# Lucien Hardy

### Teaching Affiliations

### Research Interests

I am working on operational approaches to Quantum Theory, Quantum Field Theory, General Relativity, and Quantum Gravity. Specifically I have developed an operational framework in which Quantum Theories and General Relativity can be formulated. Ultimately, I hope to formulate Quantum Gravity in this framework. In 2001 I developed an operational general probabilistic approach that provided the basis for a set of "reasonable axioms" from which the usual rules of Quantum Theory can be derived. In 2005 I set up the causaloid framework. This is an operational general probabilistic framework for physical theories having indefinite causal structure (as we would expect in a theory of Quantum Gravity). In 2010 I further developed this framework as a diagrammatic calculus, the duotensor formalism, for general circuits. In 2011 I used this framework to provide a reformulation of Quantum Theory - the operator tensor formulation and, also, provided a new set of reasonable axioms from which Quantum Theory can be reconstructed. The operator tensor reformulation motivated taking a look at the issue of composition in physics. Typically, when we study a physical object, we regard it as being built out of small objects joined together in a particular way. In 2013 I wrote a paper providing a more general theory for the use of composition in physics. Such ideas of composition may play a role across different fields in physics. In 2016 I showed how to use ideas of composition to provide an operational reformulation of General Relativity. This requires, first of all, making an assertion as to what the directly observable quantities are. For this I nominate a set of scalar fields and consider point coincidences in their values. This provides an operational space (or op-space). We can consider regions of op-space and how to glue together solutions corresponding to such regions. This leads to a diagrammatic calculus of the same nature as that used in the operator tensor formulation of Quantum Theory. I am also developing an operator tensor formulation of Quantum Field Theory. In this approach operators are associated with regions of space time. These operators must satisfy physicality conditions which guarantee that probabilities are between 0 and 1 and that signals cannot be communicated faster than light. I have made some progress in developing these physicality conditions but much remains to be done. These operational reformulations of General Relativity and Quantum Field Theory suggest approaches to solving the problem of Quantum Gravity. In 2018 I suggested routes to solving the problem of Quantum Gravity (finding a theory that reduces to General Relativity and to Quantum Field Theory in appropriate limits) that is analogous to Einstein's route to solving the problem of Relativistic Gravity (finding a theory that reduces to Newtonian gravity and to special relativistic field theories in appropriate limits). A basic idea here is the Quantum Equivalence Principle - that it is always possible to find a quantum frame of reference in which we have definite causal structure. I am hopeful that the Quantum Equivalence Principle can play a role in constructing a theory of Quantum Gravity that is similar to the role played by the equivalence principle in constructing General Relativity. Another interest I have concerns using humans to choose the settings in a Bell experiment. This would lead to a Turing-type test for certain models of mind. In 1989 I wrote preprints on this idea. I returned to this in 2015 and, in 2017, I wrote a further paper proposing that we have one hundred humans at each end of a Bell experiment that is run over a distance of 100km. With other assumptions, this may be sufficient to ensure that a significant fraction of events have the setting decided by humans at each end. In joint work in 2019, Adam Lewis and I proposed a way to implement quantum computing with machine-learning-controlled quantum stuff. The idea is to take some stuff that is, in principle, capable of quantum computing, attach wires that provide classical inputs and read off classical outputs and implement a procedure that learns how to implement arbitrary quantum circuits. The standard operational formulations are time symmetric (operations are taken to be trace non-increasing in the forward time direction). In work in 2021, I have shown how to formulate operational theories (including Quantum Theory) in a time symmetric way. This includes providing a time symmetric version of the Stinespring dilation theorem. In ongoing work, I am interested in causality constraints in operational Quantum Field Theory. If we have quantum fields over some region of spacetime then causality dictates that, if we jiggle the field in one part of the boundary of this region then no influence should spread to another part of the boundary faster than the speed of light.

### Positions Held

- Royal Society University Research Fellow, University of Oxford, 1997-2002
- Postdoctoral Researcher, La Sapienza University, 1996-1997
- Lecturer, Mathematical Sciences Department, University of Durham, 1994-1996
- Royal Society postdoctoral fellow, University of Innsbruck, 1993-1994
- Lecturer in Mathematical Physics, Maynooth College, The National University of Ireland, 1992-1993

### Awards

- The Quantum Information Structure of Spacetime (QISS) Grant, John Templeton Foundation, 2022-2025
- Discovery Grant, Natural Sciences and Engineering Research Council of Canada (NSERC), 2020
- The Quantum Information Structure of Spacetime (QISS) Grant, John Templeton Foundation, 2019-2022

