Computability for Operators on Hilbert Space

In Turing's first article on computability, he also defined a notion of computability for real numbers. This was done in terms of decimal digits. A year later, he published a correction, because addition of real numbers was not computable under this definition. This was the first indication of the importance of using the right topology when computing with real numbers. In this talk, I discuss the situation for the space B(H) of operators on a (separable) Hilbert space H. The setting is Type II computability, where there is a strong connection between continuity and computability. The continuity of the operations of addition, multiplication, adjoint and taking the expectation with respect to density matrices singles out a unique topology suitable for computation. We relate this to previous work by Schröder and Brattka.