 
Summary: Under consideration for publication in Math. Struct. in Comp. Science
A Categorical Quantum Logic
SAMSON ABRAMSKY
ROSS DUNCAN
Oxford University Computing Laboratory, Wolfson Building, Parks Road, Oxford, OX1 3QD, UK.
Received 15 December 2005
We define a strongly normalising proofnet calculus corresponding to the logic of
strongly compact closed categories with biproducts. The calculus is a full and faithful
representation of the free strongly compact closed category with biproducts on a given
category with an involution. This syntax can be used to represent and reason about
quantum processes.
1. Introduction
Recent work by Abramsky and Coecke (AC04) develops a complete axiomatization of
finite dimensional quantum mechanics in the abstract setting of strongly compact closed
categories with biproducts. This is used to formalize and verify a number of key quan
tum information protocols. In this setting, classical information flow is explicitly repre
sented by the biproduct structure, while the compact closed structure models quantum
behaviour: preparation, unitary evolution and projection, including powerful algebraic
methods for representing and reasoning about entangled states. Mediating between the
two levels is a semiring of scalars, which is an intrinsic part of the structure, and repre