### Recent Publications

- Sakharwade, N., & Hardy, L. (2024). A diagrammatic language for the Causaloid framework. arxiv:2407.01522v1
- Mrini, L., & Hardy, L. (2024). Indefinite Causal Structure and Causal Inequalities with Time-Symmetry. arxiv:2406.18489v1
- Etezad-Razavi, S., & Hardy, L. (2023). Paradox with Phase-Coupled Interferometers. arxiv:2305.14241v2
- Hardy, L. (2022). Proposal to use humans to switch settings in a Bell experiment in Consciousness and Quantum Mechanics. S. Gao (Ed.), Oxford University Press.
- Hardy, L. (2021). Time Symmetry in Operational Theories. arxiv:2104.00071v1
- Hardy, L., & Lewis, A. G. M. (2021). Quantum computation with machine-learning-controlled quantum stuff. Machine Learning: Science and Technology, 2(1), 015008. doi:10.1088/2632-2153/abb215

### Seminars

- Quantum Foundations Lecture, Quantum Foundations , 2024/02/05, PIRSA:24020027
- Quantum Foundations Lecture, Quantum Foundations , 2024/02/01, PIRSA:24020026
- Quantum Foundations Lecture, Quantum Foundations , 2024/01/31, PIRSA:24010069
- Quantum Foundations Lecture, Quantum Foundations , 2024/01/29, PIRSA:24010068
- Quantum Foundations Lecture, Quantum Foundations , 2024/01/26, PIRSA:24010067
- Quantum Foundations Lecture, Quantum Foundations , 2024/01/24, PIRSA:24010066
- Quantum Foundations Lecture, Quantum Foundations , 2024/01/22, PIRSA:24010065
- Quantum Foundations Lecture, Quantum Foundations , 2024/01/19, PIRSA:24010064
- Quantum Foundations Lecture, Quantum Foundations , 2024/01/17, PIRSA:24010063
- Quantum Foundations Lecture, Quantum Foundations , 2024/01/15, PIRSA:24010062
- Quantum Foundations Lecture, Quantum Foundations , 2024/01/12, PIRSA:24010061
- Quantum Foundations Lecture, Quantum Foundations , 2024/01/10, PIRSA:24010060
- Quantum Foundations Lecture, Quantum Foundations , 2024/01/08, PIRSA:24010059
- Causality and positivity in causally complex operational probabilistic theories, Quantum Foundations, 2023/11/16, PIRSA:23110066
- Quantum Foundations Lecture - 230206, Quantum Foundations (2022/2023), 2023/02/06, PIRSA:23020017
- Quantum Foundations Lecture - 230203, Quantum Foundations (2022/2023), 2023/02/03, PIRSA:23020016
- Quantum Foundations Lecture - 230102, Quantum Foundations (2022/2023), 2023/02/01, PIRSA:23020015
- Quantum Foundations Lecture - 230130, Quantum Foundations (2022/2023), 2023/01/30, PIRSA:23010055
- Quantum Foundations Lecture - 230127, Quantum Foundations (2022/2023), 2023/01/27, PIRSA:23010054
- Quantum Foundations Lecture - 230125, Quantum Foundations (2022/2023), 2023/01/25, PIRSA:23010053
- Quantum Foundations Lecture - 230123, Quantum Foundations (2022/2023), 2023/01/23, PIRSA:23010052
- Quantum Foundations Lecture - 230120, Quantum Foundations (2022/2023), 2023/01/20, PIRSA:23010051
- Quantum Foundations Lecture - 230119, Quantum Foundations (2022/2023), 2023/01/19, PIRSA:23010048
- Quantum Foundations Lecture - 230118, Quantum Foundations (2022/2023), 2023/01/18, PIRSA:23010050
- Quantum Foundations Lecture - 230116, Quantum Foundations (2022/2023), 2023/01/16, PIRSA:23010049
- Quantum Foundations Lecture - 230111, Quantum Foundations (2022/2023), 2023/01/11, PIRSA:23010047
- Quantum Foundations Lecture - 230109, Quantum Foundations (2022/2023), 2023/01/09, PIRSA:23010046
- Is there an arrow of time in Quantum Theory, Quantum Information Structure of Spacetime conference, University of Western Ontario, London, Ontario, 2022/01/01
- Causality constraints in Quantum Field Theory from an operational perspective, Mathematical and conceptual aspects of Quantum Theory, Banff International Research Station and taking place in Oaxaca, Mexico, 2022/01/01