On Block Structures in Quantum Computation

A block is a language construct in programming that temporarily enlarges the state space. It is typically opened by initialising some local variables, and closed via a return statement. The “scope” of these local variables is then restricted to the block in which they occur. In quantum computation such temporary extensions of the state space also play an important role. This paper axiomatises “logical” blocks in a categorical manner. Opening a block may happen via a measurement, so that the block captures the various possibilities that result from the measurement. Following work of Coecke and Pavlović we show that von Neumann projective measurements can be described as an Eilenberg-Moore coalgebra of a comonad associated with a particular kind of block structure. Closing of a block involves a collapse of options. Such blocks are investigated in non-deterministic, probabilistic, and quantum computation. In the latter setting it is shown that there are two block structures in the category of C∗-algebras, via copowers and via matrices.