. The set of effects E (H) for S is the set of operators on H satisfying 0
A I. Effects represent yesno measurements that may be unsharp (imprecise,
fuzzy). It is interesting that many of the important classes of quantum operators
