Lucien Hardy

Lucien Hardy's picture

Lucien Hardy received his PhD at Durham University in 1992 under the supervision of Professor Euan J Squires. He has held research and lecturing positions in various cities across Europe. While in Rome Lucien collaborated on an experiment to demonstrate quantum teleportation. In 1992 he found a very simple proof of non-locality in quantum theory which has become known as Hardy’s theorem.


University of Waterloo

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Research Interests

I am working on operational approaches to quantum theory and quantum gravity. Specifically I am developing an operational framework in which it is hoped that quantum theory, probabilistic general relativity, and ultimately quantum gravity can be formulate.

I provided a new reformulation of quantum theory in which operators are associated with general fragments of circuits (this unifies the treatment of states, transformations, and measurements as examples of a more general type of object in quantum theory). Additionally, I showed how quantum theory follows from a set of five natural postulates posed in operational language and applied to the circuit framework (this is a framework in which boxes representing operations are joined by wires representing the passage of systems). This builds on work from ten years earlier where I also gave a simple set of postulates for quantum theory within the context of an more basic operational framework.

This 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. Such ideas of composition may play a role across different fields in physics.

I am currently working on reformulating General Relativity as a probabilistic theory with agency. The Probabilities can be thought of as being a consequence of ignorance (the underlying theory remaining deterministic). Agency (the possibility of making different choices) is built into quantum theory (we choose which measurement to make). We can also build it into General Relativity.

The aim of reformulating both QT and GR in similar terms is, of course, to help find a theory of quantum gravity (which has QT and GR as limiting cases).

Positions Held

  • 1997-2002 Royal Society University Research Fellow, University of Oxford
  • 1996-1997 Postdoctoral position at La Sapienza University, Rome, Italy.
  • 1994-1996 Lecturer in Mathematical Sciences Department, University of Durham, UK
  • 1993-1994 Royal Society postdoctoral fellow at the University of Innsbruck, Austria
  • 1992-1993 Lecturer in Mathematical Physics, Maynooth College, The National University of Ireland.

