Some No-Go Results in Quantum Domain Theory

We show that the type of approximation used in the Solovay-Kitaev theorem cannot be carried out in quantum domain theory. Specifically, there is no countable set D of completely positive maps such that all completely positive maps can be expressed as least upper bounds of directed subsets of D, even for the case of 2ˆ2 matrices. Via a well-known isomorphism, this negative result can be carried over to the forward light cone of 4D Minkowski spacetime, considered as a domain.
We also establish that the effects (equivalently, 2-valued POVMs) on a C*-algebra form a continuous dcpo iff it is a (possibly infinite) product of finite-dimensional matrix algebras, so there are no nontrivial infinitedimensional C*-algebras that have a continuous dcpo for an effect algebra.