On the infimum for quantum effects

Abstract

The quantum effects for a physical system can be described by the set E (H) of positive operators on a complex Hilbert space H that are bounded above by the identity operator. While a general effect may be unsharp, the collection of sharp effects is described by the set of orthogonal projections P (H) ⊆E (H). Under the natural order, E (H) becomes a partially ordered set that is not a lattice if dim H≥2. A physically significant and useful characterization of the pairs A,B∈E (H) such that the infimum A∧B exists is called the infimum problem. We show that A∧P exists for all A∈E (H), P∈P (H) and give an explicit expression for A∧P. We also give a characterization of when A∧ (I-A) exists in terms of the location of the spectrum of A. We present a counterexample which shows that a recent conjecture concerning the infimum problem is false. Finally, we compare our results with the work of Ando on the infimum problem. © 2005 American Institute of Physics.