PHY644: Foundations of quantum mechanics

        [Cr:4, Lc:3, Tt:1, Lb:0]

Course Outline

• Introduction Quantum theory (QT) is empirically a very successful theory; there is however an apparent lack of understanding of the theory. This is mostly due to the fact that, unlike the space-time structure, the cut between the ontology and epistemology in QT is difficult to resolve. The two fundamental concepts–the nonlocal correlations (entanglement) between space-like separated systems and the indistinguishability (non-orthogonality) of quantum states–is widely believed to separate QT from classical theories. In this course we take a foundational approach to QT from the outside: i.e., since classical theories are completely devoid of entanglement, it is compared with various foil theories that are also nonlocal and indistinguishable in the sense of QT, such that their special nature in the theory can be quantified. The two concepts will be explained in this course through the variety of topics it has motivated in the field of quantum information and computation, or vice versa.
• Mathematical Review: The review of the Hilbert-space formulation of quantum mechanics, quantum states, quantum dynamics, and measurements qubits, block-sphere representation, Pauli algebra, pure versus mixed states, tensor-product, entanglement, purification, VECing an operator, quantum operations, LOCC, unitary versus non-unitary dynamics, decoherence, positive versus completely positive maps, Kraus decomposition
• Correlations: EPR paradox, the realism and no-signaling principle, the hidden variable theories, the violation of Bell-type inequalities by entangled states (CHSH, Mermin, and Svetlichny inequalities), Nonlocal PR box, simulating quantum correlations, shared randomness, entanglement and computational complexity
• Indistinguishability: discrimination and estimation of unknown quantum states, von Neumann versus POVM measurements, quantum tomography, nature of probabilities in QT, contextuality, Gleason's theorem, Kochen-Specker theorem, compression of information, Von Neumann entropy, accessible information and Holevo's theorem, bit commitment, efficient simulation of Hamiltonian dynamics

Recommended Reading

• A. Peres, Quantum Theory: Concepts and Methods, Kluwer Dordrecht (1995).
• J. S. Bell, Speakable and Unspeakable in Quantum Mechanics, Cambridge University Press (2004).
• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press (2000).
• J. Preskill's Lecture Notes on Quantum Information http://www.theory.caltech.edu/people/preskill/ph229/
• B. Schumacher and M. D. Westmoreland, Quantum Processes, Systems and Information, Cambridge University Press (2010).