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On 29 July 2008, Professor Anthony Thomas Sudbery – known as Tony to his friends and colleagues – celebrated his 65th birthday. To mark this occasion and to honour Tony's scientific achievements, a 2-day Symposion was held at the University of York on 29–30 September 2008 under the sponsorship of the Institute of Physics and the London Mathematical Society. The breadth of Tony's research interests was reflected in the twelve invited lectures by A Beige, I Bengtsson, K Brown, N Cerf, E Corrigan, J Ladyman, A J Macfarlane, S Majid, C Manogue, S Popescu, J Ryan and R W Tucker. This Festschrift, also made possible by the generosity of the IOP and the LMS, reproduces the majority of these contributions together with other invited papers.

Tony obtained his PhD from the University of Cambridge in 1970. His thesis, written under the guidance of Alan Macfarlane, is entitled Some aspects of chiral su(3) × su(3) symmetry in hadron dynamics. He arrived in York in 1971 with his wife Rodie, two young daughters, a lively mind and a very contemporary shock of hair. He was at that stage interested in mathematical physics and so was classed as an applied mathematician in the departmental division in place at that time. But luckily Tony did not fit into this category. His curiosity is combined with a good nose for problems and his capacity for knocking off conjectures impressed us all.

Within a short time of his arrival he was writing papers on group theory, complex analysis and combinatorics, while continuing to work on quantum mechanics. His important paper on quaternionic analysis is an example of the imagination and elegance of his ideas. By developing a derivative, he replaced the relatively obscure analytical theory of quaternions by one informed by modern complex analysis. Other interests emerged, centred round the quantum: quantum mechanics and its foundations, quantum groups and quantum information. He didn't just dabble in these areas but mastered them, gaining a national and international reputation; for instance he joined Roger Penrose in a discussion on 'The Physics of Reality' in Melvyn Bragg's radio series 'In our time'. He was much in demand for contributions to the 'News and Views' section in Nature and has written numerous book reviews in scientific and semi-popular journals as well as newspaper commentaries on important scientific developments.

Quantum foundations and quantum information have remained Tony's main professional preoccupations. As a contribution to the conceptual problems surrounding quantum measurement, he undertook a detailed analysis of the observation of decay, introducing the crucial distinction between continuous measurement and continual observation. A red thread through his work in quantum mechanics has been the critical scrutiny of different interpretations of quantum mechanics and the question of their experimental testability. As a result he has become an eloquent proponent of a version of what is commonly known as the 'many-worlds' interpretation in his most recent paper (arXiv:1009.3914), Tony explains why he considers the name 'Everett–Wheeler interpretation' as most appropriate and 'many worlds' unnecessary.

Tony started directing his research effort to quantum information theory in the late 1990s. He quickly established himself in the quantum information community, putting York on the map in this field. He was among the first people to investigate entanglement properties of multipartite states, particularly the 3-qubit states. While studying the pure 4-qubit states he mysteriously came up with what could rightly be called the 'Sudbery state':

(where ω is a primitive root of 1), which is the most entangled 4-qubit state in natural measures of entanglement. He continues to work in quantum information theory, particularly on quantum entanglement.

Not least there is also Tony the philosopher who wrote articles with such intriguing titles as 'The necessity of not doing otherwise' or 'Why am I me? and why is my world so classical?'.

Tony was a teacher of the old school. His lectures were inspiring and fun. He believed that the personal element is central to understanding and inspiration, shown by his whole-hearted commitment to small group teaching. Tony's experience of teaching quantum physics to mathematicians is reflected in his widely known text on quantum mechanics for mathematicians entitled Quantum Mechanics and the Particles of Nature (Cambridge 1986).

In 1994, Tony succeeded John Fountain as Head of Department. The 1990s had been a difficult decade for universities in Britain but John had managed to keep the Department on a sound footing, providing foundations which enabled Tony to develop a growth strategy and realise new opportunities when they arose. Despite these challenges Tony was not to be distracted from his scientific work but started to make his significant contributions to quantum information science.

