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Abstract

When quantum mechanics was formulated nearly 100 years ago, there was no clear idea what its impact in science would be and neither what it meant. It was only 40 years ago that the mainstream probabilistic interpretation of quantum mechanics was experimentally validated through Bell’s test. Those experiments corroborated the principle of quantum entanglement, the notion that two measurements can influence each other even when taken infinitely far apart.

This mini-course of four lectures will address the implications of quantum entanglement in different areas of modern research. The first lecture of the course will be devoted to interpretation of quantum mechanics. I will discuss the historical disputes between the Copenhagen interpretation, lead by Niels Bohr, Heisenberg and others, with competing theories that proposed a deterministic interpretation based on local hidden variables. In the second lecture, I will discuss some of the original ideas of Feynman on quantum computing. I will use the concept of entanglement to introduce modern subjects such as quantum cryptography and will address the surprising existence of quantum states that fail to thermalize even at infinite temperature, among other topics.

In the third lecture I will make an overview on the modern concept of order that relies on topology, a field of mathematics that deals with properties what remain invariant under continuous deformations of shapes and spaces. I will show how topology explains the remarkable quantization of the Hall conductivity and will introduce the concept of topological order through the toric code. Finally, in the last lecture I will discuss the current attempts to build fault-tolerant quantum computers with topological order, a field known as topological quantum computing. This course is intended to engage undergraduate students in contemporary research.