! Causality Violation and Nonlinear Quantum Mechanics Jacques L. Pienaar B.Sc. (Hons I) The University of Melbourne A thesis submitted for the degree of Doctor of Philosophy at The University of Queensland in 2013 arXiv:1401.0167v1 [quant-ph] 31 Dec 2013 School of Mathematics and Physics Abstract It is currently unknown whether the laws of physics permit time travel into the past. While general relativity indicates the theoretical possibility of causality violation, it is now widely accepted that a theory of quantum gravity must play an essential role in such cases. As a striking example, the logical paradoxes usually associated with causality violation can be resolved by quantum effects. We ask whether the explicit construction of a theory that allows causality violation might in turn teach us something about quantum gravity. Taking the toy model of Deutsch1 as a starting point, in Part I we argue that, despite being a nonlinear modification of quantum mechanics, the model does not imply superluminal signalling and its predictions can be operationally verified by experimenters within an appropriate ontological setting. In Part II we extend the model to relativistic quantum fields. The detailed outline is described below. Thesis outline The thesis is divided into two parts, according to the physical setting. The first part is based on the formalism of standard quantum mechanics, which might be referred to as \nonrelativistic" because all observers and (massive) systems of interest are assumed to be approximately at rest in the laboratory frame. However, the formalism does contain at least one `relativistic' element: light is assumed to travel at a finite speed between systems that are separated by large distances in space. Within this regime, therefore, it is legitimate to consider the problem of superluminal signalling between distant systems, leading us to refer to this regime simply as \low energy". The second part of the thesis is based on the formalism of relativistic quantum field theory, particularly quantum optics. This formalism extends to situations where the systems of interest are in relativistic motion with respect to each other and the laboratory, in which case it becomes desirable to use a formalism whose physical predictions are invariant under Lorentz transformations. Accordingly, we refer to this as the \high energy" regime. In each part, we first review the standard formalism and then introduce modifications of the formalism in order to accommodate possible violations of causality, in a manner consistent with Deutsch's model. The first part is primarily concerned with assessing the consistency of Deutsch's original model, as indicated by the extent to which it respects operational notions of locality and verifiability. The second part is concerned with generalising Deutsch's model to include the spatial and temporal properties of quantum fields interacting with the causality-violating region. 1Deutsch, D. Quantum mechanics near closed timelike lines. Phys. Rev. D 44, 31973217 (1991). i Part I: The low energy regime Chapter 1 gives an overview of standard quantum mechanics in the nonrelativistic regime, including the quantum circuit formalism and the operational formalism. In Chapter 2, we review Deutsch's toy model of causality violation and the surrounding literature on the topic. In order to better understand the physical interpretation of the model, we place it in the context of the broader literature on nonlinear modifications to quantum mechanics. In Chapter 3 we identify Deutsch's model as an example of a nonlinear box. We show that nonlinear boxes are amenable to an operational description, which allows us to clarify whether they allow superluminal signalling and whether their predictions can be verified in principle. We conclude that Deutsch's model is non-signalling and verifiable, the latter condition being contingent on an appropriate ontological model, an example of which is briefly discussed. Part II: The high energy regime In Chapter 4, we briefly review relativistic quantum field theory and quantum optics. Using this formalism, we describe relativistic effects in general quantum circuits. This provides a basis for future work on quantum information and quantum computation in relativistic settings, while also providing us with the formal tools needed for the following chapters. In Chapter 5 we apply these tools to quantum states of light interacting with a closed timelike curve. We find that even in trivial cases where the CTC does not produce any interactions between the past and future parts of the field, the CTC can still be exploited to violate Heisenberg's uncertainty relation for quantum optics, allowing the perfect cloning of coherent states. Finally, in Chapter 6 we propose a generalisation of Deutsch's model that extends its regime of applicability to fields whose temporal uncertainty exceeds the size of the temporal jump induced by the CTC. This allows us to consider scenarios in which the CTC becomes too small to have any experimentally observable effects, in which limit we smoothly recover standard quantum optics. We conjecture that our generalised formalism might also provide an alternative model for describing the effect of gravitational time-dilation on entangled quantum systems. Based on this conjecture, we draw a connection to the theory of \event operators" proposed earlier in the literature2, and discuss the related possibility of an experimental test in Earth's gravitational field. 