Tony's interests extend beyond mathematics, physics and philosophy: to science fiction, music (both classical and jazz) and theatre – he exercised his thespian talents in his lectures and as a member of the York Shakespeare Project. And his lively mind is matched by a lively body: Tony still plays a mean game of squash and of tennis, and his enthusiastic and acrobatic dancing is most remarkable – it would be no surprise if he were to appear on the popular British TV show 'Strictly Come Dancing'.

A man of many parts...

On behalf of all contributors to the Festschrift it remains for us to wish Tony many productive and happy years to come in this new phase of his life that he himself characterises with the word 'freedom' (and that surely doesn't match the definition of 'retirement'). There is no doubt that he will utilise this newly-gained freedom to continue to inspire and challenge his fellow scientists with his inquisitive mind and cheerful spirit.

York, October 2010

Paul Busch, Maurice Dodson and Atsushi Higuchi Stefan Weigert (editor)

All papers published in this volume of Journal of Physics: Conference Series have been peer reviewed through processes administered by the proceedings Editors. Reviews were conducted by expert referees to the professional and scientific standards expected of a proceedings journal published by IOP Publishing.

We construct exotic bialgebras that arise from multiparameter 9 × 9 R-matrices, some of which are new. We also construct the dual bialgebras of two of these exotic bialgebras.

Many quantum groups and quantum spaces of interest can be obtained by cochain (but not cocycle) twist from their corresponding classical object. This failure of the cocycle condition implies a hidden nonassociativity in the noncommutative geometry already known to be visible at the level of differential forms. We extend the cochain twist framework to connections and Riemannian structures and provide examples including twist of the S⁷ coordinate algebra to a nonassociative hyperbolic geometry in the same category as that of the octonions.

In this article, we define the infinite Dirac operator and explore some key properties, particularly its conformal invariance. En route, we also establish the conformal invariance of the p-Dirac equation. We also introduce the infinite Dirac operator on the sphere Sⁿ and establish the relationship between the two infinite Dirac operators via the Cayley transformation. Also we introduce an infinite Laplace operator on Sⁿ.

Three results are shown in this contribution. First, the diagram used in the proof of Morley's theorem is shown to provide a template for the creation of a closed solid figure. Second, it is shown how a proposition governing a class of determinants can be used to produce a rich supply of identities involving, e.g., Fibonacci and related families of numbers. Finally, it is shown in the case of an arbitrary 2 × 2 symmetric matrix A, how to obtain explicit expressions for all the elements of all the matrices involved in implementing the determination of the eigenvalues of A by means of the QR algorithm.

In 1934, Jordan et al. gave a necessary algebraic condition, the Jordan identity, for a sensible theory of quantum mechanics. All but one of the algebras that satisfy this condition can be described by Hermitian matrices over the complexes or quaternions. The remaining, exceptional Jordan algebra can be described by 3 × 3 Hermitian matrices over the octonions.

We first review properties of the octonions and the exceptional Jordan algebra, including our previous work on the octonionic Jordan eigenvalue problem. We then examine a particular real, noncompact form of the Lie group E₆, which preserves determinants in the exceptional Jordan algebra.

Finally, we describe a possible symmetry-breaking scenario within E₆: first choose one of the octonionic directions to be special, then choose one of the 2×2 submatrices inside the 3×3 matrices to be special. Making only these two choices, we are able to describe many properties of leptons in a natural way. We further speculate on the ways in which quarks might be similarly encoded.

