Topics on Causal Structures in Quantum Theory

This project falls within the EPSRC Physical Science research area. It is aimed at expanding the results and ideas developed in my master's thesis, which studied the categorical structure of a process theoretical framework modelling the interactions between causally respecting and causally violating regions of spacetime.

In the process theoretical Quantum mechanical framework there is an assumed asymmetry, an assumed direction of the arrow of time. In general relativity the structure of the spacetime is however taken to be a dynamical variable, it is therefore an important task to free the quantum mechanical level from an assumed direction of the arrow of time. The project aims to develop a process theoretical framework which takes into account the possibility of local violations of the one-way time direction.

In 1949, Godel proposed a solution to the Einstein's equations of general relativity which admits the existence of closed timelike curves (CTC), chronology-violating regions of the space time. Those solutions are by many considered to be unphysical: in classical physics the existence of the CTCs imposes some constraints which are paradoxical. However they are not in principle ruled out by scientific considerations [2]. In fact, it has been proposed by David Deutsch that at the level of quantum mechanics none of the 'classical' pathologies occur. Deutsch in the 1991 paper "Quantum mechanics near closed timelike lines" [1] uses the methods derived from the quantum theory of computation to construct a theory that would explain what happens if we include to a 'traditional' quantum circuit a chronology-violating region, representing the existence of an hypothetical CTC. Twenty years later Seth Lloyd proposed a different approach in an attempt to fix some problematic aspects of the otherwise elegant solution proposed by Deutsch [2]. Lloyd model is based on a phenomenon (post-selected teleportation) which is particularly amenable to an analysis using the methods and the approach provided by categorical quantum mechanics.

One of the aims of the project will therefore be to translate the two approaches into a categorical and process theoretical setting. This will allow us to construct a formal mathematical framework which would formalise the otherwise often too discursive analysis of the quantum phenomena near CTCs and of the paradoxes described by the two authors. Moreover, we can try to build on the theory of causality in process theories developed by Coecke and Kissinger to see where and how the idea of a causal theory can be retained while taking into account the existence of CTCs. A particularly promising aspects lies in the fact that the models developed by Deutsch and Lloyd
[1,2] are mutually incompatible, however, Lloyd doesn't manage to rule out the physicality of the older model. A compositional and categorical approach of the phenomena should shed light on this aspect and may even allow us to either rule out one of the two models and more importantly construct a possible alternative starting from compositional observations. One of the main strength of the project lies in the inherent novelty of the approach, in the above mentioned papers [1,2] there isn't a formal setting that would allow us to speak about the causal structure of quantum processes, this is in fact provided by the tools developed by the categorical approach to
quantum mechanics.

Bibliography
[1] Quantum mechanics near closed timelike lines - Deutsch, David Phys. Rev. D44 3197-3217
(1991)
[2] The quantum mechanics of time travel through post-selected teleportation - Lloyd et al. Phys.
Rev. D84 025007 (2011)