Recent Publications

  • Hardy, L. Bell's theorem with retarded settings. To be published in Quantum [Un]Speakables II: 50 Years of Bell's Theorem, edited by Anton Zeilinger and Reinhold Bertlmann.
  • Hardy, L. Quantum theory with bold operator tensors. Phil. Trans. R. Soc. A, 373(2047), 20140239. (2015)
  • On the theory of composition in physics , in Computation, Logic, Games, and Quantum Foundations. The Many Facets of Samson Abramsky. Lecture Notes in Computer Science Volume 7860, 2013, pp 83-106
  • Hardy, L The operator tensor formulation of quantum theory, Phil. Trans. R. Soc. A 28 vol. 370 no. 1971 pages 3385-3417 (special issue in honour of Alan Turing) (2012) also at arXiv: 1201.4390.
  • Abramsky, S., and Hardy, L. Logical Bell inequalities, Phys. Rev. A 85, 062114 (2012) [11 pages] also at arXiv: 1203.1352
  • Genkina, D., Chiribella, G., and Hardy, L. Optimal Probabilistic Simulation of Quantum Channels from the Future to the Past, Phys. Rev. A 85, 022330 (2012) also in arXiv: 1112.1469
  • Hardy, L., and Wootters, W. K., Limited holism and real vector space quantum theory, Volume 42, Number 3 (2012), 454-473.
  • Towards Quantum Gravity: A Framework for Probabilistic Theories with Non-Fixed Causal Structure}, J. Phys. A 40, 3081 (2007), gr-qc/0608043.
  • Hardy, L; Kent, A; {\sl Cheat sensitive quantum bit commitment}, Phys. Rev. Lett. 92, 157901 (2004).
  • Hardy, L; {\sl Quantum ontological excess baggage}, Stud. Hist. Phil. Sci. B 35 267 (2004)
  • Jonathan Barrett (Universite Libre de Bruxelles), Lucien Hardy (Perimeter Institute), Adrian Kent (Centre for Quantum Computation, DAMTP, University of Cambridge), No Signalling and Quantum Key Distribution, Phys. Rev. Lett. 95, 010503 (2005), arxiv:quant-ph/405101
  • Galvao, E; Hardy, L; Substituting a qubit for an arbitrarily large number of classical bits, Phys. Rev. Lett. 90, 087902 (2003)
  • Walgate, J; Hardy, L; Nonlocality, asymmetry, and distinguishing bipar- tite states, Phys. Rev. Lett. 89 147901 (2002).
  • Walgate, J; Short, AJ; Hardy, L; Vedral,V. Local Distinguishability of Multipartite Orthogonal Quantum States Authors, Phys. Rev. Lett. 85, 4972 (2000).
  • Hardy, L; Song, DD. Entanglement swapping chains for general pure states, Phys. Rev. A 62, 052315 (2000).
  • Kwiat, PG; Hardy, L. The mystery of the quantum cakes, Amer. J. Phys. 68, 33 (2000).
  • Boschi, D; Branca, S; De Martini, F; Hardy, L; Popescu, S. Experimental realization of teleporting an unknown pure quantum state via dual classical and Einstein-Podolsky-Rosen channels, Phys. Rev. Lett. 80, 1121 (1998).
  • Blasi, B; Hardy, L. Realism and time symmetry in quantum mechanics, Phys. Lett. A 207, 3 (1995).
  • Hardy, L. Nonlocality of a single-photon revisited, Phys. Rev. Lett. 73, 2279 (1994).
  • Hardy, L. Nonlocality for 2 particles without inequalities for almost-all entangled states, Phys. Rev. Lett. 71, 1665 (1993).
  • Hardy, L. On the existence of empty waves in quantum-theory - reply, Phys. Lett. A 175, 259 (1993).
  • Hardy, L. On the existence of empty waves in quantum-theory, Phys. Lett. A 167, 11 (1992).
  • Hardy, L. Quantum-mechanics, local realistic theories, and lorentz- in- variant realistic theories, Phys. Rev. Lett. 68, 2981 (1992).
  • Hardy, L. Can classical wave theory explain the photon anticorrelation e(R)ect on a beam splitter, Europhys. Lett. 15, 591 (1991).
  • Reformulating and reconstructing quantum theory, arXiv: 1104.2066
  • A formalism-local framework for general probabilistic theories including quantum theory, Lucien Hardy, arXiv: 1005.5164
  • Why Physics Needs Quantum Foundations, Lucien Hardy and Robert Spekkens arXiv: 1003.5008
  • Foliable Operational Structures for General Probabilistic Theories, arXiv: 0912.4740
  • Entropy for theories with indefinite causal structure, Sonia Markes and Lucien Hardy, arXiv: 0910.1323
  • Probability Theories with Dynamic Causal Structure: A New Framework for Quantum Gravity, Lucien Hardy, arxiv:gr-qc/509120
  • Quantum Theory from five reasonable axioms, Lucien Hardy, arXiv: quant-ph/0101012
  • Disentangling nonlocality and teleportation, Lucien Hardy, arXiv: quant-ph/9906123
  • Quantum gravity computers: On the theory of computation with indefinite causal structure}, quant-ph/0701019, for proceedings of "Quantum Reality, Relativistic Causality, and Closing the Epistemic Circle: An International Conference in Honour of Abner Shimony" (2007).
  • Hardy, L; Quantum gravity computers: On the theory of computation with inde?nite causal structure, quant-ph/0701019, for proceedings of "Quantum Reality, Relativistic Causality, and Closing the Epistemic Circle: An International Conference in Honour of Abner Shimony" (2007).
  • Reconstructing quantum theory to appear in "Quantum Theory: Informational Foundations and Foils" edited by Guilio Chiribella and Robert Spekkens (2016)
  • Formalism Locality in Quantum Theory and Quantum Gravity, (also at arXiv: 0804.0054), To appear in "Philosophy of Quantum Information and Entanglement" Eds A. Bokulich and G. Jaeger (CUP).
  • Why is nature described by quantum theory? in Science and Ultimate Reality, eds. John Barrow, Paul Davies, and Charles Harper, (Cambridge, 2004).


