PERIOD: OCTOBER Professors: Chiara Marletto - Vlatko Vedral - Oxford University We will explore the foundations of quantum information with special emphasis on quantum coherence and witnesses of entanglement in many-body systems. We will explore frameworks to extend quantum information to post-quantum theories.

1. The qubit We introduce the qubit as a generalisation of a classical bit; and discuss unitary transformations as quantum gates. 2. Interference experiment as a quantum computation. We will describe how the Mach-Zehnder interferometry works; and how it can be used as a method to realise any one-qubit gate. 3. Entanglement We will define entangled systems and explore their properties with the formalism of density operators. We will also explain how entanglement is generated, with one and two-qubit gates. 4. Heisenberg and Schrödinger picture We will introduce the Heisenberg picture as a useful tool to describe the generation of entanglement in an explicitly local way. 5. Witnesses of entanglement We will explain how expected values of certain observables can classify entanglement in bipartite systems, for pure and mixed states; we also mention possible applications to real systems, e.g. in quantum biology. 6. EPR and No-signalling We will express the EPR paradox in information-theoretic terms; and explain how it does not violate locality (despite the vexed terminology ‘ quantum non-locality’ referring to entangled systems). 7. No-Cloning and its consequences We will prove the no-cloning theorem and show how it leads to a number of central properties of quantum theory. 8. Deriving quantum theory from principles. We will explore how most qualitative properties of quantum theories can be derived from information-theoretic principles, leading us to consider post-quantum theories. 9. What does it mean that a system is quantum? We will explore a number of inequivalent ways in which a system can be non-classical. We will explain why these are relevant for seeking signature of quantum effects in macroscopic systems. 10. Witnessing quantum effects in gravity using entanglement. We will discuss a recently proposed experiment, to witness quantum effects in the gravitational field without directly measuring observables of the field, based on a powerful information-theoretic argument.

The course will begin on Thursday 17th in aula Perucca (entrance 1 DISAT), and will follow the schedule: Thu 17th, 24th: 14:30-16:00 Fri 18th, 25th: 14:30-17:30 Mon 21st, 28th: 16:00-17:30 Tue 22nd, 29th: 14:30-16:00 Wed 23rd, 30th: 14:30-16:00

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