Note: Course descriptions are provided for informational purposes
only and may vary. Links to exercises, lecture notes, handouts,
and past exam papers are updated as the course progresses.
Objective and Summary
: This course offers a step-by-step introduction to the
mathematical foundations of quantum information, starting from
simple postulates within finite-dimensional systems.
The course is intended for a broad audience at Tongji
university, including students, professionals, and enthusiasts
from all mathematical fields who wish to develop a solid
theoretical understanding of the principles underlying quantum
computing and quantum information.
Prerequisite : As a
prerequisite, participants are recommended to be familiar with
linear algebra, complex numbers, and elementary probability
theory and information theory. It should be noted that, although
the course does not presuppose knowledge of Hilbert spaces, it
does presuppose a fairly comprehensive knowledge of
finite-dimensional complex scalar product spaces (also known as
Hermitian spaces, the natural generalization of Euclidean spaces
over the complex numbers). See below documents to read
before class.
This course does not presuppose any knowledge on quantum physics
but focuses on the mathematics of quantum information.
Syllabus : This course
offers a step-by-step introduction to the mathematical
foundations of quantum information, starting from simple
postulates within finite-dimensional systems. A central
emphasis is placed on quantum measurement and its inherently
probabilistic nature, from which the linear evolution of quantum
states and the derivation of the Schrödinger differential
equation for isolated systems can be mathematically
derived. The course then addresses more realistic
scenarios involving noise or interactions with an external
environment. In such settings, quantum states are
described by density matrices, while their transformations are
modeled by quantum channels, together with a general framework
for quantum measurements. If time permits, the course will
also explore mathematical tools used to define and compare
distances, divergences or similarities between quantum states,
highlighting their connections with quantum channels and
measurement processes.
Quantum states; Measurements; Observables
Uncertainty Principle; Unitary evolution and Schrödinger's
equation
Generalized measurement, POVM
Mixed states and density matrices
Quantum channels
Trace distance and fidelity
(if time permits) Entropy