2Ralph, T. C., Milburn, G. J. & Downes, T. Quantum connectivity of space-time and gravitationally induced decorrelation of entanglement. Phys. Rev. A 79, 022121 (2009). ii Declaration by author This thesis is composed of my original work, and contains no material previously published or written by another person except where due reference has been made in the text. I have clearly stated the contribution by others to jointly-authored works that I have included in my thesis. I have clearly stated the contribution of others to my thesis as a whole, including statistical assistance, survey design, data analysis, significant technical procedures, professional editorial advice, and any other original research work used or reported in my thesis. The content of my thesis is the result of work I have carried out since the commencement of my research higher degree candidature and does not include a substantial part of work that has been submitted to qualify for the award of any other degree or diploma in any university or other tertiary insti- tution. I have clearly stated which parts of my thesis, if any, have been submitted to qualify for another award. I acknowledge that an electronic copy of my thesis must be lodged with the University Library and, subject to the General Award Rules of The University of Queensland, immediately made available for research and study in accordance with the Copyright Act 1968. I acknowledge that copyright of all material contained in my thesis resides with the copyright holder(s) of that material. Where appropriate I have obtained copyright permission from the copyright holder to reproduce material in this thesis. iii Publications during candidature Peer-reviewed publications 1. Space-time qubits, Pienaar, J. L., Myers, C. R., Ralph, T. C. Physical Review A 84, 022315 (2011). (arXiv:1101.4250v2) 2. Quantum fields on closed timelike curves, Pienaar, J. L., Myers, C. R., Ralph, T. C. Physical Review A 84, 062316 (2011). (arXiv:1110.3582v2) 3. Open timelike curves violate Heisenbergs uncertainty principle, Pienaar, J. L., Myers, C. R., Ralph, T. C. Physical Review Letters 110, 060501 (2013). (arXiv:1206.5485v2) Online e-print publications 4. The preparation problem in nonlinear extensions of quantum theory, e-print arXiv:1206.2725v1, (2012). Cavalcanti, E. G., Menicucci, N. C., Pienaar, J. L. Publications included in this thesis No publications included. Contributions by others to this thesis The results described in Chapter 3 are based on joint work undertaken with Dr. Nicolas Menicucci and Dr. Eric Cavalcanti of the University of Sydney (see Ref. [4] above). All three authors contributed equally to this work. For the remaining work in this thesis, credit is to be shared equally between the author and his advisory team, Dr. Casey Myers and Prof. Timothy Ralph. Statement of parts of the thesis submitted to qualify for the award of another degree None. iv Acknowledgements Let me begin by acknowledging the support of the Australian Government for awarding me an Australian Postgraduate Award (APA), without which this work would not have been possible. In addition, I owe much to the advice and support of my parents; thanks Mum and Dad. Thanks to Professors Lloyd Hollenberg, Andrew Greentree and Jeffrey McCallum at the University of Melbourne for giving me excellent advice about pursuing a PhD. Thanks to uncle Bart and aunt Pat for looking after me until I found my feet in Brisbane. Special thanks to Aggie Branczyk and Charles Meaney for stimulating discussions, Rob Pfeifer for providing the expertise in our physics reading group and Matt Broome, Leif Humbert and the rest of the motley crew from Teach at the Beach. Thanks to Guifre Vidal for his hospitality and humour, thanks to Andrew Doherty for running the QFT reading group, thanks to Gerard Milburn for being an excellent host and for hanging out with me one time at the James Squire brewhouse. I owe a debt of gratitude to Professor Ping Koy Lam for being kind enough to host me at the Australian National University during my stay in Canberra; I cannot express how valuable this experience was to the completion of my PhD. Thanks also to the quantum theory group at ANU for making me feel welcome, particularly Craig Savage for enlightening discussions, Helen Chrzanowski for lending me her laptop and to Michael Hush for introducing me to the subtleties of the many-worlds interpretation. Thanks to Nick Menicucci and Eric Cavalcanti for hosting me at the University of Sydney and for being good mentors and friends, not to mention excellent collaborators.