Since 2002 Perimeter Institute has been recording seminars, conference talks, and public outreach events using video cameras installed in our lecture theatres. Perimeter now has 7 formal presentation spaces for its many scientific conferences, seminars, workshops and educational outreach activities, all with advanced audio-visual technical capabilities. Recordings of events in these areas are all available On-Demand from this Video Library and on Perimeter Institute Recorded Seminar Archive (PIRSA). PIRSA is a permanent, free, searchable, and citable archive of recorded seminars from relevant bodies in physics. This resource has been partially modelled after Cornell University's arXiv.org.
Hopf-Algebras and their Representations
In order to consolidate the above motivation, we have to introduce Hopf-algebras on a mathematical footing. We define Hopf-algebras, discuss duality and especially we will have a closer look at the question why coproducts induce a multiplication on the dual algebra but not the other why around. With these preparations we close this unit by the discussion of representations and corepresentations - and how these are related for dual Hopf-algebras.
Motivation: From Quantum Mechanics to Quantum Groups
The notion of 'quantization' commonly used in textbooks of quantum mechanics has to be specified in order to turn it into a defined mathematical operation. We discuss that on the trails of Weyl's phase space deformation, i.e. we introduce the Weyl-Moyal starproduct and the deformation of Poisson-manifolds.
Generalizing from this, we understand, why Hopf-algebras are the most genuine way to apply 'quantization'
to various other algebraic objects - and why this has direct physical applications.
A multi-partite entanglement measure is constructed via the distance or angle of the pure state to its nearest unentangled state.
If spacetime is "quantized" (discrete), then any equation of motion compatible with the Lorentz transformations is necessarily non-local. I will present evidence that this sort of nonlocality survives on length scales much greater than Planckian, yielding for example a nonlocal effective wave-equation for a scalar field propagating on an underlying causal set. Nonlocality of our effective field theories may thus provide a characteristic signature of quantum gravity.
5-qubit code, logical Pauli group for stabilizer codes, classical linear codes (generator and parity check matrices, Hamming codes), CSS codes (definition, 7-qubit code)
Stabilizer codes (definition of stabilizer, basic properties of stabilizer, binary vector representation of stabilizer)
I begin with a brief description of the black strings in backgrounds with compact circle, the Gregory-Laflamme instability and the resulting phase transition, and the critical dimensions.Then I describe a Landau-Ginzburg thermodynamic perspective on the instability and on the order of the phase transition. Next, the approach is generalized from a circle compactification to an arbitrary torus compactification. It is shown that the transition order depends only on the number of extended dimensions.
Kolmogorov complexity is a measure of the information contained in a binary string. We investigate the notion of quantum Kolmogorov complexity, a measure of the information required to describe a quantum state. We show that for any definition of quantum Kolmogorov complexity measuring the number of classical bits required to describe a pure quantum state, there exists a pure n-qubit state which requires exponentially many bits of description. This is shown by relating the classical communication complexity to the quantum Kolmogorov complexity.
Graduate Course on Standard Model & Quantum Field Theory
A full analysis of QCD, the fundamental theory of subnuclear structure and interactions, relies upon numerical simulations and the lattice approximation. After being stalled for almost 30 years, recent breakthroughs in lattice QCD allow us for the first time to analyze the low-energy structure of QCD nonperturbatively with few-percent precision. This talk will present a non-technical overview of the history leading up to these breakthroughs, and survey the wide array of applications that have been enabled by them.