Composing causal orderings

          @misc{ pirsa_PIRSA:19120020,
            doi = {10.48660/19120020},
            url = {https://pirsa.org/19120020},
            author = {Kissinger, Aleks},
            keywords = {Quantum Foundations, Quantum Gravity, Quantum Information},
            language = {en},
            title = {Composing causal orderings},
            publisher = {Perimeter Institute},
            year = {2019},
            month = {dec},
            note = {PIRSA:19120020 see, \url{https://pirsa.org}}
          }
          

Abstract

When studying (definite or indefinite) causal orderings of processes, it is often useful to consider higher-order processes, i.e. processes which take other processes as their input. However, as a recent no-go result of Guerin et al indicates, our naive first-order notions of "composition" of processes become ill-defined at higher-order. Unlike state spaces, there are multiple non-equivalent notions of "joint system" for process spaces and many different ways one might attempt to plug processes together, with only some giving well-defined (i.e. normalised) processes as outputs. While this starts to look a bit like the Wild West, I'll show in this talk that we can get quite a bit of mileage from considering just two kinds of joint systems: a "non-signalling" tensor product, and a (de Morgan dual) "signalling" product. The interaction between these two products has in fact been well-understood by logicians since the 1980s in a very different disguise: multiplicative linear logic. Using this connection, I'll show how a set of "contractibility" criteria due to Danos and Regnier give a relatively simple, dimension-independent technique for determining whether an arbitrary plugging of higher-order processes is well-defined.