Scientific Series

There is a deep relation between Loop Quantum Gravity and notions from category theory, which have been pointed out by many researchers, such as Baez or Velhinho. Concepts like holonomies, connections and gauge transformations can be naturally formulated in that language. In this formulation, the (spatial) diffeomorphisms appear as the path grouopid automorphisms. We investigate the effect of extending the diffeomorphisms to all such automorphisms, which can be viewed as \"distributional diffeomorphisms\". We also give a notion of \"categorial holonomy-flux-algebra\", and present the construction of the automorphism-invariant Hilbert space for abelian gauge groups, which will be entirely combinatorial.

We will compare quantum phase estimation from the point of view of quantum computation and quantum metrology. In the simplest cases, the former can be simplified to a sequential (unentangled) protocol, while the latter is parallel (entangled). We show that both protocols can be formally related with circuit identities and that they respond in exactly the same way to decoherence. We present sequential protocols for optimal estimation and frame synchronization in DQC1. Finally, we introduce new estimation protocols based on nonlinear Hamiltonians. We show that both optimal input states and product states with separable measurements improve the scaling of linear Hamiltonians. We will comment on the effect of decoherence in nonlinear protocols, and the role of entanglement in nonlinear protocols with product states.

The Universe offers environments with extreme physical conditions that cannot be realized in laboratories on Earth. These environments provide unprecedented tests for extensions of the Standard Model. I will describe three such \"astrophysical laboratories\", which are likely to represent new frontiers in cosmology and astrophysics over the next decade. One provides a novel probe of the initial conditions from inflation and the nature of the dark matter, based on 3D mapping of the distribution of cosmic hydrogen through its resonant 21cm line. The second allows to constrain the metric around supermassive black holes based on direct imaging or the detection of gravitational waves. The third involves the acceleration of high-energy particles in cosmological shock waves. I will describe past and future observations of these environments and some related theoretical work.

A new microcanonical equilibrium state is introduced for quantum systems with finite-dimensional state spaces. Equilibrium is characterised by a uniform distribution on a level surface of the expectation value of the Hamiltonian. The distinguishing feature of the proposed equilibrium state is that the corresponding density of states is a continuous function of the energy, and hence thermodynamic functions are well defined for finite quantum systems. The density of states, however, is not in general an analytic function. It is demonstrated that generic quantum systems therefore exhibit second-order phase transitions at finite temperatures. The talk is based on work carried out in collaboration with D.W. Hook and L.P. Hughston.

We consider pure three dimensional quantum gravity with a negative cosmological constant. The torus partition function can be computed exactly as a sum over geometries, including all known quantum corrections. The answer provides important clues about the structure of quantum gravity; in particular, in order for the theory to be a proper quantum mechanical system some extra ingredients are needed beyond the usual real geometries considered in general relativity. One possiblity is that complex geometries need to be included; this leads to holomorphically factorized partition functions. These partition functions provide a wealth of information about black hole microphysics. For example, the Hawking-page phase transition can be studied exactly; it is a phase transition of the type described by Lee and Yang, which is associated with a condensation of zeros in the complex temperature plane.

What if the second law of thermodynamics, in the hierarchy of physical laws, were at the same level as the fundamental laws of mechanics, such as the great conservation principles? What if entropy were an intrinsic property of matter at the same level as energy is universally understood to be? What if irreversibility were an intrinsic feature of the microscopic dynamical law of all physical objects, including an individual qubit or qudit? This talk will show how positive answers to these questions need not contradict any of the known results of quantum mechanics. We construct a logically consistent, mathematically sound and definite, physically intriguing, non-relativistic and non-statistical quantum theory, in which the second law of thermodynamics is embedded as a fundamental microscopical law. The theory hinges upon a nonlinear extension of unitary Hamiltonian dynamics which for uncorrelated and noninteracting systems reduces to the usual Schroedinger equation for the zero entropy states, but in general generates a group (not a semi group) of irreversible time evolutions, where the non-Hamiltonian entropy generating term in the evolution equation attracts the state towards the direction of maximal entropy increase. Various examples and features of this highly non-conventional dynamical theory are discussed. References available at http://www.quantumthermodynamics.org/

Renner\'s global quantum de Finetti theorem establishes that if the state of a quantum system is invariant under permutations of its systems, then almost all of its subsystems are almost in the same state and independent of each other. Motivated by this result, we show that the most straightforward classical analogue of Renner\'s theorem is false. Joint work with Matthias Christandl (Cambridge).

This will be an introductory talk about Topological Quantum Computation. TQC is attractive because it is intrinsicaly decoherence free. We introduce the basic notions, such as non abelian anyons, quantum symmetries and topological order. A topologically ordered phase is a gapped phase in which the basic degrees of freedom are of a topological nature (denoted as anyons), charactetized by their fusion and braiding properties. If time permits possible implementations based on Quantum Hall systems will be discussed as well.

I will discuss a new method of inflaton potential reconstruction that combines the flow formalism, which is a stochastic method of inflationary model generation, with an exact numerical calculation of the mode equations of quantum fluctuations. This technique allows one to explore regions of the inflationary parameter space yielding spectra that are not well parameterized as power-laws. We use this method to generate an ensemble of generalized spectral shapes that provide equally good fits to current CMB and LSS as data as do simpler power-law spectra. Within this ensemble are spectra that exhibit a strong running on large angular scales (where cosmic variance is large) that turns off on small scales. Such strongly running spectra are accompanied by large tensor components that lie outside the 1 and 2 sigma limits of current WMAP3 and SDSS data. This demonstrates that the generalization of the spectral shape adversely impacts our ability to constrain key inflationary observables. The inflationary models giving rise to such spectra are characterized by an initially fast rolling inflaton, in marked contrast to the dominant paradigm of slow roll inflation.