## Mathematical Physics (MAP): Quantum Logic and Quantum Probability

Instructor:  Dr. Mihály Weiner

Text: handouts and Chapter VIII and IX of T. Matolcsi: A Concept of Mathematical Physics, Models in Mechanics.

Prerequisites: basics of classical probability theory and linear algebra.

Course description: the course is about the non-classical calculus of probability which is behind Quantum Physics. (Read this short summary written in a "Q&A" form about the essence of Quantum Physics.) The emphasis will be on the mathematical, information-theoretical and philosophycal aspects (but not directly on physics). In the first part of the course the neccessary mathematicals tools are introduced, while in the second part some simple physicial systems as well as quantum computers and some "paradoxes" (such as the "EPR" paradox) are discussed.

Topics:

1st part (the mathematical tools):
finite dimensional Hilbert spaces, orthogonal projections, operator norms, normal operators, self-adjoint operators, unitary operators, spectral resolution, operator-calculus, positive operators, tensorial products
ortho-lattices and probability laws, distributive and non-distributive probability spaces, dispersion free and pure states, measurable quantites
the ortho-lattice of projections, Gleason's theory (without proof), operations between measurable quantites

2nd part (applications):
spin systems, the "EPR" paradox, quantum cryptography (the protocol of Bennett and Brassard), state changes, symmetries operations and Wigner's theorem, dense coding, no-clone theorem, quantum bits and quantum computers, complexity and quantum complexity, an example of an algorithm for a quantum computer (either Grover's search algorithm or Shor's algorithm for factorizing numbers)