Infima of Hilbert space effects

The quantum effects for a physical system are usually described by the set E(H) of positive operators on a Hilbert space H that are bounded above by the unit operator. Under a natural order, E(H) becomes a partially ordered set that is not a lattice unless dim H less than or equal to 1. A characterization of the pairs A,B is an element of E(H) such that the infimum A boolean AND B exists is an open problem called the infimum problem. Methods of linear algebra are employed to solve the infimum problem when dim H < infinity. These methods do not appear to generalize to infinite dimensions so the general problem remains open. Progress toward solution of the general problem is discussed. (C) 1999 Elsevier Science Inc. All rights reserved.