On the infimum of quantum effects

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 $dimH⩾2$⁠. 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.