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The talk is based on my recent work with Ryan Aziz. We find a dual version of a previous double-bosonisation theorem whereby each finite-dimensional braided-Hopf algebra in the category of corepresentations of a coquasitriangular Hopf algebra gives a new larger coquasitriangular Hopf algebra, for example taking c_q[SL_2] to c_q[SL_3] for these quantum groups reduced at certain odd roots of unity. As an application we find new generators for c_q[SL2] with the remarkable property that their monomials are essentially a dual basis to the standard PBW basis of the reduced quantum enveloping algebra u_q(sl2). This allows one to calculate  Fourier transform and other results for such quantum groups. Our method also works for even roots of unity where we obtain new finite-dimensional quantum groups, including an 8-dimensional one at q=-1. Our method
can be used to construct  many other new finite-dimensional  quasitriangular Hopf algebras and their duals that could be fed into applications in quantum gravity and quantum computing.

One of the basic puzzles of black hole thermodynamics is the simplicity and universality of the  Bekenstein-Hawking entropy. The idea that this entropy might be governed by a symmetry at the horizon is an old one, but until now efforts have focused on conformal symmetries, either at infinity or on a "stretched horizon."  I argue that a better approach uses a BMS-like symmetry of the horizon itself.  This avoids the limitations of previous attempts (including my own), and explains the entropy in terms of a generalization of the Cardy formula for the density of states.

In this talk I am going to describe Spekkens’ toy model, a non-contextual hidden variable model with an epistemic restriction, a constraint on what an observer can know about reality. The aim of the model, developed for continuous and discrete prime degrees of freedom, is to advocate the epistemic view of quantum theory, where quantum states are states of incomplete knowledge about a deeper underlying reality. In spite of its classical flavour, many aspects that were thought to belong only to quantum mechanics can be reproduced in the model. 

I am going to describe our results regarding the formulation of rules for the update of states after measurement. I will do it for systems of discrete prime dimensions and I will then give the idea on how to proceed in the non-prime dimensional case. 

I will also depict the relationship between Spekkens’ model, stabiliser quantum mechanics and Gross' theory of discrete Wigner functions (they are equivalent theories in odd dimensions) in terms of measurement update rules.

I will conclude by briefly discussing a project we have been recently working on that consists of characterising sub theories of Spekkens’ model that are operationally equivalent to sub theories of QM (in particular in the case of qubits) and use them to represent the non-contextual classically simulable part of state-injection schemes of computation with contextuality as a resource.

A recent breakthrough in the condensed matter community is the identification and characterization of a rich set of ordered states, known as symmetry protected topological (SPT) phases. These phases are not only fascinating from the perspective of fundamental physics but have also found powerful applications in quantum computation. Very little is known about the thermal stability of SPT ordered systems, or whether their associated computational properties may survive at non-zero temperature. In this talk I will give a brief introduction to SPT phases of matter, and address the question of whether these phases can survive at non-zero temperature. In particular, I will focus on understanding the thermal stability and fault-tolerant properties of models studied in the theory of quantum computation.