Basic linear algebra is enough. A bit of number theory can be useful but not necessary.
My aim is to present mathematical methods for quantum information processing. As in most applications it is enough to work with qubits and systems of qubits, mathematical methods originate from linear algebra, which is usually one of first curses taught. It makes quantum information accessible for very 'fresh' students. I would like to convince students that quantum information processing is useful, interesting, counter-intuitive, sometimes seemingly as mysterious as the Schroedinger cat.
Mathematical formalism of quantum mechanics.
Postulates of quantum mechanics.
Quantum information: quantum gates, no-go theorems, measurement.
Quantum entanglement: mathematical basis.
Selected applications: teleportation, dense coding.
Quantum cloning and applications.
Basic protocols for quantum cryptography: BB84, B92.
Quantum nonlocality: Bell and Leggett-Garg inequalities, contextuality.
Dynamics of quantum systems, open quantum systems.
Quantum error correction.
Quantum Computation and Quantum Information by Michael A. Nielsen & Isaac L. Chuang
Lecture notes by John Preskill http://www.theory.caltech.edu/people/preskill/ph229/