A self-contained formulation of Maxwell's theory of electromagnetic fields and sources is presented in the language of distributional forms. Properties of a fundamental double-form of bi-degree (p,p) for p ≥ 0 are reviewed in order to establish a computational framework for analysing equations involving the Hodge-de Rham operator for p–forms on space or spacetime and singular sources. A constructive approach to Dirac distributions on (moving) submanifolds embedded in R³ is developed in terms of (Leray) forms generated by the geometry of the embedding and illustrative examples presented from problems in electrostatics, magnetostatics and radiating point charges. The formulation offers a straightforward analysis of the relativistic jump conditions across static and moving interfaces where certain fields become discontinuous and provides a general methodology for electromagnetic modeling where possibly time dependent sources of certain physical attributes, such as electric charge, electric current and polarization or magnetization, are concentrated on localized regions in space or spacetime.

This is a survey of some very old knowledge about Mutually Unbiased Bases (MUB) and Symmetric Informationally Complete POVMs (SIC). In prime dimensions the former are closely tied to an elliptic normal curve symmetric under the Heisenberg group, while the latter are believed to be orbits under the Heisenberg group in all dimensions. In dimensions 3 and 4 the SICs are understandable in terms of elliptic curves, but a general statement escapes us. The geometry of the SICs in 3 and 4 dimensions is discussed in some detail.

A complex Hilbert space of dimension six supports at least three but not more than seven mutually unbiased bases. Two computer-aided analytical methods to tighten these bounds are reviewed, based on a discretization of parameter space and on Gröbner bases. A third algorithmic approach is presented: the non-existence of more than three mutually unbiased bases in composite dimensions can be decided by a global optimization method known as semidefinite programming. The method is used to confirm that the spectral matrix cannot be part of a complete set of seven mutually unbiased bases in dimension six.

Many quantum control tasks aim at manipulating the state of a quantum mechanical system within a finite subspace of states. However, couplings to the outside are often inevitable. Here we discuss strategies which keep the system in the controlled subspace by applying strong interactions onto the outside. This is done by drawing analogies to simple toy models and to the quantum Zeno effect. Special attention is paid to the constructive use of dissipation in the protection of subspaces.

In this chapter we discuss methods to calculate the entanglement of a system using density-functional theory. We firstly introduce density-functional theory and the local-density approximation (LDA). We then discuss the concept of the 'interacting LDA system'. This is characterised by an interacting many-body Hamiltonian which reproduces, uniquely and exactly, the ground state density obtained from the single-particle Kohn-Sham equations of density-functional theory when the local-density approximation is used. We motivate why this idea can be useful for appraising the local-density approximation in many-body physics particularly with regards to entanglement and related quantum information applications. Using an iterative scheme, we find the Hamiltonian characterising the interacting LDA system in relation to the test systems of Hooke's atom and helium-like atoms. The interacting LDA system ground state wavefunction is then used to calculate the spatial entanglement and the results are compared and contrasted with the exact entanglement for the two test systems. For Hooke's atom we also compare the entanglement to our previous estimates of an LDA entanglement. These were obtained using a combination of evolutionary algorithm and gradient descent, and using an LDA-based perturbative approach. We finally discuss if the position-space information entropy of the density—which can be obtained directly from the system density and hence easily from density-functional theory methods—can be considered as a proxy measure for the spatial entanglement for the test systems.

According to Hudson's theorem, the only quantum pure states with a positive Wigner function are Gaussian states. We summarize and compare some recent attempts at extending this theorem to the space of quantum mixed states, and complement them with new results obtained from numerical observations. Specifically, we look for upper bounds on the admissible non-Gaussianity of mixed states with a positive Wigner function.

There is perfect state transfer between two vertices of a graph, if a single excitation can travel with fidelity one between the corresponding sites of a spin system modeled by the graph. When the excitation is back at the initial site, for all sites at the same time, the graph is said to be periodic. A graph is cubic if each of its vertices has a neighbourhood of size exactly three. We prove that the 3-dimensional cube is the only periodic, connected cubic graph with perfect state transfer. We conjecture that this is also the only connected cubic graph with perfect state transfer.

In this paper I review Tony Sudberys foundational work with a view to tracing the evolution of his ideas about the measurement problem and the interpretation of quantum mechanics.