Talks

If a large quantum computer (QC) existed today, what type of physical problems could we efficiently simulate on it that we could not simulate on a conventional computer? In this talk, I argue that a QC could solve some relevant physical "questions" more efficiently. First, I will focus on the quantum simulation of quantum systems satisfying different particle statistics (e.g., anyons), using a QC made of two-level physical systems or qubits. The existence of one-to-one mappings between different algebras of observables or between different Hilbert spaces allow us to represent and imitate any physical system by any other one (e.g., a bosonic system by a spin-1/2 system). We explain how these mappings can be performed showing quantum networks useful for the efficient evaluation of some physical properties, such as correlation functions and energy spectra. Second, I will focus on the quantum simulation of classical systems. Interestingly, the thermodynamic properties of any d-dimensional classical system can be obtained by studying the zero-temperature properties of an associated d-dimensional quantum system. This classical-quantum correspondence allows us to understand classical annealing procedures as slow (adiabatic) evolutions of the lowest-energy state of the corresponding quantum system. Since many of these problems are NP-hard and therefore difficult to solve, is worth investigating if a QC would be a better device to find the corresponding solutions.

The mathematical formalism of quantum theory has many features whose physical origin remains obscure. In this paper, we attempt to systematically investigate the possibility that the concept of information may play a key role in understanding some of these features. We formulate a set of assumptions, based on generalizations of experimental facts that are representative of quantum phenomena and physically comprehensible theoretical ideas and principles, and show that it is possible to deduce the finite-dimensional quantum formalism from these assumptions. The concept of information, via an information-theoretic invariance principle, plays a central role in the derivation, and gives rise to some of the central structural features of the quantum formalism.

Universal Enveloping Algebras and dual Algebras of Functions The two most relevant types of Hopf-algebras for applications in physics are discussed in this unit. Most central notion will be their duality and representation.
Motivation: From Quantum Mechanics to Quantum GroupsThe 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.

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. The extention to mixed states is made via the convex-hull construction, as is done in the case of entanglement of formation. This geometric measure is shown to be a monotone. It can be calculated for various states, including arbitrary two-qubit states, generalized Werner and isotropic states in bi-partite systems. It is also calculated for various multi-partite pure and mixed states, including ground states of some physical models and states generated from quantum alogrithms, such as Grover's. A specific application to a spin model with quantum phase transistions will be presented in detail.The connection of the geometric measure to other entanglement properties will also be discussed.

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.

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. I end up with outlining several open questions and puzzles related to the outcome of the Gregory-Laflamme instability.

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. Furthermore we give some examples of how quantum Kolmogorov complexity can be applied to prove results in different fields, such as quantum computation and communication.