Quantum information theory and many-body physics
Freek Gerrit Witteveen

Abstract:
In this dissertation we study the fascinating interaction between quantum information theory and many-body physics. Many-body physics broadly are physical systems which are made up of a large number of subsystems. While the laws and principles of quantum mechanics are well known and can be formulated in compact equations, understanding many-body physics and the emergent phenomena related to it gives rise to a whole new set of challenges. In this dissertation we contribute to this field by showing how in three different domains perspectives from quantum information theory and computation can be useful in the study of many-body physics.
  In Part I we investigate quantum systems in one spatial dimension. We show that the ground state, the lowest energy state, of a class of free systems can be prepared efficiently on a quantum computer using a certain circuit structure. Free quantum systems can be simulated efficiently by classical computers; so the possibility of preparing such ground states on a quantum computer is well-known. However, we show that this is possible using a method which implements an important physical principle: real-space renormalization. This method is known as entanglement renormalization. There is an associated tensor network ansatz, the multiscale entanglement renormalization ansatz (MERA). We show that entanglement renormalization in free systems is closely related to the theory of wavelets. Wavelets allow a decomposition of functions into a basis set of functions, similar to Fourier analysis. However, in the case of wavelets, the basis functions are not plane waves but localized ‘wave packets’. Quantization of appropriately chosen wavelets yields an entanglement renormalization scheme. We explain how to find such wavelets, and we show that if one has wavelets with the right properties one can obtain accurate MERA approximation. We show that this has a natural continuum limit, which is related to free quantum field theories.
  In Part II we study a different aspect of many-body quantum mechanics. Whereas in the first part we focussed on finding ground states, here we are interested in the dynamics of one-dimensional quantum systems. This means that we have a system with a state changing over time. Closed systems have unitary time evolution. Moreover, in physical systems with local interactions, one finds that time evolution also conserves a certain amount of locality. Based on these two general principles we dynamics which consist of a single time step which are both unitary and (approximately) locality preserving, which we call approximately locality preserving unitary (ALPU). If one imposes strict locality this is known as a quantum cellular automaton. In one spatial dimension these have been classified by an index which measures an information flow. We show that this classification extends to approximately local dynamics, using tools from the theory of operator algebras. We prove that a one-dimensional ALPU can be connected by a continuous path of ALPUs to the identity if and only if the ALPU has no net information flow to the left or the right.
  Finally, Part III is inspired by the interaction between quantum information theory and gravity. Developing a theory of quantum gravity which can describe our universe is one the great outstanding challenges of theoretical physical. An important physical phenomenon where such a theory would be relevant are black holes. Based on general physical principles, it appears that black holes have an entropy (that is, a number of degrees of freedom) corresponding to their area. This has lead to the development of holographic quantum gravity, where a d + 1-dimensional gravitational theory is dual to a d -dimensional non-gravitational quantum theory, which lives on the boundary of the gravitational spacetime. We study a simple toy model for this phenomenon: while it is certainly not describing a theory of quantum gravity, it shows in a surprisingly accurate way certain mechanisms which are crucial in holographic gravity. The model is a tensor network model where we choose uniformly random tensors. We generalize this model by adapting the usual PEPS model to use different link states and we deduce properties of the entanglement spectrum. These results closely mirror conjectured properties of holographic systems.