ngc341_201 : 2016/2/16(10:47) New Generation Computing, 34(2016)69-85 Ohmsha, Ltd. and Springer Terminality Implies No-signalling ...and Much More Than That Bob COECKE Department of Computer Science, University of Oxford, Wolfson Bd., Parks Rd., Oxford OX1 3QD UK bob.coecke@cs.ox.ac.uk Received 21 January 2014 Revised manuscript received 20 October 2015 Abstract A ‘process theory’ is any theory of systems and processes which admits sequential and parallel composition. ‘Terminality’ unifies nor- malisation for pure states, trace-preservation for positive superoperators, and adding up to identity of positive operators in quantum theory, and it moreover generalises this to arbitrary process theories. We show that terminality and no-signalling coincide in any process theory, provided one makes causal struc- ture explicit. In fact, making causal structure explicit is necessary to even make sense of no-signalling in process theories. We conclude that because of its much simpler mathematical form, terminality should be taken to be a more fundamental notion than no-signalling. We also point out that termi- nality imposes many other nice features upon process theories, for example, it even imposes relativistic covariance. §1 Introduction Causality related notions are prominent in many areas of physics, and the relationships between these are by no means obvious: C1 Relativistic space-time is often abstracted as a partial ordering, called causal structure. 16) A discrete counterpart to such a causal structure is prominent in modern theories of probabilistic inference. 15) C2 In quantum information, for example in the context of generalised proba- bilistic theories, 2) one often relies on the notion of no-signalling, by means of which one intends to implement these relativistic constraints for spa- tially distributed information-processing devices. C3 In quantum foundations, an axiom called causality has recently been put forward in 4) by Chiribella, D’Ariano and Perinotti. For process theories, 7–9) that is, in category-theoretic terms, symmetric monoidal cat-