**Quantum Statistics**

**Autumn Semester, 2008**

“QS”, Fridays, 11:15—13:00, Snellius 401, Nielsbohrweg 1, Leiden. Advanced Bachelor’s level — Master’s level.

Quantum physics is the framework within which we describe the physics of microscopically small systems: how they evolve in time on their own, and how they interact with the macroscopic world. In isolation from the macroscopic world, the evolution is strictly deterministic: it is governed by a differential equation called Schrödinger’s equation. On the other hand, the interaction with the macroscopic world is stochastic. In particular, when we measure an individual quantum system the outcome, i.e., what we get to see in the laboratory, is random. On being measured, the state of the quantum system itself makes a random jump. Quantum physics allows us only to compute the probability of each particular measurement outcome (and corresponding new state of the quantum system). This probability is determined according to a precise rule called Born’s law.

Till a few years ago this picture of measuring a single microscopic quantum system was purely a thought experiment. In recent years however enormous progress has been made and now physicists are able to manipulate and measure single quantum systems, e.g., a single ion held in an ion trap for weeks on end. At the same time, a mathematical theory of communication and computation based on operations on single quantum systems has been developed.

For finite dimensional quantum systems – roughly speaking, systems where only a finite number of basic states are involved, e.g., spin can be up or down, polarization can be vertical or horizontal) – the mathematical structure of quantum theory is built on a combination of elementary linear algebra (finite dimensional complex Hilbert spaces) and elementary probability theory. Within this elementary mathematical framework, one can formulate and in some cases solve deep results about the possibilities of quantum communication and quantum computation; quantum state reconstruction and quantum stochastic processes.

The course will give an introduction to this field, focussed on the mathematical structure and in particular the role of probability and statistics.

**Literature**:

Draft chapters of a book on quantum statistics by R.D. Gill and others, together with the “bible of quantum information”, the book

Mada Guta's slides for his lecture course on quantum statistics

Quantum Computation and Quantum Information by Michael A. Nielsen and Isaac L. Chuang (2000), Cambridge University Press.

For a rapid introduction, see the paper Teleportation into quantum statistics by R.D. Gill (2001).

gill@math.leidenuniv.nl