Infima of Hilbert space effects Original Research Article

The quantum effects for a physical system are usually described by the set script capital e(H) of positive operators on a Hilbert spaceH that are bounded above by the unit operator. Under a natural order, script capital e(H) becomes a partially ordered set that is not a lattice unless dimH less-than-or-equals, slant 1. A characterization of the pairsA, B var epsilon script capital e(H) such that the infimumA logical and B exists is an open problem called the infimum problem. Methods of linear algebra are employed to solve the infimum problem when dimH < ∞. 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.