  • Probabilistic General Relativity with Agency in an Operational Framework, at Quantum Probability and Logic 2015, Oxford (invited speaker).
  • Steps on route to Probabilistic General Relativity with Agency (PAGeR), at Materials Department, University of Oxford.
  • Duotensors, operator tensors, and quantum theory, at the workshop "New geometric concepts in the foundations of quantum physics" held at Chicheley Hall, England.
  • Operationalism, probabilities, agency, and composition on the road to quantum gravity, at the QMAC(Quantum Mathematics and Computation) symposium for the opening of the Clay Building in Oxford.
  • On the theory of composition in physics, Computer science department
  • The operator tensor formulation of quantum theory, Materials department, Oxford
  • On the theory of composition in physics, Physics department, Imperial College
  • On the theory of composition in physics, Askloster at Oxford (workshop)
  • operational approach to quantum theory, Quantum Foundations Workshop at Bellairs, Barbados
  • The operator tensor formulation of quantum theory, Q+ hangout.
  • Reformulating quantum theory, Department of Computer Science, Oxford
  • Reformulating quantum theory, Materials Department, Oxford
  • Quantum Theory from reasonable postulates, at New frontiers in quantum foundations, CUPI 2011, Clemson, South Carolina.
  • Why we should consider situating the de Broglie-Bohm model in the context of a much more general framework for hidden variable theories, 21st-Century directions in de Broglie-Bohm theory and beyond, The Towler Institute, Vallico Sotto, Tuscany, Italy.
  • Operational Computing with Quantum Stuff, Imperial College
  • Operational Computing with Quantum Stuff, Developments in Computational Models 2010, Causality, Computation and Physics, FLoC workshop, Edinburgh.
  • A formalism-local framework for general probabilistic theories including quantum theory, seminar at ComLab, University of Oxford
  • Operational computing with quantum stuff, at Information Theory, Quantum Mechanics, and Security, Bellairs, Barbados
  • Computing with quantum stuff, at Conference on quantum information and quantum control III, Toronto
  • Natural postulates for quantum theory, at On the nature of the laws of physics, Beyond Center, Arizona
  • Natural postulates for quantum theory, at Workshop on informatic phenomena, New Orleans
  • An operational approach to Quantum Gravity}, University of Oxford, Computer Science Dept.
  • The causaloid approach to quantum theory and quantum gravity, at Logic, Physics and Quantum Information Theory, Bellairs, Barbados.
  • The Causaloid Framework: A tentative framework for Quantum Gravity, at The UK Foundations Conference, Leeds March 2007.
  • Quantum foundations and quantum gravity}, at PIAF workshop in quantum foundations, Sydney.
  • An Operational Framework for a Theory of Quantum Gravity, at Deep Beauty, Princeton
  • A foil for quantum gravity, at Foils workshop, Cambridge
  • The Causaloid Framework: A tentative framework for Quantum Gravity, at Loops 07, Morellia Mexico.
  • Time symmetry and quantum gravity, at Weak values and weak measurements, Arizona.
  • The Causaloid Framework: A tentative framework for Quantum Gravity at Philosophical and formal foundations of quantum physics, Les Treilles.
  • Quantum theory from reasonable postulates}, University of Birmingham, Physics Dept.
  • Beyond quantum theory: Information and entanglement in general probabilistic frameworks, at Foundations of quantum information and entanglement, Boston.
  • Probability theories with non-fixed causal structure: a new framework for quantum gravity}, University of Cambridge
  • From quantum foundations to quantum gravity, at APS March meeting, Focus session: Foundations of Quantum Theory, Baltimore.
  • Generalized Probability theories and how to go beyond Quantum Theory}, Wayne State University, Detroit.
  • Generalized probability theories with dynamic causal structure: a new approach to quantum gravity, at Being Baysian in a quantum world, Konstanz.
  • Why quantum theory?, at Endophysics, time, quantum and the subjective, Bielefeld.
  • Probabilistic theories with dynamical causal structure, at - On the present status of quantum mechanics, Losinj, Croatia.
  • The what and the why of quantum theory}, Philosophy dept, Dalhousie University, Halifax.
  • Composite systems in quantum theory and beyond, at the 13th UK Meeting on the Foundations of Physics, York UK.
  • Cheat sensitive quantum bit commitment, at Special Week on Quantum Cryptography, Newton Institute, Cambridge.
  • Are the laws of quantum theory a consequence of the human condition?, at Subtle Technologies, Toronto.
  • Quantum theory as an example of a probability theory with special focus on composite systems}, at - New Directions in the Foundations of Physics, Washington.
  • Cheat sensitive bit commitment}, at CIAR Quantum Information Processing Program Meeting, Banff, Alberta.
  • Quantum theory from reasonable axioms}, Philosophy Dept, University of Western Ontario.
  • Quantum theory from reasonable axioms, at Quantum foundations in the light of quantum information, Montreal