23 March 2004
2:30 - 4:00 Bob Coecke
Abstract quantum mechanics (Samson Abramsky and Bob Coecke)
We recast the standard axiomatic presentation of quantum mechanics, due to von Neumann, at a more abstract level, of strongly compact closed categories with biproducts. We show how the essential structures found in key quantum information protocols such as quantum teleportation can be captured at this abstract level. Moreover, from the combination of the --apparently purely qualitative-- structures of compact closure and biproducts there emerge `scalars', `inner-products' and even a `Born rule to calculate the quantum probabilities'.
This abstract point of view opens up new possibilities for describing and reasoning about quantum systems. It also shows the degrees of axiomatic freedom: we can show what requirements are placed on the (semi)ring of scalars C(I,I), where C is the category and I is the tensor unit, in order to perform various protocols such as teleportation. Our formalism captures both the information-flow aspect of the protocols (see quant-ph/0402014), and the branching due to quantum indeterminism. This contrasts with the standard accounts, in which the classical information flows are `outside' the usual quantum-mechanical formalism. Hence the abstract formalism can be conceived as `quantum mechanics extended with (both classical and quantum) information flow'.