## 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)